/* Subroutine */ int zhpcon_(char *uplo, integer *n, doublecomplex *ap,
integer *ipiv, doublereal *anorm, doublereal *rcond, doublecomplex *
work, integer *info)
{
/* System generated locals */
integer i__1, i__2;
/* Local variables */
integer i__, ip, kase;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *), xerbla_(
char *, integer *);
doublereal ainvnm;
extern /* Subroutine */ int zhptrs_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZHPCON estimates the reciprocal of the condition number of a complex */
/* Hermitian packed matrix A using the factorization A = U*D*U**H or */
/* A = L*D*L**H computed by ZHPTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* Specifies whether the details of the factorization are stored */
/* as an upper or lower triangular matrix. */
/* = 'U': Upper triangular, form is A = U*D*U**H; */
/* = 'L': Lower triangular, form is A = L*D*L**H. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2) */
/* The block diagonal matrix D and the multipliers used to */
/* obtain the factor U or L as computed by ZHPTRF, stored as a */
/* packed triangular matrix. */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by ZHPTRF. */
/* ANORM (input) DOUBLE PRECISION */
/* The 1-norm of the original matrix A. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--work;
--ipiv;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*anorm < 0.) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZHPCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0) {
*rcond = 1.;
return 0;
} else if (*anorm <= 0.) {
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper) {
/* Upper triangular storage: examine D from bottom to top */
ip = *n * (*n + 1) / 2;
for (i__ = *n; i__ >= 1; --i__) {
i__1 = ip;
if (ipiv[i__] > 0 && (ap[i__1].r == 0. && ap[i__1].i == 0.)) {
return 0;
}
ip -= i__;
/* L10: */
}
} else {
/* Lower triangular storage: examine D from top to bottom. */
ip = 1;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = ip;
if (ipiv[i__] > 0 && (ap[i__2].r == 0. && ap[i__2].i == 0.)) {
return 0;
}
ip = ip + *n - i__ + 1;
/* L20: */
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
/* Multiply by inv(L*D*L') or inv(U*D*U'). */
zhptrs_(uplo, n, &c__1, &ap[1], &ipiv[1], &work[1], n, info);
goto L30;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / ainvnm / *anorm;
}
return 0;
/* End of ZHPCON */
} /* zhpcon_ */
doublereal zla_gercond_c__(char *trans, integer *n, doublecomplex *a, integer
*lda, doublecomplex *af, integer *ldaf, integer *ipiv, doublereal *
c__, logical *capply, integer *info, doublecomplex *work, doublereal *
rwork, ftnlen trans_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2, i__3, i__4;
doublereal ret_val, d__1, d__2;
doublecomplex z__1;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer i__, j;
doublereal tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
doublereal anorm;
extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *), xerbla_(
char *, integer *);
doublereal ainvnm;
extern /* Subroutine */ int zgetrs_(char *, integer *, integer *,
doublecomplex *, integer *, integer *, doublecomplex *, integer *,
integer *);
logical notrans;
/* -- LAPACK routine (version 3.2.1) -- */
/* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/* -- Jason Riedy of Univ. of California Berkeley. -- */
/* -- April 2009 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley and NAG Ltd. -- */
/* .. */
/* .. Scalar Aguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZLA_GERCOND_C computes the infinity norm condition number of */
/* op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate Transpose = Transpose) */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* A (input) COMPLEX*16 array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) COMPLEX*16 array, dimension (LDAF,N) */
/* The factors L and U from the factorization */
/* A = P*L*U as computed by ZGETRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from the factorization A = P*L*U */
/* as computed by ZGETRF; row i of the matrix was interchanged */
/* with row IPIV(i). */
/* C (input) DOUBLE PRECISION array, dimension (N) */
/* The vector C in the formula op(A) * inv(diag(C)). */
/* CAPPLY (input) LOGICAL */
/* If .TRUE. then access the vector C in the formula above. */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) COMPLEX*16 array, dimension (2*N). */
/* Workspace. */
/* RWORK (input) DOUBLE PRECISION array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function Definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
--c__;
--work;
--rwork;
/* Function Body */
ret_val = 0.;
*info = 0;
notrans = lsame_(trans, "N");
if (! notrans && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
} else if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZLA_GERCOND_C", &i__1);
return ret_val;
}
/* Compute norm of op(A)*op2(C). */
anorm = 0.;
if (notrans) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*capply) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + j * a_dim1]), abs(d__2))) / c__[j];
}
} else {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + j * a_dim1]), abs(d__2));
}
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*capply) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
j + i__ * a_dim1]), abs(d__2))) / c__[j];
}
} else {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
j + i__ * a_dim1]), abs(d__2));
}
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0) {
ret_val = 1.;
return ret_val;
} else if (anorm == 0.) {
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.;
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
if (notrans) {
zgetrs_("No transpose", n, &c__1, &af[af_offset], ldaf, &ipiv[
1], &work[1], n, info);
} else {
zgetrs_("Conjugate transpose", n, &c__1, &af[af_offset], ldaf,
&ipiv[1], &work[1], n, info);
}
/* Multiply by inv(C). */
if (*capply) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = c__[i__4] * work[i__3].r, z__1.i = c__[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
} else {
/* Multiply by inv(C'). */
if (*capply) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = c__[i__4] * work[i__3].r, z__1.i = c__[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
if (notrans) {
zgetrs_("Conjugate transpose", n, &c__1, &af[af_offset], ldaf,
&ipiv[1], &work[1], n, info);
} else {
zgetrs_("No transpose", n, &c__1, &af[af_offset], ldaf, &ipiv[
1], &work[1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
ret_val = 1. / ainvnm;
}
return ret_val;
} /* zla_gercond_c__ */
/* Subroutine */ int zgbrfs_(char *trans, integer *n, integer *kl, integer *
ku, integer *nrhs, doublecomplex *ab, integer *ldab, doublecomplex *
afb, integer *ldafb, integer *ipiv, doublecomplex *b, integer *ldb,
doublecomplex *x, integer *ldx, doublereal *ferr, doublereal *berr,
doublecomplex *work, doublereal *rwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset,
x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5, i__6, i__7;
doublereal d__1, d__2, d__3, d__4;
doublecomplex z__1;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer i__, j, k;
doublereal s;
integer kk;
doublereal xk;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int zgbmv_(char *, integer *, integer *, integer *
, integer *, doublecomplex *, doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *);
integer count;
extern /* Subroutine */ int zcopy_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *), zaxpy_(integer *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, integer *), zlacn2_(
integer *, doublecomplex *, doublecomplex *, doublereal *,
integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
logical notran;
char transn[1], transt[1];
doublereal lstres;
extern /* Subroutine */ int zgbtrs_(char *, integer *, integer *, integer
*, integer *, doublecomplex *, integer *, integer *,
doublecomplex *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZGBRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is banded, and provides */
/* error bounds and backward error estimates for the solution. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate transpose) */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KL (input) INTEGER */
/* The number of subdiagonals within the band of A. KL >= 0. */
/* KU (input) INTEGER */
/* The number of superdiagonals within the band of A. KU >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* AB (input) COMPLEX*16 array, dimension (LDAB,N) */
/* The original band matrix A, stored in rows 1 to KL+KU+1. */
/* The j-th column of A is stored in the j-th column of the */
/* array AB as follows: */
/* AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KL+KU+1. */
/* AFB (input) COMPLEX*16 array, dimension (LDAFB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by ZGBTRF. U is stored as an upper triangular band */
/* matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and */
/* the multipliers used during the factorization are stored in */
/* rows KL+KU+2 to 2*KL+KU+1. */
/* LDAFB (input) INTEGER */
/* The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from ZGBTRF; for 1<=i<=N, row i of the */
/* matrix was interchanged with row IPIV(i). */
/* B (input) COMPLEX*16 array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by ZGBTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The estimated forward error bound for each solution vector */
/* X(j) (the j-th column of the solution matrix X). */
/* If XTRUE is the true solution corresponding to X(j), FERR(j) */
/* is an estimated upper bound for the magnitude of the largest */
/* element in (X(j) - XTRUE) divided by the magnitude of the */
/* largest element in X(j). The estimate is as reliable as */
/* the estimate for RCOND, and is almost always a slight */
/* overestimate of the true error. */
/* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Internal Parameters */
/* =================== */
/* ITMAX is the maximum number of steps of iterative refinement. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
notran = lsame_(trans, "N");
if (! notran && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kl < 0) {
*info = -3;
} else if (*ku < 0) {
*info = -4;
} else if (*nrhs < 0) {
*info = -5;
} else if (*ldab < *kl + *ku + 1) {
*info = -7;
} else if (*ldafb < (*kl << 1) + *ku + 1) {
*info = -9;
} else if (*ldb < max(1,*n)) {
*info = -12;
} else if (*ldx < max(1,*n)) {
*info = -14;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGBRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.;
berr[j] = 0.;
/* L10: */
}
return 0;
}
if (notran) {
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
} else {
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
/* Computing MIN */
i__1 = *kl + *ku + 2, i__2 = *n + 1;
nz = min(i__1,i__2);
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
count = 1;
lstres = 3.;
L20:
/* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
zcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
z__1.r = -1., z__1.i = -0.;
zgbmv_(trans, n, n, kl, ku, &z__1, &ab[ab_offset], ldab, &x[j *
x_dim1 + 1], &c__1, &c_b1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 = d_imag(&b[
i__ + j * b_dim1]), abs(d__2));
/* L30: */
}
/* Compute abs(op(A))*abs(X) + abs(B). */
if (notran) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
kk = *ku + 1 - k;
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[k + j *
x_dim1]), abs(d__2));
/* Computing MAX */
i__3 = 1, i__4 = k - *ku;
/* Computing MIN */
i__6 = *n, i__7 = k + *kl;
i__5 = min(i__6,i__7);
for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) {
i__3 = kk + i__ + k * ab_dim1;
rwork[i__] += ((d__1 = ab[i__3].r, abs(d__1)) + (d__2 =
d_imag(&ab[kk + i__ + k * ab_dim1]), abs(d__2))) *
xk;
/* L40: */
}
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
kk = *ku + 1 - k;
/* Computing MAX */
i__5 = 1, i__3 = k - *ku;
/* Computing MIN */
i__6 = *n, i__7 = k + *kl;
i__4 = min(i__6,i__7);
for (i__ = max(i__5,i__3); i__ <= i__4; ++i__) {
i__5 = kk + i__ + k * ab_dim1;
i__3 = i__ + j * x_dim1;
s += ((d__1 = ab[i__5].r, abs(d__1)) + (d__2 = d_imag(&ab[
kk + i__ + k * ab_dim1]), abs(d__2))) * ((d__3 =
x[i__3].r, abs(d__3)) + (d__4 = d_imag(&x[i__ + j
* x_dim1]), abs(d__4)));
/* L60: */
}
rwork[k] += s;
/* L70: */
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__4 = i__;
d__3 = s, d__4 = ((d__1 = work[i__4].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2))) / rwork[i__];
s = max(d__3,d__4);
} else {
/* Computing MAX */
i__4 = i__;
d__3 = s, d__4 = ((d__1 = work[i__4].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__]
+ safe1);
s = max(d__3,d__4);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2. <= lstres && count <= 5) {
/* Update solution and try again. */
zgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1]
, &work[1], n, info);
zaxpy_(n, &c_b1, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(op(A)))* */
/* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(op(A)) is the inverse of op(A) */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(op(A))*abs(X) + abs(B) is less than SAFE2. */
/* Use ZLACN2 to estimate the infinity-norm of the matrix */
/* inv(op(A)) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__4 = i__;
rwork[i__] = (d__1 = work[i__4].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
;
} else {
i__4 = i__;
rwork[i__] = (d__1 = work[i__4].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
+ safe1;
}
/* L90: */
}
kase = 0;
L100:
zlacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**H). */
zgbtrs_(transt, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &
ipiv[1], &work[1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__4 = i__;
i__5 = i__;
i__3 = i__;
z__1.r = rwork[i__5] * work[i__3].r, z__1.i = rwork[i__5]
* work[i__3].i;
work[i__4].r = z__1.r, work[i__4].i = z__1.i;
/* L110: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__4 = i__;
i__5 = i__;
i__3 = i__;
z__1.r = rwork[i__5] * work[i__3].r, z__1.i = rwork[i__5]
* work[i__3].i;
work[i__4].r = z__1.r, work[i__4].i = z__1.i;
/* L120: */
}
zgbtrs_(transn, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &
ipiv[1], &work[1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__4 = i__ + j * x_dim1;
d__3 = lstres, d__4 = (d__1 = x[i__4].r, abs(d__1)) + (d__2 =
d_imag(&x[i__ + j * x_dim1]), abs(d__2));
lstres = max(d__3,d__4);
/* L130: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of ZGBRFS */
} /* zgbrfs_ */
/* Subroutine */ int zgecon_(char *norm, integer *n, doublecomplex *a,
integer *lda, doublereal *anorm, doublereal *rcond, doublecomplex *
work, doublereal *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
doublereal d__1, d__2;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
doublereal sl;
integer ix;
doublereal su;
integer kase, kase1;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal ainvnm;
extern integer izamax_(integer *, doublecomplex *, integer *);
logical onenrm;
extern /* Subroutine */ int zdrscl_(integer *, doublereal *,
doublecomplex *, integer *);
char normin[1];
doublereal smlnum;
extern /* Subroutine */ int zlatrs_(char *, char *, char *, char *,
integer *, doublecomplex *, integer *, doublecomplex *,
doublereal *, doublereal *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZGECON estimates the reciprocal of the condition number of a general */
/* complex matrix A, in either the 1-norm or the infinity-norm, using */
/* the LU factorization computed by ZGETRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input) COMPLEX*16 array, dimension (LDA,N) */
/* The factors L and U from the factorization A = P*L*U */
/* as computed by ZGETRF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* ANORM (input) DOUBLE PRECISION */
/* If NORM = '1' or 'O', the 1-norm of the original matrix A. */
/* If NORM = 'I', the infinity-norm of the original matrix A. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*anorm < 0.) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGECON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0) {
*rcond = 1.;
return 0;
} else if (*anorm == 0.) {
return 0;
}
smlnum = dlamch_("Safe minimum");
/* Estimate the norm of inv(A). */
ainvnm = 0.;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(L). */
zlatrs_("Lower", "No transpose", "Unit", normin, n, &a[a_offset],
lda, &work[1], &sl, &rwork[1], info);
/* Multiply by inv(U). */
zlatrs_("Upper", "No transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &su, &rwork[*n + 1], info);
} else {
/* Multiply by inv(U'). */
zlatrs_("Upper", "Conjugate transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &su, &rwork[*n + 1], info);
/* Multiply by inv(L'). */
zlatrs_("Lower", "Conjugate transpose", "Unit", normin, n, &a[
a_offset], lda, &work[1], &sl, &rwork[1], info);
}
/* Divide X by 1/(SL*SU) if doing so will not cause overflow. */
scale = sl * su;
*(unsigned char *)normin = 'Y';
if (scale != 1.) {
ix = izamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((d__1 = work[i__1].r, abs(d__1)) + (d__2 = d_imag(&
work[ix]), abs(d__2))) * smlnum || scale == 0.) {
goto L20;
}
zdrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / ainvnm / *anorm;
}
L20:
return 0;
/* End of ZGECON */
} /* zgecon_ */
/* Subroutine */ int ztpcon_(char *norm, char *uplo, char *diag, integer *n,
doublecomplex *ap, doublereal *rcond, doublecomplex *work, doublereal
*rwork, integer *info)
{
/* System generated locals */
integer i__1;
doublereal d__1, d__2;
/* Local variables */
integer ix, kase, kase1;
doublereal scale;
integer isave[3];
doublereal anorm;
logical upper;
doublereal xnorm;
doublereal ainvnm;
logical onenrm;
char normin[1];
doublereal smlnum;
logical nounit;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* ZTPCON estimates the reciprocal of the condition number of a packed */
/* triangular matrix A, in either the 1-norm or the infinity-norm. */
/* The norm of A is computed and an estimate is obtained for */
/* norm(inv(A)), then the reciprocal of the condition number is */
/* computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': A is upper triangular; */
/* = 'L': A is lower triangular. */
/* DIAG (input) CHARACTER*1 */
/* = 'N': A is non-unit triangular; */
/* = 'U': A is unit triangular. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2) */
/* The upper or lower triangular matrix A, packed columnwise in */
/* a linear array. The j-th column of A is stored in the array */
/* AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */
/* If DIAG = 'U', the diagonal elements of A are not referenced */
/* and are assumed to be 1. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
--rwork;
--work;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
nounit = lsame_(diag, "N");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (! upper && ! lsame_(uplo, "L")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTPCON", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*rcond = 1.;
return 0;
}
*rcond = 0.;
smlnum = dlamch_("Safe minimum") * (doublereal) max(1,*n);
/* Compute the norm of the triangular matrix A. */
anorm = zlantp_(norm, uplo, diag, n, &ap[1], &rwork[1]);
/* Continue only if ANORM > 0. */
if (anorm > 0.) {
/* Estimate the norm of the inverse of A. */
ainvnm = 0.;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(A). */
zlatps_(uplo, "No transpose", diag, normin, n, &ap[1], &work[
1], &scale, &rwork[1], info);
} else {
/* Multiply by inv(A'). */
zlatps_(uplo, "Conjugate transpose", diag, normin, n, &ap[1],
&work[1], &scale, &rwork[1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.) {
ix = izamax_(n, &work[1], &c__1);
i__1 = ix;
xnorm = (d__1 = work[i__1].r, abs(d__1)) + (d__2 = d_imag(&
work[ix]), abs(d__2));
if (scale < xnorm * smlnum || scale == 0.) {
goto L20;
}
zdrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / anorm / ainvnm;
}
}
L20:
return 0;
/* End of ZTPCON */
} /* ztpcon_ */
int ztbcon_(char *norm, char *uplo, char *diag, int *n,
int *kd, doublecomplex *ab, int *ldab, double *rcond,
doublecomplex *work, double *rwork, int *info)
{
/* System generated locals */
int ab_dim1, ab_offset, i__1;
double d__1, d__2;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
int ix, kase, kase1;
double scale;
extern int lsame_(char *, char *);
int isave[3];
double anorm;
int upper;
double xnorm;
extern int zlacn2_(int *, doublecomplex *,
doublecomplex *, double *, int *, int *);
extern double dlamch_(char *);
extern int xerbla_(char *, int *);
double ainvnm;
extern int izamax_(int *, doublecomplex *, int *);
extern double zlantb_(char *, char *, char *, int *, int *,
doublecomplex *, int *, double *);
int onenrm;
extern int zlatbs_(char *, char *, char *, char *,
int *, int *, doublecomplex *, int *, doublecomplex *,
double *, double *, int *), zdrscl_(int *, double *, doublecomplex *,
int *);
char normin[1];
double smlnum;
int nounit;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZTBCON estimates the reciprocal of the condition number of a */
/* triangular band matrix A, in either the 1-norm or the infinity-norm. */
/* The norm of A is computed and an estimate is obtained for */
/* norm(inv(A)), then the reciprocal of the condition number is */
/* computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': A is upper triangular; */
/* = 'L': A is lower triangular. */
/* DIAG (input) CHARACTER*1 */
/* = 'N': A is non-unit triangular; */
/* = 'U': A is unit triangular. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals or subdiagonals of the */
/* triangular band matrix A. KD >= 0. */
/* AB (input) COMPLEX*16 array, dimension (LDAB,N) */
/* The upper or lower triangular band matrix A, stored in the */
/* first kd+1 rows of the array. The j-th column of A is stored */
/* in the j-th column of the array AB as follows: */
/* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for MAX(1,j-kd)<=i<=j; */
/* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=MIN(n,j+kd). */
/* If DIAG = 'U', the diagonal elements of A are not referenced */
/* and are assumed to be 1. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD+1. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
nounit = lsame_(diag, "N");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (! upper && ! lsame_(uplo, "L")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*kd < 0) {
*info = -5;
} else if (*ldab < *kd + 1) {
*info = -7;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTBCON", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*rcond = 1.;
return 0;
}
*rcond = 0.;
smlnum = dlamch_("Safe minimum") * (double) MAX(*n,1);
/* Compute the 1-norm of the triangular matrix A or A'. */
anorm = zlantb_(norm, uplo, diag, n, kd, &ab[ab_offset], ldab, &rwork[1]);
/* Continue only if ANORM > 0. */
if (anorm > 0.) {
/* Estimate the 1-norm of the inverse of A. */
ainvnm = 0.;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(A). */
zlatbs_(uplo, "No transpose", diag, normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scale, &rwork[1], info);
} else {
/* Multiply by inv(A'). */
zlatbs_(uplo, "Conjugate transpose", diag, normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scale, &rwork[1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.) {
ix = izamax_(n, &work[1], &c__1);
i__1 = ix;
xnorm = (d__1 = work[i__1].r, ABS(d__1)) + (d__2 = d_imag(&
work[ix]), ABS(d__2));
if (scale < xnorm * smlnum || scale == 0.) {
goto L20;
}
zdrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / anorm / ainvnm;
}
}
L20:
return 0;
/* End of ZTBCON */
} /* ztbcon_ */
doublereal zla_porcond_c__(char *uplo, integer *n, doublecomplex *a, integer *
lda, doublecomplex *af, integer *ldaf, doublereal *c__, logical *
capply, integer *info, doublecomplex *work, doublereal *rwork, ftnlen
uplo_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2, i__3, i__4;
doublereal ret_val, d__1, d__2;
doublecomplex z__1;
/* Local variables */
integer i__, j;
logical up;
doublereal tmp;
integer kase;
integer isave[3];
doublereal anorm;
doublereal ainvnm;
/* -- LAPACK routine (version 3.2.1) -- */
/* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/* -- Jason Riedy of Univ. of California Berkeley. -- */
/* -- April 2009 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley and NAG Ltd. -- */
/* Purpose */
/* ======= */
/* ZLA_PORCOND_C Computes the infinity norm condition number of */
/* op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* A (input) COMPLEX*16 array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) COMPLEX*16 array, dimension (LDAF,N) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**T*U or A = L*L**T, as computed by ZPOTRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* C (input) DOUBLE PRECISION array, dimension (N) */
/* The vector C in the formula op(A) * inv(diag(C)). */
/* CAPPLY (input) LOGICAL */
/* If .TRUE. then access the vector C in the formula above. */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) COMPLEX*16 array, dimension (2*N). */
/* Workspace. */
/* RWORK (input) DOUBLE PRECISION array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--c__;
--work;
--rwork;
/* Function Body */
ret_val = 0.;
*info = 0;
if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZLA_PORCOND_C", &i__1);
return ret_val;
}
up = FALSE_;
if (lsame_(uplo, "U")) {
up = TRUE_;
}
/* Compute norm of op(A)*op2(C). */
anorm = 0.;
if (up) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*capply) {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
j + i__ * a_dim1]), abs(d__2))) / c__[j];
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + j * a_dim1]), abs(d__2))) / c__[j];
}
} else {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
j + i__ * a_dim1]), abs(d__2));
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + j * a_dim1]), abs(d__2));
}
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*capply) {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + j * a_dim1]), abs(d__2))) / c__[j];
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
j + i__ * a_dim1]), abs(d__2))) / c__[j];
}
} else {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + j * a_dim1]), abs(d__2));
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a[
j + i__ * a_dim1]), abs(d__2));
}
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0) {
ret_val = 1.;
return ret_val;
} else if (anorm == 0.) {
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.;
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
if (up) {
zpotrs_("U", n, &c__1, &af[af_offset], ldaf, &work[1], n,
info);
} else {
zpotrs_("L", n, &c__1, &af[af_offset], ldaf, &work[1], n,
info);
}
/* Multiply by inv(C). */
if (*capply) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = c__[i__4] * work[i__3].r, z__1.i = c__[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
} else {
/* Multiply by inv(C'). */
if (*capply) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = c__[i__4] * work[i__3].r, z__1.i = c__[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
if (up) {
zpotrs_("U", n, &c__1, &af[af_offset], ldaf, &work[1], n,
info);
} else {
zpotrs_("L", n, &c__1, &af[af_offset], ldaf, &work[1], n,
info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
ret_val = 1. / ainvnm;
}
return ret_val;
} /* zla_porcond_c__ */
doublereal zla_gbrcond_x__(char *trans, integer *n, integer *kl, integer *ku,
doublecomplex *ab, integer *ldab, doublecomplex *afb, integer *ldafb,
integer *ipiv, doublecomplex *x, integer *info, doublecomplex *work,
doublereal *rwork, ftnlen trans_len)
{
/* System generated locals */
integer ab_dim1, ab_offset, afb_dim1, afb_offset, i__1, i__2, i__3, i__4;
doublereal ret_val, d__1, d__2;
doublecomplex z__1, z__2;
/* Builtin functions */
double d_imag(doublecomplex *);
void z_div(doublecomplex *, doublecomplex *, doublecomplex *);
/* Local variables */
integer i__, j, kd, ke;
doublereal tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
doublereal anorm;
extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *), xerbla_(
char *, integer *);
doublereal ainvnm;
extern /* Subroutine */ int zgbtrs_(char *, integer *, integer *, integer
*, integer *, doublecomplex *, integer *, integer *,
doublecomplex *, integer *, integer *);
logical notrans;
/* -- LAPACK routine (version 3.2.1) -- */
/* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/* -- Jason Riedy of Univ. of California Berkeley. -- */
/* -- April 2009 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley and NAG Ltd. -- */
/* .. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* Purpose */
/* ======= */
/* ZLA_GBRCOND_X Computes the infinity norm condition number of */
/* op(A) * diag(X) where X is a COMPLEX*16 vector. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate Transpose = Transpose) */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* KL (input) INTEGER */
/* The number of subdiagonals within the band of A. KL >= 0. */
/* KU (input) INTEGER */
/* The number of superdiagonals within the band of A. KU >= 0. */
/* AB (input) COMPLEX*16 array, dimension (LDAB,N) */
/* On entry, the matrix A in band storage, in rows 1 to KL+KU+1. */
/* The j-th column of A is stored in the j-th column of the */
/* array AB as follows: */
/* AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KL+KU+1. */
/* AFB (input) COMPLEX*16 array, dimension (LDAFB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by ZGBTRF. U is stored as an upper triangular */
/* band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, */
/* and the multipliers used during the factorization are stored */
/* in rows KL+KU+2 to 2*KL+KU+1. */
/* LDAFB (input) INTEGER */
/* The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from the factorization A = P*L*U */
/* as computed by ZGBTRF; row i of the matrix was interchanged */
/* with row IPIV(i). */
/* X (input) COMPLEX*16 array, dimension (N) */
/* The vector X in the formula op(A) * diag(X). */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) COMPLEX*16 array, dimension (2*N). */
/* Workspace. */
/* RWORK (input) DOUBLE PRECISION array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function Definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_offset;
--ipiv;
--x;
--work;
--rwork;
/* Function Body */
ret_val = 0.;
*info = 0;
notrans = lsame_(trans, "N");
if (! notrans && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kl < 0 || *kl > *n - 1) {
*info = -3;
} else if (*ku < 0 || *ku > *n - 1) {
*info = -4;
} else if (*ldab < *kl + *ku + 1) {
*info = -6;
} else if (*ldafb < (*kl << 1) + *ku + 1) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZLA_GBRCOND_X", &i__1);
return ret_val;
}
/* Compute norm of op(A)*op2(C). */
kd = *ku + 1;
ke = *kl + 1;
anorm = 0.;
if (notrans) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
/* Computing MAX */
i__2 = i__ - *kl;
/* Computing MIN */
i__4 = i__ + *ku;
i__3 = min(i__4,*n);
for (j = max(i__2,1); j <= i__3; ++j) {
i__2 = kd + i__ - j + j * ab_dim1;
i__4 = j;
z__2.r = ab[i__2].r * x[i__4].r - ab[i__2].i * x[i__4].i,
z__2.i = ab[i__2].r * x[i__4].i + ab[i__2].i * x[i__4]
.r;
z__1.r = z__2.r, z__1.i = z__2.i;
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1),
abs(d__2));
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
/* Computing MAX */
i__3 = i__ - *kl;
/* Computing MIN */
i__4 = i__ + *ku;
i__2 = min(i__4,*n);
for (j = max(i__3,1); j <= i__2; ++j) {
i__3 = ke - i__ + j + i__ * ab_dim1;
i__4 = j;
z__2.r = ab[i__3].r * x[i__4].r - ab[i__3].i * x[i__4].i,
z__2.i = ab[i__3].r * x[i__4].i + ab[i__3].i * x[i__4]
.r;
z__1.r = z__2.r, z__1.i = z__2.i;
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1),
abs(d__2));
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0) {
ret_val = 1.;
return ret_val;
} else if (anorm == 0.) {
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.;
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
if (notrans) {
zgbtrs_("No transpose", n, kl, ku, &c__1, &afb[afb_offset],
ldafb, &ipiv[1], &work[1], n, info);
} else {
zgbtrs_("Conjugate transpose", n, kl, ku, &c__1, &afb[
afb_offset], ldafb, &ipiv[1], &work[1], n, info);
}
/* Multiply by inv(X). */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
z_div(&z__1, &work[i__], &x[i__]);
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
} else {
/* Multiply by inv(X'). */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
z_div(&z__1, &work[i__], &x[i__]);
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
if (notrans) {
zgbtrs_("Conjugate transpose", n, kl, ku, &c__1, &afb[
afb_offset], ldafb, &ipiv[1], &work[1], n, info);
} else {
zgbtrs_("No transpose", n, kl, ku, &c__1, &afb[afb_offset],
ldafb, &ipiv[1], &work[1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
ret_val = 1. / ainvnm;
}
return ret_val;
} /* zla_gbrcond_x__ */
/* Subroutine */ int ztrsna_(char *job, char *howmny, logical *select,
integer *n, doublecomplex *t, integer *ldt, doublecomplex *vl,
integer *ldvl, doublecomplex *vr, integer *ldvr, doublereal *s,
doublereal *sep, integer *mm, integer *m, doublecomplex *work,
integer *ldwork, doublereal *rwork, integer *info)
{
/* System generated locals */
integer t_dim1, t_offset, vl_dim1, vl_offset, vr_dim1, vr_offset,
work_dim1, work_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2;
doublecomplex z__1;
/* Builtin functions */
double z_abs(doublecomplex *), d_imag(doublecomplex *);
/* Local variables */
integer i__, j, k, ks, ix;
doublereal eps, est;
integer kase, ierr;
doublecomplex prod;
doublereal lnrm, rnrm, scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *);
doublecomplex dummy[1];
logical wants;
doublereal xnorm;
extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *), dlabad_(
doublereal *, doublereal *);
extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_(
char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal bignum;
logical wantbh;
extern integer izamax_(integer *, doublecomplex *, integer *);
logical somcon;
extern /* Subroutine */ int zdrscl_(integer *, doublereal *,
doublecomplex *, integer *);
char normin[1];
extern /* Subroutine */ int zlacpy_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *);
doublereal smlnum;
logical wantsp;
extern /* Subroutine */ int zlatrs_(char *, char *, char *, char *,
integer *, doublecomplex *, integer *, doublecomplex *,
doublereal *, doublereal *, integer *), ztrexc_(char *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, integer *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZTRSNA estimates reciprocal condition numbers for specified */
/* eigenvalues and/or right eigenvectors of a complex upper triangular */
/* matrix T (or of any matrix Q*T*Q**H with Q unitary). */
/* Arguments */
/* ========= */
/* JOB (input) CHARACTER*1 */
/* Specifies whether condition numbers are required for */
/* eigenvalues (S) or eigenvectors (SEP): */
/* = 'E': for eigenvalues only (S); */
/* = 'V': for eigenvectors only (SEP); */
/* = 'B': for both eigenvalues and eigenvectors (S and SEP). */
/* HOWMNY (input) CHARACTER*1 */
/* = 'A': compute condition numbers for all eigenpairs; */
/* = 'S': compute condition numbers for selected eigenpairs */
/* specified by the array SELECT. */
/* SELECT (input) LOGICAL array, dimension (N) */
/* If HOWMNY = 'S', SELECT specifies the eigenpairs for which */
/* condition numbers are required. To select condition numbers */
/* for the j-th eigenpair, SELECT(j) must be set to .TRUE.. */
/* If HOWMNY = 'A', SELECT is not referenced. */
/* N (input) INTEGER */
/* The order of the matrix T. N >= 0. */
/* T (input) COMPLEX*16 array, dimension (LDT,N) */
/* The upper triangular matrix T. */
/* LDT (input) INTEGER */
/* The leading dimension of the array T. LDT >= max(1,N). */
/* VL (input) COMPLEX*16 array, dimension (LDVL,M) */
/* If JOB = 'E' or 'B', VL must contain left eigenvectors of T */
/* (or of any Q*T*Q**H with Q unitary), corresponding to the */
/* eigenpairs specified by HOWMNY and SELECT. The eigenvectors */
/* must be stored in consecutive columns of VL, as returned by */
/* ZHSEIN or ZTREVC. */
/* If JOB = 'V', VL is not referenced. */
/* LDVL (input) INTEGER */
/* The leading dimension of the array VL. */
/* LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. */
/* VR (input) COMPLEX*16 array, dimension (LDVR,M) */
/* If JOB = 'E' or 'B', VR must contain right eigenvectors of T */
/* (or of any Q*T*Q**H with Q unitary), corresponding to the */
/* eigenpairs specified by HOWMNY and SELECT. The eigenvectors */
/* must be stored in consecutive columns of VR, as returned by */
/* ZHSEIN or ZTREVC. */
/* If JOB = 'V', VR is not referenced. */
/* LDVR (input) INTEGER */
/* The leading dimension of the array VR. */
/* LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. */
/* S (output) DOUBLE PRECISION array, dimension (MM) */
/* If JOB = 'E' or 'B', the reciprocal condition numbers of the */
/* selected eigenvalues, stored in consecutive elements of the */
/* array. Thus S(j), SEP(j), and the j-th columns of VL and VR */
/* all correspond to the same eigenpair (but not in general the */
/* j-th eigenpair, unless all eigenpairs are selected). */
/* If JOB = 'V', S is not referenced. */
/* SEP (output) DOUBLE PRECISION array, dimension (MM) */
/* If JOB = 'V' or 'B', the estimated reciprocal condition */
/* numbers of the selected eigenvectors, stored in consecutive */
/* elements of the array. */
/* If JOB = 'E', SEP is not referenced. */
/* MM (input) INTEGER */
/* The number of elements in the arrays S (if JOB = 'E' or 'B') */
/* and/or SEP (if JOB = 'V' or 'B'). MM >= M. */
/* M (output) INTEGER */
/* The number of elements of the arrays S and/or SEP actually */
/* used to store the estimated condition numbers. */
/* If HOWMNY = 'A', M is set to N. */
/* WORK (workspace) COMPLEX*16 array, dimension (LDWORK,N+6) */
/* If JOB = 'E', WORK is not referenced. */
/* LDWORK (input) INTEGER */
/* The leading dimension of the array WORK. */
/* LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* If JOB = 'E', RWORK is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* The reciprocal of the condition number of an eigenvalue lambda is */
/* defined as */
/* S(lambda) = |v'*u| / (norm(u)*norm(v)) */
/* where u and v are the right and left eigenvectors of T corresponding */
/* to lambda; v' denotes the conjugate transpose of v, and norm(u) */
/* denotes the Euclidean norm. These reciprocal condition numbers always */
/* lie between zero (very badly conditioned) and one (very well */
/* conditioned). If n = 1, S(lambda) is defined to be 1. */
/* An approximate error bound for a computed eigenvalue W(i) is given by */
/* EPS * norm(T) / S(i) */
/* where EPS is the machine precision. */
/* The reciprocal of the condition number of the right eigenvector u */
/* corresponding to lambda is defined as follows. Suppose */
/* T = ( lambda c ) */
/* ( 0 T22 ) */
/* Then the reciprocal condition number is */
/* SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) */
/* where sigma-min denotes the smallest singular value. We approximate */
/* the smallest singular value by the reciprocal of an estimate of the */
/* one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is */
/* defined to be abs(T(1,1)). */
/* An approximate error bound for a computed right eigenvector VR(i) */
/* is given by */
/* EPS * norm(T) / SEP(i) */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/* Parameter adjustments */
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--s;
--sep;
work_dim1 = *ldwork;
work_offset = 1 + work_dim1;
work -= work_offset;
--rwork;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
somcon = lsame_(howmny, "S");
/* Set M to the number of eigenpairs for which condition numbers are */
/* to be computed. */
if (somcon) {
*m = 0;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (select[j]) {
++(*m);
}
/* L10: */
}
} else {
*m = *n;
}
*info = 0;
if (! wants && ! wantsp) {
*info = -1;
} else if (! lsame_(howmny, "A") && ! somcon) {
*info = -2;
} else if (*n < 0) {
*info = -4;
} else if (*ldt < max(1,*n)) {
*info = -6;
} else if (*ldvl < 1 || wants && *ldvl < *n) {
*info = -8;
} else if (*ldvr < 1 || wants && *ldvr < *n) {
*info = -10;
} else if (*mm < *m) {
*info = -13;
} else if (*ldwork < 1 || wantsp && *ldwork < *n) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTRSNA", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (somcon) {
if (! select[1]) {
return 0;
}
}
if (wants) {
s[1] = 1.;
}
if (wantsp) {
sep[1] = z_abs(&t[t_dim1 + 1]);
}
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S") / eps;
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
ks = 1;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (somcon) {
if (! select[k]) {
goto L50;
}
}
if (wants) {
/* Compute the reciprocal condition number of the k-th */
/* eigenvalue. */
zdotc_(&z__1, n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks * vl_dim1 +
1], &c__1);
prod.r = z__1.r, prod.i = z__1.i;
rnrm = dznrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
lnrm = dznrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
s[ks] = z_abs(&prod) / (rnrm * lnrm);
}
if (wantsp) {
/* Estimate the reciprocal condition number of the k-th */
/* eigenvector. */
/* Copy the matrix T to the array WORK and swap the k-th */
/* diagonal element to the (1,1) position. */
zlacpy_("Full", n, n, &t[t_offset], ldt, &work[work_offset],
ldwork);
ztrexc_("No Q", n, &work[work_offset], ldwork, dummy, &c__1, &k, &
c__1, &ierr);
/* Form C = T22 - lambda*I in WORK(2:N,2:N). */
i__2 = *n;
for (i__ = 2; i__ <= i__2; ++i__) {
i__3 = i__ + i__ * work_dim1;
i__4 = i__ + i__ * work_dim1;
i__5 = work_dim1 + 1;
z__1.r = work[i__4].r - work[i__5].r, z__1.i = work[i__4].i -
work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
/* L20: */
}
/* Estimate a lower bound for the 1-norm of inv(C'). The 1st */
/* and (N+1)th columns of WORK are used to store work vectors. */
sep[ks] = 0.;
est = 0.;
kase = 0;
*(unsigned char *)normin = 'N';
L30:
i__2 = *n - 1;
zlacn2_(&i__2, &work[(*n + 1) * work_dim1 + 1], &work[work_offset]
, &est, &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve C'*x = scale*b */
i__2 = *n - 1;
zlatrs_("Upper", "Conjugate transpose", "Nonunit", normin,
&i__2, &work[(work_dim1 << 1) + 2], ldwork, &
work[work_offset], &scale, &rwork[1], &ierr);
} else {
/* Solve C*x = scale*b */
i__2 = *n - 1;
zlatrs_("Upper", "No transpose", "Nonunit", normin, &i__2,
&work[(work_dim1 << 1) + 2], ldwork, &work[
work_offset], &scale, &rwork[1], &ierr);
}
*(unsigned char *)normin = 'Y';
if (scale != 1.) {
/* Multiply by 1/SCALE if doing so will not cause */
/* overflow. */
i__2 = *n - 1;
ix = izamax_(&i__2, &work[work_offset], &c__1);
i__2 = ix + work_dim1;
xnorm = (d__1 = work[i__2].r, abs(d__1)) + (d__2 = d_imag(
&work[ix + work_dim1]), abs(d__2));
if (scale < xnorm * smlnum || scale == 0.) {
goto L40;
}
zdrscl_(n, &scale, &work[work_offset], &c__1);
}
goto L30;
}
sep[ks] = 1. / max(est,smlnum);
}
L40:
++ks;
L50:
;
}
return 0;
/* End of ZTRSNA */
} /* ztrsna_ */
/* Subroutine */ int zpbcon_(char *uplo, integer *n, integer *kd,
doublecomplex *ab, integer *ldab, doublereal *anorm, doublereal *
rcond, doublecomplex *work, doublereal *rwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1;
doublereal d__1, d__2;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer ix, kase;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal scalel, scaleu;
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal ainvnm;
extern integer izamax_(integer *, doublecomplex *, integer *);
extern /* Subroutine */ int zlatbs_(char *, char *, char *, char *,
integer *, integer *, doublecomplex *, integer *, doublecomplex *,
doublereal *, doublereal *, integer *), zdrscl_(integer *, doublereal *, doublecomplex *,
integer *);
char normin[1];
doublereal smlnum;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZPBCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a complex Hermitian positive definite band matrix using */
/* the Cholesky factorization A = U**H*U or A = L*L**H computed by */
/* ZPBTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangular factor stored in AB; */
/* = 'L': Lower triangular factor stored in AB. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals of the matrix A if UPLO = 'U', */
/* or the number of sub-diagonals if UPLO = 'L'. KD >= 0. */
/* AB (input) COMPLEX*16 array, dimension (LDAB,N) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**H*U or A = L*L**H of the band matrix A, stored in the */
/* first KD+1 rows of the array. The j-th column of U or L is */
/* stored in the j-th column of the array AB as follows: */
/* if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j; */
/* if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd). */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD+1. */
/* ANORM (input) DOUBLE PRECISION */
/* The 1-norm (or infinity-norm) of the Hermitian band matrix A. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kd < 0) {
*info = -3;
} else if (*ldab < *kd + 1) {
*info = -5;
} else if (*anorm < 0.) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZPBCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0) {
*rcond = 1.;
return 0;
} else if (*anorm == 0.) {
return 0;
}
smlnum = dlamch_("Safe minimum");
/* Estimate the 1-norm of the inverse. */
kase = 0;
*(unsigned char *)normin = 'N';
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (upper) {
/* Multiply by inv(U'). */
zlatbs_("Upper", "Conjugate transpose", "Non-unit", normin, n, kd,
&ab[ab_offset], ldab, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
zlatbs_("Upper", "No transpose", "Non-unit", normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scaleu, &rwork[1], info);
} else {
/* Multiply by inv(L). */
zlatbs_("Lower", "No transpose", "Non-unit", normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L'). */
zlatbs_("Lower", "Conjugate transpose", "Non-unit", normin, n, kd,
&ab[ab_offset], ldab, &work[1], &scaleu, &rwork[1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.) {
ix = izamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((d__1 = work[i__1].r, abs(d__1)) + (d__2 = d_imag(&
work[ix]), abs(d__2))) * smlnum || scale == 0.) {
goto L20;
}
zdrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / ainvnm / *anorm;
}
L20:
return 0;
/* End of ZPBCON */
} /* zpbcon_ */
int zgtcon_(char *norm, int *n, doublecomplex *dl,
doublecomplex *d__, doublecomplex *du, doublecomplex *du2, int *
ipiv, double *anorm, double *rcond, doublecomplex *work,
int *info)
{
/* System generated locals */
int i__1, i__2;
/* Local variables */
int i__, kase, kase1;
extern int lsame_(char *, char *);
int isave[3];
extern int zlacn2_(int *, doublecomplex *,
doublecomplex *, double *, int *, int *), xerbla_(
char *, int *);
double ainvnm;
int onenrm;
extern int zgttrs_(char *, int *, int *,
doublecomplex *, doublecomplex *, doublecomplex *, doublecomplex *
, int *, doublecomplex *, int *, int *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZGTCON estimates the reciprocal of the condition number of a complex */
/* tridiagonal matrix A using the LU factorization as computed by */
/* ZGTTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* DL (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) multipliers that define the matrix L from the */
/* LU factorization of A as computed by ZGTTRF. */
/* D (input) COMPLEX*16 array, dimension (N) */
/* The n diagonal elements of the upper triangular matrix U from */
/* the LU factorization of A. */
/* DU (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) elements of the first superdiagonal of U. */
/* DU2 (input) COMPLEX*16 array, dimension (N-2) */
/* The (n-2) elements of the second superdiagonal of U. */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices; for 1 <= i <= n, row i of the matrix was */
/* interchanged with row IPIV(i). IPIV(i) will always be either */
/* i or i+1; IPIV(i) = i indicates a row interchange was not */
/* required. */
/* ANORM (input) DOUBLE PRECISION */
/* If NORM = '1' or 'O', the 1-norm of the original matrix A. */
/* If NORM = 'I', the infinity-norm of the original matrix A. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments. */
/* Parameter adjustments */
--work;
--ipiv;
--du2;
--du;
--d__;
--dl;
/* Function Body */
*info = 0;
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*anorm < 0.) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGTCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0) {
*rcond = 1.;
return 0;
} else if (*anorm == 0.) {
return 0;
}
/* Check that D(1:N) is non-zero. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
if (d__[i__2].r == 0. && d__[i__2].i == 0.) {
return 0;
}
/* L10: */
}
ainvnm = 0.;
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L20:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(U)*inv(L). */
zgttrs_("No transpose", n, &c__1, &dl[1], &d__[1], &du[1], &du2[1]
, &ipiv[1], &work[1], n, info);
} else {
/* Multiply by inv(L')*inv(U'). */
zgttrs_("Conjugate transpose", n, &c__1, &dl[1], &d__[1], &du[1],
&du2[1], &ipiv[1], &work[1], n, info);
}
goto L20;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / ainvnm / *anorm;
}
return 0;
/* End of ZGTCON */
} /* zgtcon_ */
/* ===================================================================== */
doublereal zla_gbrcond_x_(char *trans, integer *n, integer *kl, integer *ku, doublecomplex *ab, integer *ldab, doublecomplex *afb, integer *ldafb, integer *ipiv, doublecomplex *x, integer *info, doublecomplex *work, doublereal *rwork)
{
/* System generated locals */
integer ab_dim1, ab_offset, afb_dim1, afb_offset, i__1, i__2, i__3, i__4;
doublereal ret_val, d__1, d__2;
doublecomplex z__1, z__2;
/* Builtin functions */
double d_imag(doublecomplex *);
void z_div(doublecomplex *, doublecomplex *, doublecomplex *);
/* Local variables */
integer i__, j, kd, ke;
doublereal tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
doublereal anorm;
extern /* Subroutine */
int zlacn2_(integer *, doublecomplex *, doublecomplex *, doublereal *, integer *, integer *), xerbla_( char *, integer *);
doublereal ainvnm;
extern /* Subroutine */
int zgbtrs_(char *, integer *, integer *, integer *, integer *, doublecomplex *, integer *, integer *, doublecomplex *, integer *, integer *);
logical notrans;
/* -- LAPACK computational routine (version 3.4.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* September 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function Definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_offset;
--ipiv;
--x;
--work;
--rwork;
/* Function Body */
ret_val = 0.;
*info = 0;
notrans = lsame_(trans, "N");
if (! notrans && ! lsame_(trans, "T") && ! lsame_( trans, "C"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*kl < 0 || *kl > *n - 1)
{
*info = -3;
}
else if (*ku < 0 || *ku > *n - 1)
{
*info = -4;
}
else if (*ldab < *kl + *ku + 1)
{
*info = -6;
}
else if (*ldafb < (*kl << 1) + *ku + 1)
{
*info = -8;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZLA_GBRCOND_X", &i__1);
return ret_val;
}
/* Compute norm of op(A)*op2(C). */
kd = *ku + 1;
ke = *kl + 1;
anorm = 0.;
if (notrans)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.;
/* Computing MAX */
i__2 = i__ - *kl;
/* Computing MIN */
i__4 = i__ + *ku;
i__3 = min(i__4,*n);
for (j = max(i__2,1);
j <= i__3;
++j)
{
i__2 = kd + i__ - j + j * ab_dim1;
i__4 = j;
z__2.r = ab[i__2].r * x[i__4].r - ab[i__2].i * x[i__4].i;
z__2.i = ab[i__2].r * x[i__4].i + ab[i__2].i * x[i__4] .r; // , expr subst
z__1.r = z__2.r;
z__1.i = z__2.i; // , expr subst
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1), abs(d__2));
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
else
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.;
/* Computing MAX */
i__3 = i__ - *kl;
/* Computing MIN */
i__4 = i__ + *ku;
i__2 = min(i__4,*n);
for (j = max(i__3,1);
j <= i__2;
++j)
{
i__3 = ke - i__ + j + i__ * ab_dim1;
i__4 = j;
z__2.r = ab[i__3].r * x[i__4].r - ab[i__3].i * x[i__4].i;
z__2.i = ab[i__3].r * x[i__4].i + ab[i__3].i * x[i__4] .r; // , expr subst
z__1.r = z__2.r;
z__1.i = z__2.i; // , expr subst
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1), abs(d__2));
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0)
{
ret_val = 1.;
return ret_val;
}
else if (anorm == 0.)
{
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.;
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (kase == 2)
{
/* Multiply by R. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r;
z__1.i = rwork[i__4] * work[i__3].i; // , expr subst
work[i__2].r = z__1.r;
work[i__2].i = z__1.i; // , expr subst
}
if (notrans)
{
zgbtrs_("No transpose", n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1], &work[1], n, info);
}
else
{
zgbtrs_("Conjugate transpose", n, kl, ku, &c__1, &afb[ afb_offset], ldafb, &ipiv[1], &work[1], n, info);
}
/* Multiply by inv(X). */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
z_div(&z__1, &work[i__], &x[i__]);
work[i__2].r = z__1.r;
work[i__2].i = z__1.i; // , expr subst
}
}
else
{
/* Multiply by inv(X**H). */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
z_div(&z__1, &work[i__], &x[i__]);
work[i__2].r = z__1.r;
work[i__2].i = z__1.i; // , expr subst
}
if (notrans)
{
zgbtrs_("Conjugate transpose", n, kl, ku, &c__1, &afb[ afb_offset], ldafb, &ipiv[1], &work[1], n, info);
}
else
{
zgbtrs_("No transpose", n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1], &work[1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r;
z__1.i = rwork[i__4] * work[i__3].i; // , expr subst
work[i__2].r = z__1.r;
work[i__2].i = z__1.i; // , expr subst
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.)
{
ret_val = 1. / ainvnm;
}
return ret_val;
}
/* Subroutine */ int ztrsen_(char *job, char *compq, logical *select, integer
*n, doublecomplex *t, integer *ldt, doublecomplex *q, integer *ldq,
doublecomplex *w, integer *m, doublereal *s, doublereal *sep,
doublecomplex *work, integer *lwork, integer *info)
{
/* System generated locals */
integer q_dim1, q_offset, t_dim1, t_offset, i__1, i__2, i__3;
/* Local variables */
integer k, n1, n2, nn, ks;
doublereal est;
integer kase, ierr;
doublereal scale;
integer isave[3], lwmin;
logical wantq, wants;
doublereal rnorm, rwork[1];
logical wantbh;
logical wantsp;
logical lquery;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* ZTRSEN reorders the Schur factorization of a complex matrix */
/* A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in */
/* the leading positions on the diagonal of the upper triangular matrix */
/* T, and the leading columns of Q form an orthonormal basis of the */
/* corresponding right invariant subspace. */
/* Optionally the routine computes the reciprocal condition numbers of */
/* the cluster of eigenvalues and/or the invariant subspace. */
/* Arguments */
/* ========= */
/* JOB (input) CHARACTER*1 */
/* Specifies whether condition numbers are required for the */
/* cluster of eigenvalues (S) or the invariant subspace (SEP): */
/* = 'N': none; */
/* = 'E': for eigenvalues only (S); */
/* = 'V': for invariant subspace only (SEP); */
/* = 'B': for both eigenvalues and invariant subspace (S and */
/* SEP). */
/* COMPQ (input) CHARACTER*1 */
/* = 'V': update the matrix Q of Schur vectors; */
/* = 'N': do not update Q. */
/* SELECT (input) LOGICAL array, dimension (N) */
/* SELECT specifies the eigenvalues in the selected cluster. To */
/* N (input) INTEGER */
/* The order of the matrix T. N >= 0. */
/* T (input/output) COMPLEX*16 array, dimension (LDT,N) */
/* On entry, the upper triangular matrix T. */
/* On exit, T is overwritten by the reordered matrix T, with the */
/* selected eigenvalues as the leading diagonal elements. */
/* LDT (input) INTEGER */
/* The leading dimension of the array T. LDT >= max(1,N). */
/* Q (input/output) COMPLEX*16 array, dimension (LDQ,N) */
/* On entry, if COMPQ = 'V', the matrix Q of Schur vectors. */
/* On exit, if COMPQ = 'V', Q has been postmultiplied by the */
/* unitary transformation matrix which reorders T; the leading M */
/* columns of Q form an orthonormal basis for the specified */
/* invariant subspace. */
/* If COMPQ = 'N', Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. */
/* LDQ >= 1; and if COMPQ = 'V', LDQ >= N. */
/* W (output) COMPLEX*16 array, dimension (N) */
/* The reordered eigenvalues of T, in the same order as they */
/* appear on the diagonal of T. */
/* M (output) INTEGER */
/* The dimension of the specified invariant subspace. */
/* 0 <= M <= N. */
/* S (output) DOUBLE PRECISION */
/* If JOB = 'E' or 'B', S is a lower bound on the reciprocal */
/* condition number for the selected cluster of eigenvalues. */
/* S cannot underestimate the true reciprocal condition number */
/* by more than a factor of sqrt(N). If M = 0 or N, S = 1. */
/* If JOB = 'N' or 'V', S is not referenced. */
/* SEP (output) DOUBLE PRECISION */
/* If JOB = 'V' or 'B', SEP is the estimated reciprocal */
/* condition number of the specified invariant subspace. If */
/* M = 0 or N, SEP = norm(T). */
/* If JOB = 'N' or 'E', SEP is not referenced. */
/* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. */
/* If JOB = 'N', LWORK >= 1; */
/* if JOB = 'E', LWORK = max(1,M*(N-M)); */
/* if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)). */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* ZTRSEN first collects the selected eigenvalues by computing a unitary */
/* transformation Z to move them to the top left corner of T. In other */
/* words, the selected eigenvalues are the eigenvalues of T11 in: */
/* Z'*T*Z = ( T11 T12 ) n1 */
/* ( 0 T22 ) n2 */
/* n1 n2 */
/* where N = n1+n2 and Z' means the conjugate transpose of Z. The first */
/* n1 columns of Z span the specified invariant subspace of T. */
/* If T has been obtained from the Schur factorization of a matrix */
/* A = Q*T*Q', then the reordered Schur factorization of A is given by */
/* A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the */
/* corresponding invariant subspace of A. */
/* The reciprocal condition number of the average of the eigenvalues of */
/* T11 may be returned in S. S lies between 0 (very badly conditioned) */
/* and 1 (very well conditioned). It is computed as follows. First we */
/* compute R so that */
/* P = ( I R ) n1 */
/* ( 0 0 ) n2 */
/* n1 n2 */
/* is the projector on the invariant subspace associated with T11. */
/* R is the solution of the Sylvester equation: */
/* T11*R - R*T22 = T12. */
/* Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote */
/* the two-norm of M. Then S is computed as the lower bound */
/* (1 + F-norm(R)**2)**(-1/2) */
/* on the reciprocal of 2-norm(P), the true reciprocal condition number. */
/* S cannot underestimate 1 / 2-norm(P) by more than a factor of */
/* sqrt(N). */
/* An approximate error bound for the computed average of the */
/* eigenvalues of T11 is */
/* EPS * norm(T) / S */
/* where EPS is the machine precision. */
/* The reciprocal condition number of the right invariant subspace */
/* spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. */
/* SEP is defined as the separation of T11 and T22: */
/* sep( T11, T22 ) = sigma-min( C ) */
/* where sigma-min(C) is the smallest singular value of the */
/* n1*n2-by-n1*n2 matrix */
/* C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) */
/* I(m) is an m by m identity matrix, and kprod denotes the Kronecker */
/* product. We estimate sigma-min(C) by the reciprocal of an estimate of */
/* the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) */
/* cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). */
/* When SEP is small, small changes in T can cause large changes in */
/* the invariant subspace. An approximate bound on the maximum angular */
/* error in the computed right invariant subspace is */
/* EPS * norm(T) / SEP */
/* ===================================================================== */
/* Decode and test the input parameters. */
/* Parameter adjustments */
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--w;
--work;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
wantq = lsame_(compq, "V");
/* Set M to the number of selected eigenvalues. */
*m = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (select[k]) {
++(*m);
}
}
n1 = *m;
n2 = *n - *m;
nn = n1 * n2;
*info = 0;
lquery = *lwork == -1;
if (wantsp) {
/* Computing MAX */
i__1 = 1, i__2 = nn << 1;
lwmin = max(i__1,i__2);
} else if (lsame_(job, "N")) {
lwmin = 1;
} else if (lsame_(job, "E")) {
lwmin = max(1,nn);
}
if (! lsame_(job, "N") && ! wants && ! wantsp) {
*info = -1;
} else if (! lsame_(compq, "N") && ! wantq) {
*info = -2;
} else if (*n < 0) {
*info = -4;
} else if (*ldt < max(1,*n)) {
*info = -6;
} else if (*ldq < 1 || wantq && *ldq < *n) {
*info = -8;
} else if (*lwork < lwmin && ! lquery) {
*info = -14;
}
if (*info == 0) {
work[1].r = (doublereal) lwmin, work[1].i = 0.;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTRSEN", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*m == *n || *m == 0) {
if (wants) {
*s = 1.;
}
if (wantsp) {
*sep = zlange_("1", n, n, &t[t_offset], ldt, rwork);
}
goto L40;
}
/* Collect the selected eigenvalues at the top left corner of T. */
ks = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (select[k]) {
++ks;
/* Swap the K-th eigenvalue to position KS. */
if (k != ks) {
ztrexc_(compq, n, &t[t_offset], ldt, &q[q_offset], ldq, &k, &
ks, &ierr);
}
}
}
if (wants) {
/* Solve the Sylvester equation for R: */
/* T11*R - R*T22 = scale*T12 */
zlacpy_("F", &n1, &n2, &t[(n1 + 1) * t_dim1 + 1], ldt, &work[1], &n1);
ztrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 + 1 + (n1
+ 1) * t_dim1], ldt, &work[1], &n1, &scale, &ierr);
/* Estimate the reciprocal of the condition number of the cluster */
/* of eigenvalues. */
rnorm = zlange_("F", &n1, &n2, &work[1], &n1, rwork);
if (rnorm == 0.) {
*s = 1.;
} else {
*s = scale / (sqrt(scale * scale / rnorm + rnorm) * sqrt(rnorm));
}
}
if (wantsp) {
/* Estimate sep(T11,T22). */
est = 0.;
kase = 0;
L30:
zlacn2_(&nn, &work[nn + 1], &work[1], &est, &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve T11*R - R*T22 = scale*X. */
ztrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 +
1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &
ierr);
} else {
/* Solve T11'*R - R*T22' = scale*X. */
ztrsyl_("C", "C", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 +
1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &
ierr);
}
goto L30;
}
*sep = scale / est;
}
L40:
/* Copy reordered eigenvalues to W. */
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = k;
i__3 = k + k * t_dim1;
w[i__2].r = t[i__3].r, w[i__2].i = t[i__3].i;
}
work[1].r = (doublereal) lwmin, work[1].i = 0.;
return 0;
/* End of ZTRSEN */
} /* ztrsen_ */
/* Subroutine */ int zgbcon_(char *norm, integer *n, integer *kl, integer *ku,
doublecomplex *ab, integer *ldab, integer *ipiv, doublereal *anorm,
doublereal *rcond, doublecomplex *work, doublereal *rwork, integer *
info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1, i__2, i__3;
doublereal d__1, d__2;
doublecomplex z__1, z__2;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer j;
doublecomplex t;
integer kd, lm, jp, ix, kase, kase1;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *);
logical lnoti;
extern /* Subroutine */ int zaxpy_(integer *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, integer *), zlacn2_(
integer *, doublecomplex *, doublecomplex *, doublereal *,
integer *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal ainvnm;
extern integer izamax_(integer *, doublecomplex *, integer *);
logical onenrm;
extern /* Subroutine */ int zlatbs_(char *, char *, char *, char *,
integer *, integer *, doublecomplex *, integer *, doublecomplex *,
doublereal *, doublereal *, integer *), zdrscl_(integer *, doublereal *, doublecomplex *,
integer *);
char normin[1];
doublereal smlnum;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZGBCON estimates the reciprocal of the condition number of a complex */
/* general band matrix A, in either the 1-norm or the infinity-norm, */
/* using the LU factorization computed by ZGBTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KL (input) INTEGER */
/* The number of subdiagonals within the band of A. KL >= 0. */
/* KU (input) INTEGER */
/* The number of superdiagonals within the band of A. KU >= 0. */
/* AB (input) COMPLEX*16 array, dimension (LDAB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by ZGBTRF. U is stored as an upper triangular band */
/* matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and */
/* the multipliers used during the factorization are stored in */
/* rows KL+KU+2 to 2*KL+KU+1. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= 2*KL+KU+1. */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices; for 1 <= i <= N, row i of the matrix was */
/* interchanged with row IPIV(i). */
/* ANORM (input) DOUBLE PRECISION */
/* If NORM = '1' or 'O', the 1-norm of the original matrix A. */
/* If NORM = 'I', the infinity-norm of the original matrix A. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--ipiv;
--work;
--rwork;
/* Function Body */
*info = 0;
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kl < 0) {
*info = -3;
} else if (*ku < 0) {
*info = -4;
} else if (*ldab < (*kl << 1) + *ku + 1) {
*info = -6;
} else if (*anorm < 0.) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGBCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0) {
*rcond = 1.;
return 0;
} else if (*anorm == 0.) {
return 0;
}
smlnum = dlamch_("Safe minimum");
/* Estimate the norm of inv(A). */
ainvnm = 0.;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kd = *kl + *ku + 1;
lnoti = *kl > 0;
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(L). */
if (lnoti) {
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
i__2 = *kl, i__3 = *n - j;
lm = min(i__2,i__3);
jp = ipiv[j];
i__2 = jp;
t.r = work[i__2].r, t.i = work[i__2].i;
if (jp != j) {
i__2 = jp;
i__3 = j;
work[i__2].r = work[i__3].r, work[i__2].i = work[i__3]
.i;
i__2 = j;
work[i__2].r = t.r, work[i__2].i = t.i;
}
z__1.r = -t.r, z__1.i = -t.i;
zaxpy_(&lm, &z__1, &ab[kd + 1 + j * ab_dim1], &c__1, &
work[j + 1], &c__1);
/* L20: */
}
}
/* Multiply by inv(U). */
i__1 = *kl + *ku;
zlatbs_("Upper", "No transpose", "Non-unit", normin, n, &i__1, &
ab[ab_offset], ldab, &work[1], &scale, &rwork[1], info);
} else {
/* Multiply by inv(U'). */
i__1 = *kl + *ku;
zlatbs_("Upper", "Conjugate transpose", "Non-unit", normin, n, &
i__1, &ab[ab_offset], ldab, &work[1], &scale, &rwork[1],
info);
/* Multiply by inv(L'). */
if (lnoti) {
for (j = *n - 1; j >= 1; --j) {
/* Computing MIN */
i__1 = *kl, i__2 = *n - j;
lm = min(i__1,i__2);
i__1 = j;
i__2 = j;
zdotc_(&z__2, &lm, &ab[kd + 1 + j * ab_dim1], &c__1, &
work[j + 1], &c__1);
z__1.r = work[i__2].r - z__2.r, z__1.i = work[i__2].i -
z__2.i;
work[i__1].r = z__1.r, work[i__1].i = z__1.i;
jp = ipiv[j];
if (jp != j) {
i__1 = jp;
t.r = work[i__1].r, t.i = work[i__1].i;
i__1 = jp;
i__2 = j;
work[i__1].r = work[i__2].r, work[i__1].i = work[i__2]
.i;
i__1 = j;
work[i__1].r = t.r, work[i__1].i = t.i;
}
/* L30: */
}
}
}
/* Divide X by 1/SCALE if doing so will not cause overflow. */
*(unsigned char *)normin = 'Y';
if (scale != 1.) {
ix = izamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((d__1 = work[i__1].r, abs(d__1)) + (d__2 = d_imag(&
work[ix]), abs(d__2))) * smlnum || scale == 0.) {
goto L40;
}
zdrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / ainvnm / *anorm;
}
L40:
return 0;
/* End of ZGBCON */
} /* zgbcon_ */
/* Subroutine */ int zgtrfs_(char *trans, integer *n, integer *nrhs,
doublecomplex *dl, doublecomplex *d__, doublecomplex *du,
doublecomplex *dlf, doublecomplex *df, doublecomplex *duf,
doublecomplex *du2, integer *ipiv, doublecomplex *b, integer *ldb,
doublecomplex *x, integer *ldx, doublereal *ferr, doublereal *berr,
doublecomplex *work, doublereal *rwork, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5,
i__6, i__7, i__8, i__9;
doublereal d__1, d__2, d__3, d__4, d__5, d__6, d__7, d__8, d__9, d__10,
d__11, d__12, d__13, d__14;
doublecomplex z__1;
/* Local variables */
integer i__, j;
doublereal s;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
integer isave[3], count;
doublereal safmin;
logical notran;
char transn[1], transt[1];
doublereal lstres;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* ZGTRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is tridiagonal, and provides */
/* error bounds and backward error estimates for the solution. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate transpose) */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrix B. NRHS >= 0. */
/* DL (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) subdiagonal elements of A. */
/* D (input) COMPLEX*16 array, dimension (N) */
/* The diagonal elements of A. */
/* DU (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) superdiagonal elements of A. */
/* DLF (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) multipliers that define the matrix L from the */
/* LU factorization of A as computed by ZGTTRF. */
/* DF (input) COMPLEX*16 array, dimension (N) */
/* The n diagonal elements of the upper triangular matrix U from */
/* the LU factorization of A. */
/* DUF (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) elements of the first superdiagonal of U. */
/* DU2 (input) COMPLEX*16 array, dimension (N-2) */
/* The (n-2) elements of the second superdiagonal of U. */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices; for 1 <= i <= n, row i of the matrix was */
/* interchanged with row IPIV(i). IPIV(i) will always be either */
/* i or i+1; IPIV(i) = i indicates a row interchange was not */
/* required. */
/* B (input) COMPLEX*16 array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by ZGTTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The estimated forward error bound for each solution vector */
/* X(j) (the j-th column of the solution matrix X). */
/* If XTRUE is the true solution corresponding to X(j), FERR(j) */
/* is an estimated upper bound for the magnitude of the largest */
/* element in (X(j) - XTRUE) divided by the magnitude of the */
/* largest element in X(j). The estimate is as reliable as */
/* the estimate for RCOND, and is almost always a slight */
/* overestimate of the true error. */
/* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Internal Parameters */
/* =================== */
/* ITMAX is the maximum number of steps of iterative refinement. */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
--dl;
--d__;
--du;
--dlf;
--df;
--duf;
--du2;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
notran = lsame_(trans, "N");
if (! notran && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*ldb < max(1,*n)) {
*info = -13;
} else if (*ldx < max(1,*n)) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGTRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.;
berr[j] = 0.;
}
return 0;
}
if (notran) {
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
} else {
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = 4;
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
count = 1;
lstres = 3.;
L20:
/* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
zcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
zlagtm_(trans, n, &c__1, &c_b18, &dl[1], &d__[1], &du[1], &x[j *
x_dim1 + 1], ldx, &c_b19, &work[1], n);
/* Compute abs(op(A))*abs(x) + abs(b) for use in the backward */
/* error bound. */
if (notran) {
if (*n == 1) {
i__2 = j * b_dim1 + 1;
i__3 = j * x_dim1 + 1;
rwork[1] = (d__1 = b[i__2].r, abs(d__1)) + (d__2 = d_imag(&b[
j * b_dim1 + 1]), abs(d__2)) + ((d__3 = d__[1].r, abs(
d__3)) + (d__4 = d_imag(&d__[1]), abs(d__4))) * ((
d__5 = x[i__3].r, abs(d__5)) + (d__6 = d_imag(&x[j *
x_dim1 + 1]), abs(d__6)));
} else {
i__2 = j * b_dim1 + 1;
i__3 = j * x_dim1 + 1;
i__4 = j * x_dim1 + 2;
rwork[1] = (d__1 = b[i__2].r, abs(d__1)) + (d__2 = d_imag(&b[
j * b_dim1 + 1]), abs(d__2)) + ((d__3 = d__[1].r, abs(
d__3)) + (d__4 = d_imag(&d__[1]), abs(d__4))) * ((
d__5 = x[i__3].r, abs(d__5)) + (d__6 = d_imag(&x[j *
x_dim1 + 1]), abs(d__6))) + ((d__7 = du[1].r, abs(
d__7)) + (d__8 = d_imag(&du[1]), abs(d__8))) * ((d__9
= x[i__4].r, abs(d__9)) + (d__10 = d_imag(&x[j *
x_dim1 + 2]), abs(d__10)));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
i__4 = i__ - 1;
i__5 = i__ - 1 + j * x_dim1;
i__6 = i__;
i__7 = i__ + j * x_dim1;
i__8 = i__;
i__9 = i__ + 1 + j * x_dim1;
rwork[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 =
d_imag(&b[i__ + j * b_dim1]), abs(d__2)) + ((d__3
= dl[i__4].r, abs(d__3)) + (d__4 = d_imag(&dl[i__
- 1]), abs(d__4))) * ((d__5 = x[i__5].r, abs(d__5)
) + (d__6 = d_imag(&x[i__ - 1 + j * x_dim1]), abs(
d__6))) + ((d__7 = d__[i__6].r, abs(d__7)) + (
d__8 = d_imag(&d__[i__]), abs(d__8))) * ((d__9 =
x[i__7].r, abs(d__9)) + (d__10 = d_imag(&x[i__ +
j * x_dim1]), abs(d__10))) + ((d__11 = du[i__8].r,
abs(d__11)) + (d__12 = d_imag(&du[i__]), abs(
d__12))) * ((d__13 = x[i__9].r, abs(d__13)) + (
d__14 = d_imag(&x[i__ + 1 + j * x_dim1]), abs(
d__14)));
}
i__2 = *n + j * b_dim1;
i__3 = *n - 1;
i__4 = *n - 1 + j * x_dim1;
i__5 = *n;
i__6 = *n + j * x_dim1;
rwork[*n] = (d__1 = b[i__2].r, abs(d__1)) + (d__2 = d_imag(&b[
*n + j * b_dim1]), abs(d__2)) + ((d__3 = dl[i__3].r,
abs(d__3)) + (d__4 = d_imag(&dl[*n - 1]), abs(d__4)))
* ((d__5 = x[i__4].r, abs(d__5)) + (d__6 = d_imag(&x[*
n - 1 + j * x_dim1]), abs(d__6))) + ((d__7 = d__[i__5]
.r, abs(d__7)) + (d__8 = d_imag(&d__[*n]), abs(d__8)))
* ((d__9 = x[i__6].r, abs(d__9)) + (d__10 = d_imag(&
x[*n + j * x_dim1]), abs(d__10)));
}
} else {
if (*n == 1) {
i__2 = j * b_dim1 + 1;
i__3 = j * x_dim1 + 1;
rwork[1] = (d__1 = b[i__2].r, abs(d__1)) + (d__2 = d_imag(&b[
j * b_dim1 + 1]), abs(d__2)) + ((d__3 = d__[1].r, abs(
d__3)) + (d__4 = d_imag(&d__[1]), abs(d__4))) * ((
d__5 = x[i__3].r, abs(d__5)) + (d__6 = d_imag(&x[j *
x_dim1 + 1]), abs(d__6)));
} else {
i__2 = j * b_dim1 + 1;
i__3 = j * x_dim1 + 1;
i__4 = j * x_dim1 + 2;
rwork[1] = (d__1 = b[i__2].r, abs(d__1)) + (d__2 = d_imag(&b[
j * b_dim1 + 1]), abs(d__2)) + ((d__3 = d__[1].r, abs(
d__3)) + (d__4 = d_imag(&d__[1]), abs(d__4))) * ((
d__5 = x[i__3].r, abs(d__5)) + (d__6 = d_imag(&x[j *
x_dim1 + 1]), abs(d__6))) + ((d__7 = dl[1].r, abs(
d__7)) + (d__8 = d_imag(&dl[1]), abs(d__8))) * ((d__9
= x[i__4].r, abs(d__9)) + (d__10 = d_imag(&x[j *
x_dim1 + 2]), abs(d__10)));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
i__4 = i__ - 1;
i__5 = i__ - 1 + j * x_dim1;
i__6 = i__;
i__7 = i__ + j * x_dim1;
i__8 = i__;
i__9 = i__ + 1 + j * x_dim1;
rwork[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 =
d_imag(&b[i__ + j * b_dim1]), abs(d__2)) + ((d__3
= du[i__4].r, abs(d__3)) + (d__4 = d_imag(&du[i__
- 1]), abs(d__4))) * ((d__5 = x[i__5].r, abs(d__5)
) + (d__6 = d_imag(&x[i__ - 1 + j * x_dim1]), abs(
d__6))) + ((d__7 = d__[i__6].r, abs(d__7)) + (
d__8 = d_imag(&d__[i__]), abs(d__8))) * ((d__9 =
x[i__7].r, abs(d__9)) + (d__10 = d_imag(&x[i__ +
j * x_dim1]), abs(d__10))) + ((d__11 = dl[i__8].r,
abs(d__11)) + (d__12 = d_imag(&dl[i__]), abs(
d__12))) * ((d__13 = x[i__9].r, abs(d__13)) + (
d__14 = d_imag(&x[i__ + 1 + j * x_dim1]), abs(
d__14)));
}
i__2 = *n + j * b_dim1;
i__3 = *n - 1;
i__4 = *n - 1 + j * x_dim1;
i__5 = *n;
i__6 = *n + j * x_dim1;
rwork[*n] = (d__1 = b[i__2].r, abs(d__1)) + (d__2 = d_imag(&b[
*n + j * b_dim1]), abs(d__2)) + ((d__3 = du[i__3].r,
abs(d__3)) + (d__4 = d_imag(&du[*n - 1]), abs(d__4)))
* ((d__5 = x[i__4].r, abs(d__5)) + (d__6 = d_imag(&x[*
n - 1 + j * x_dim1]), abs(d__6))) + ((d__7 = d__[i__5]
.r, abs(d__7)) + (d__8 = d_imag(&d__[*n]), abs(d__8)))
* ((d__9 = x[i__6].r, abs(d__9)) + (d__10 = d_imag(&
x[*n + j * x_dim1]), abs(d__10)));
}
}
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2))) / rwork[i__];
s = max(d__3,d__4);
} else {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__]
+ safe1);
s = max(d__3,d__4);
}
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2. <= lstres && count <= 5) {
/* Update solution and try again. */
zgttrs_(trans, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &ipiv[
1], &work[1], n, info);
zaxpy_(n, &c_b26, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(op(A)))* */
/* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(op(A)) is the inverse of op(A) */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(op(A))*abs(X) + abs(B) is less than SAFE2. */
/* Use ZLACN2 to estimate the infinity-norm of the matrix */
/* inv(op(A)) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
;
} else {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
+ safe1;
}
}
kase = 0;
L70:
zlacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**H). */
zgttrs_(transt, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &
ipiv[1], &work[1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
}
zgttrs_(transn, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &
ipiv[1], &work[1], n, info);
}
goto L70;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = i__ + j * x_dim1;
d__3 = lstres, d__4 = (d__1 = x[i__3].r, abs(d__1)) + (d__2 =
d_imag(&x[i__ + j * x_dim1]), abs(d__2));
lstres = max(d__3,d__4);
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
}
return 0;
/* End of ZGTRFS */
} /* zgtrfs_ */
/* Subroutine */ int ztprfs_(char *uplo, char *trans, char *diag, integer *n,
integer *nrhs, doublecomplex *ap, doublecomplex *b, integer *ldb,
doublecomplex *x, integer *ldx, doublereal *ferr, doublereal *berr,
doublecomplex *work, doublereal *rwork, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3, d__4;
doublecomplex z__1;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer i__, j, k;
doublereal s;
integer kc;
doublereal xk;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */ int zcopy_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *), zaxpy_(integer *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, integer *), ztpmv_(
char *, char *, char *, integer *, doublecomplex *, doublecomplex
*, integer *), ztpsv_(char *, char *,
char *, integer *, doublecomplex *, doublecomplex *, integer *), zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
logical notran;
char transn[1], transt[1];
logical nounit;
doublereal lstres;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZTPRFS provides error bounds and backward error estimates for the */
/* solution to a system of linear equations with a triangular packed */
/* coefficient matrix. */
/* The solution matrix X must be computed by ZTPTRS or some other */
/* means before entering this routine. ZTPRFS does not do iterative */
/* refinement because doing so cannot improve the backward error. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': A is upper triangular; */
/* = 'L': A is lower triangular. */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate transpose) */
/* DIAG (input) CHARACTER*1 */
/* = 'N': A is non-unit triangular; */
/* = 'U': A is unit triangular. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2) */
/* The upper or lower triangular matrix A, packed columnwise in */
/* a linear array. The j-th column of A is stored in the array */
/* AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */
/* If DIAG = 'U', the diagonal elements of A are not referenced */
/* and are assumed to be 1. */
/* B (input) COMPLEX*16 array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input) COMPLEX*16 array, dimension (LDX,NRHS) */
/* The solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The estimated forward error bound for each solution vector */
/* X(j) (the j-th column of the solution matrix X). */
/* If XTRUE is the true solution corresponding to X(j), FERR(j) */
/* is an estimated upper bound for the magnitude of the largest */
/* element in (X(j) - XTRUE) divided by the magnitude of the */
/* largest element in X(j). The estimate is as reliable as */
/* the estimate for RCOND, and is almost always a slight */
/* overestimate of the true error. */
/* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
notran = lsame_(trans, "N");
nounit = lsame_(diag, "N");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "T") && !
lsame_(trans, "C")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*nrhs < 0) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -8;
} else if (*ldx < max(1,*n)) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTPRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.;
berr[j] = 0.;
/* L10: */
}
return 0;
}
if (notran) {
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
} else {
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
/* Compute residual R = B - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
zcopy_(n, &x[j * x_dim1 + 1], &c__1, &work[1], &c__1);
ztpmv_(uplo, trans, diag, n, &ap[1], &work[1], &c__1);
z__1.r = -1., z__1.i = -0.;
zaxpy_(n, &z__1, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 = d_imag(&b[
i__ + j * b_dim1]), abs(d__2));
/* L20: */
}
if (notran) {
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&
x[k + j * x_dim1]), abs(d__2));
i__3 = k;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
rwork[i__] += ((d__1 = ap[i__4].r, abs(d__1)) + (
d__2 = d_imag(&ap[kc + i__ - 1]), abs(
d__2))) * xk;
/* L30: */
}
kc += k;
/* L40: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&
x[k + j * x_dim1]), abs(d__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
rwork[i__] += ((d__1 = ap[i__4].r, abs(d__1)) + (
d__2 = d_imag(&ap[kc + i__ - 1]), abs(
d__2))) * xk;
/* L50: */
}
rwork[k] += xk;
kc += k;
/* L60: */
}
}
} else {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&
x[k + j * x_dim1]), abs(d__2));
i__3 = *n;
for (i__ = k; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
rwork[i__] += ((d__1 = ap[i__4].r, abs(d__1)) + (
d__2 = d_imag(&ap[kc + i__ - k]), abs(
d__2))) * xk;
/* L70: */
}
kc = kc + *n - k + 1;
/* L80: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&
x[k + j * x_dim1]), abs(d__2));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
rwork[i__] += ((d__1 = ap[i__4].r, abs(d__1)) + (
d__2 = d_imag(&ap[kc + i__ - k]), abs(
d__2))) * xk;
/* L90: */
}
rwork[k] += xk;
kc = kc + *n - k + 1;
/* L100: */
}
}
}
} else {
/* Compute abs(A**H)*abs(X) + abs(B). */
if (upper) {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
i__3 = k;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 =
d_imag(&ap[kc + i__ - 1]), abs(d__2))) * (
(d__3 = x[i__5].r, abs(d__3)) + (d__4 =
d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L110: */
}
rwork[k] += s;
kc += k;
/* L120: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
s = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[
k + j * x_dim1]), abs(d__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 =
d_imag(&ap[kc + i__ - 1]), abs(d__2))) * (
(d__3 = x[i__5].r, abs(d__3)) + (d__4 =
d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L130: */
}
rwork[k] += s;
kc += k;
/* L140: */
}
}
} else {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
i__3 = *n;
for (i__ = k; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 =
d_imag(&ap[kc + i__ - k]), abs(d__2))) * (
(d__3 = x[i__5].r, abs(d__3)) + (d__4 =
d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L150: */
}
rwork[k] += s;
kc = kc + *n - k + 1;
/* L160: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
s = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[
k + j * x_dim1]), abs(d__2));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 =
d_imag(&ap[kc + i__ - k]), abs(d__2))) * (
(d__3 = x[i__5].r, abs(d__3)) + (d__4 =
d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L170: */
}
rwork[k] += s;
kc = kc + *n - k + 1;
/* L180: */
}
}
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2))) / rwork[i__];
s = max(d__3,d__4);
} else {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__]
+ safe1);
s = max(d__3,d__4);
}
/* L190: */
}
berr[j] = s;
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(op(A)))* */
/* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(op(A)) is the inverse of op(A) */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(op(A))*abs(X) + abs(B) is less than SAFE2. */
/* Use ZLACN2 to estimate the infinity-norm of the matrix */
/* inv(op(A)) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
;
} else {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
+ safe1;
}
/* L200: */
}
kase = 0;
L210:
zlacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**H). */
ztpsv_(uplo, transt, diag, n, &ap[1], &work[1], &c__1);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
/* L220: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
/* L230: */
}
ztpsv_(uplo, transn, diag, n, &ap[1], &work[1], &c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = i__ + j * x_dim1;
d__3 = lstres, d__4 = (d__1 = x[i__3].r, abs(d__1)) + (d__2 =
d_imag(&x[i__ + j * x_dim1]), abs(d__2));
lstres = max(d__3,d__4);
/* L240: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of ZTPRFS */
} /* ztprfs_ */
/* Subroutine */
int zhprfs_(char *uplo, integer *n, integer *nrhs, doublecomplex *ap, doublecomplex *afp, integer *ipiv, doublecomplex * b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *ferr, doublereal *berr, doublecomplex *work, doublereal *rwork, integer * info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3, d__4;
doublecomplex z__1;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer i__, j, k;
doublereal s;
integer ik, kk;
doublereal xk;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3], count;
logical upper;
extern /* Subroutine */
int zcopy_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), zhpmv_(char *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), zaxpy_( integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *), zlacn2_(integer *, doublecomplex *, doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */
int xerbla_(char *, integer *);
doublereal lstres;
extern /* Subroutine */
int zhptrs_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *);
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
--afp;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*nrhs < 0)
{
*info = -3;
}
else if (*ldb < max(1,*n))
{
*info = -8;
}
else if (*ldx < max(1,*n))
{
*info = -10;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZHPRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0)
{
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
ferr[j] = 0.;
berr[j] = 0.;
/* L10: */
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
count = 1;
lstres = 3.;
L20: /* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - A * X */
zcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
z__1.r = -1.;
z__1.i = -0.; // , expr subst
zhpmv_(uplo, n, &z__1, &ap[1], &x[j * x_dim1 + 1], &c__1, &c_b1, & work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( f2c_abs(R(i)) / ( f2c_abs(A)*f2c_abs(X) + f2c_abs(B) )(i) ) */
/* where f2c_abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__ + j * b_dim1;
rwork[i__] = (d__1 = b[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&b[ i__ + j * b_dim1]), f2c_abs(d__2));
/* L30: */
}
/* Compute f2c_abs(A)*f2c_abs(X) + f2c_abs(B). */
kk = 1;
if (upper)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.;
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&x[k + j * x_dim1]), f2c_abs(d__2));
ik = kk;
i__3 = k - 1;
for (i__ = 1;
i__ <= i__3;
++i__)
{
i__4 = ik;
rwork[i__] += ((d__1 = ap[i__4].r, f2c_abs(d__1)) + (d__2 = d_imag(&ap[ik]), f2c_abs(d__2))) * xk;
i__4 = ik;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, f2c_abs(d__1)) + (d__2 = d_imag(&ap[ ik]), f2c_abs(d__2))) * ((d__3 = x[i__5].r, f2c_abs(d__3)) + (d__4 = d_imag(&x[i__ + j * x_dim1]), f2c_abs(d__4) ));
++ik;
/* L40: */
}
i__3 = kk + k - 1;
rwork[k] = rwork[k] + (d__1 = ap[i__3].r, f2c_abs(d__1)) * xk + s;
kk += k;
/* L50: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.;
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&x[k + j * x_dim1]), f2c_abs(d__2));
i__3 = kk;
rwork[k] += (d__1 = ap[i__3].r, f2c_abs(d__1)) * xk;
ik = kk + 1;
i__3 = *n;
for (i__ = k + 1;
i__ <= i__3;
++i__)
{
i__4 = ik;
rwork[i__] += ((d__1 = ap[i__4].r, f2c_abs(d__1)) + (d__2 = d_imag(&ap[ik]), f2c_abs(d__2))) * xk;
i__4 = ik;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, f2c_abs(d__1)) + (d__2 = d_imag(&ap[ ik]), f2c_abs(d__2))) * ((d__3 = x[i__5].r, f2c_abs(d__3)) + (d__4 = d_imag(&x[i__ + j * x_dim1]), f2c_abs(d__4) ));
++ik;
/* L60: */
}
rwork[k] += s;
kk += *n - k + 1;
/* L70: */
}
}
s = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
/* Computing MAX */
i__3 = i__;
d__3 = s;
d__4 = ((d__1 = work[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&work[i__]), f2c_abs(d__2))) / rwork[i__]; // , expr subst
s = max(d__3,d__4);
}
else
{
/* Computing MAX */
i__3 = i__;
d__3 = s;
d__4 = ((d__1 = work[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&work[i__]), f2c_abs(d__2)) + safe1) / (rwork[i__] + safe1); // , expr subst
s = max(d__3,d__4);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2. <= lstres && count <= 5)
{
/* Update solution and try again. */
zhptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
zaxpy_(n, &c_b1, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( f2c_abs(inv(A))* */
/* ( f2c_abs(R) + NZ*EPS*( f2c_abs(A)*f2c_abs(X)+f2c_abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(A) is the inverse of A */
/* f2c_abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of f2c_abs(R)+NZ*EPS*(f2c_abs(A)*f2c_abs(X)+f2c_abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* f2c_abs(A)*f2c_abs(X) + f2c_abs(B) is less than SAFE2. */
/* Use ZLACN2 to estimate the infinity-norm of the matrix */
/* inv(A) * diag(W), */
/* where W = f2c_abs(R) + NZ*EPS*( f2c_abs(A)*f2c_abs(X)+f2c_abs(B) ))) */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&work[i__]), f2c_abs(d__2)) + nz * eps * rwork[i__] ;
}
else
{
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&work[i__]), f2c_abs(d__2)) + nz * eps * rwork[i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
zlacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Multiply by diag(W)*inv(A**H). */
zhptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r;
z__1.i = rwork[i__4] * work[i__5].i; // , expr subst
work[i__3].r = z__1.r;
work[i__3].i = z__1.i; // , expr subst
/* L110: */
}
}
else if (kase == 2)
{
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r;
z__1.i = rwork[i__4] * work[i__5].i; // , expr subst
work[i__3].r = z__1.r;
work[i__3].i = z__1.i; // , expr subst
/* L120: */
}
zhptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
/* Computing MAX */
i__3 = i__ + j * x_dim1;
d__3 = lstres;
d__4 = (d__1 = x[i__3].r, f2c_abs(d__1)) + (d__2 = d_imag(&x[i__ + j * x_dim1]), f2c_abs(d__2)); // , expr subst
lstres = max(d__3,d__4);
/* L130: */
}
if (lstres != 0.)
{
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of ZHPRFS */
}
/* Subroutine */
int ztrsen_(char *job, char *compq, logical *select, integer *n, doublecomplex *t, integer *ldt, doublecomplex *q, integer *ldq, doublecomplex *w, integer *m, doublereal *s, doublereal *sep, doublecomplex *work, integer *lwork, integer *info)
{
/* System generated locals */
integer q_dim1, q_offset, t_dim1, t_offset, i__1, i__2, i__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer k, n1, n2, nn, ks;
doublereal est;
integer kase, ierr;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3], lwmin;
logical wantq, wants;
doublereal rnorm, rwork[1];
extern /* Subroutine */
int zlacn2_(integer *, doublecomplex *, doublecomplex *, doublereal *, integer *, integer *), xerbla_( char *, integer *);
extern doublereal zlange_(char *, integer *, integer *, doublecomplex *, integer *, doublereal *);
logical wantbh;
extern /* Subroutine */
int zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *);
logical wantsp;
extern /* Subroutine */
int ztrexc_(char *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *, integer *, integer *);
logical lquery;
extern /* Subroutine */
int ztrsyl_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, integer *);
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters. */
/* Parameter adjustments */
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--w;
--work;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
wantq = lsame_(compq, "V");
/* Set M to the number of selected eigenvalues. */
*m = 0;
i__1 = *n;
for (k = 1;
k <= i__1;
++k)
{
if (select[k])
{
++(*m);
}
/* L10: */
}
n1 = *m;
n2 = *n - *m;
nn = n1 * n2;
*info = 0;
lquery = *lwork == -1;
if (wantsp)
{
/* Computing MAX */
i__1 = 1;
i__2 = nn << 1; // , expr subst
lwmin = max(i__1,i__2);
}
else if (lsame_(job, "N"))
{
lwmin = 1;
}
else if (lsame_(job, "E"))
{
lwmin = max(1,nn);
}
if (! lsame_(job, "N") && ! wants && ! wantsp)
{
*info = -1;
}
else if (! lsame_(compq, "N") && ! wantq)
{
*info = -2;
}
else if (*n < 0)
{
*info = -4;
}
else if (*ldt < max(1,*n))
{
*info = -6;
}
else if (*ldq < 1 || wantq && *ldq < *n)
{
*info = -8;
}
else if (*lwork < lwmin && ! lquery)
{
*info = -14;
}
if (*info == 0)
{
work[1].r = (doublereal) lwmin;
work[1].i = 0.; // , expr subst
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZTRSEN", &i__1);
return 0;
}
else if (lquery)
{
return 0;
}
/* Quick return if possible */
if (*m == *n || *m == 0)
{
if (wants)
{
*s = 1.;
}
if (wantsp)
{
*sep = zlange_("1", n, n, &t[t_offset], ldt, rwork);
}
goto L40;
}
/* Collect the selected eigenvalues at the top left corner of T. */
ks = 0;
i__1 = *n;
for (k = 1;
k <= i__1;
++k)
{
if (select[k])
{
++ks;
/* Swap the K-th eigenvalue to position KS. */
if (k != ks)
{
ztrexc_(compq, n, &t[t_offset], ldt, &q[q_offset], ldq, &k, & ks, &ierr);
}
}
/* L20: */
}
if (wants)
{
/* Solve the Sylvester equation for R: */
/* T11*R - R*T22 = scale*T12 */
zlacpy_("F", &n1, &n2, &t[(n1 + 1) * t_dim1 + 1], ldt, &work[1], &n1);
ztrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 + 1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &ierr);
/* Estimate the reciprocal of the condition number of the cluster */
/* of eigenvalues. */
rnorm = zlange_("F", &n1, &n2, &work[1], &n1, rwork);
if (rnorm == 0.)
{
*s = 1.;
}
else
{
*s = scale / (sqrt(scale * scale / rnorm + rnorm) * sqrt(rnorm));
}
}
if (wantsp)
{
/* Estimate sep(T11,T22). */
est = 0.;
kase = 0;
L30:
zlacn2_(&nn, &work[nn + 1], &work[1], &est, &kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Solve T11*R - R*T22 = scale*X. */
ztrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 + 1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, & ierr);
}
else
{
/* Solve T11**H*R - R*T22**H = scale*X. */
ztrsyl_("C", "C", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 + 1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, & ierr);
}
goto L30;
}
*sep = scale / est;
}
L40: /* Copy reordered eigenvalues to W. */
i__1 = *n;
for (k = 1;
k <= i__1;
++k)
{
i__2 = k;
i__3 = k + k * t_dim1;
w[i__2].r = t[i__3].r;
w[i__2].i = t[i__3].i; // , expr subst
/* L50: */
}
work[1].r = (doublereal) lwmin;
work[1].i = 0.; // , expr subst
return 0;
/* End of ZTRSEN */
}
/* Subroutine */
int ztbrfs_(char *uplo, char *trans, char *diag, integer *n, integer *kd, integer *nrhs, doublecomplex *ab, integer *ldab, doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *ferr, doublereal *berr, doublecomplex *work, doublereal * rwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3, d__4;
doublecomplex z__1;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer i__, j, k;
doublereal s, xk;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */
int ztbmv_(char *, char *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *), zcopy_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), ztbsv_(char *, char *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *), zaxpy_( integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *), zlacn2_(integer *, doublecomplex *, doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */
int xerbla_(char *, integer *);
logical notran;
char transn[1], transt[1];
logical nounit;
doublereal lstres;
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
notran = lsame_(trans, "N");
nounit = lsame_(diag, "N");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (! notran && ! lsame_(trans, "T") && ! lsame_(trans, "C"))
{
*info = -2;
}
else if (! nounit && ! lsame_(diag, "U"))
{
*info = -3;
}
else if (*n < 0)
{
*info = -4;
}
else if (*kd < 0)
{
*info = -5;
}
else if (*nrhs < 0)
{
*info = -6;
}
else if (*ldab < *kd + 1)
{
*info = -8;
}
else if (*ldb < max(1,*n))
{
*info = -10;
}
else if (*ldx < max(1,*n))
{
*info = -12;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZTBRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0)
{
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
ferr[j] = 0.;
berr[j] = 0.;
/* L10: */
}
return 0;
}
if (notran)
{
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
}
else
{
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *kd + 2;
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
/* Compute residual R = B - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
zcopy_(n, &x[j * x_dim1 + 1], &c__1, &work[1], &c__1);
ztbmv_(uplo, trans, diag, n, kd, &ab[ab_offset], ldab, &work[1], & c__1);
z__1.r = -1.;
z__1.i = -0.; // , expr subst
zaxpy_(n, &z__1, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__ + j * b_dim1;
rwork[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 = d_imag(&b[ i__ + j * b_dim1]), abs(d__2));
/* L20: */
}
if (notran)
{
/* Compute abs(A)*abs(X) + abs(B). */
if (upper)
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(& x[k + j * x_dim1]), abs(d__2));
/* Computing MAX */
i__3 = 1;
i__4 = k - *kd; // , expr subst
i__5 = k;
for (i__ = max(i__3,i__4);
i__ <= i__5;
++i__)
{
i__3 = *kd + 1 + i__ - k + k * ab_dim1;
rwork[i__] += ((d__1 = ab[i__3].r, abs(d__1)) + ( d__2 = d_imag(&ab[*kd + 1 + i__ - k + k * ab_dim1]), abs(d__2))) * xk;
/* L30: */
}
/* L40: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__5 = k + j * x_dim1;
xk = (d__1 = x[i__5].r, abs(d__1)) + (d__2 = d_imag(& x[k + j * x_dim1]), abs(d__2));
/* Computing MAX */
i__5 = 1;
i__3 = k - *kd; // , expr subst
i__4 = k - 1;
for (i__ = max(i__5,i__3);
i__ <= i__4;
++i__)
{
i__5 = *kd + 1 + i__ - k + k * ab_dim1;
rwork[i__] += ((d__1 = ab[i__5].r, abs(d__1)) + ( d__2 = d_imag(&ab[*kd + 1 + i__ - k + k * ab_dim1]), abs(d__2))) * xk;
/* L50: */
}
rwork[k] += xk;
/* L60: */
}
}
}
else
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__4 = k + j * x_dim1;
xk = (d__1 = x[i__4].r, abs(d__1)) + (d__2 = d_imag(& x[k + j * x_dim1]), abs(d__2));
/* Computing MIN */
i__5 = *n;
i__3 = k + *kd; // , expr subst
i__4 = min(i__5,i__3);
for (i__ = k;
i__ <= i__4;
++i__)
{
i__5 = i__ + 1 - k + k * ab_dim1;
rwork[i__] += ((d__1 = ab[i__5].r, abs(d__1)) + ( d__2 = d_imag(&ab[i__ + 1 - k + k * ab_dim1]), abs(d__2))) * xk;
/* L70: */
}
/* L80: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__4 = k + j * x_dim1;
xk = (d__1 = x[i__4].r, abs(d__1)) + (d__2 = d_imag(& x[k + j * x_dim1]), abs(d__2));
/* Computing MIN */
i__5 = *n;
i__3 = k + *kd; // , expr subst
i__4 = min(i__5,i__3);
for (i__ = k + 1;
i__ <= i__4;
++i__)
{
i__5 = i__ + 1 - k + k * ab_dim1;
rwork[i__] += ((d__1 = ab[i__5].r, abs(d__1)) + ( d__2 = d_imag(&ab[i__ + 1 - k + k * ab_dim1]), abs(d__2))) * xk;
/* L90: */
}
rwork[k] += xk;
/* L100: */
}
}
}
}
else
{
/* Compute abs(A**H)*abs(X) + abs(B). */
if (upper)
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.;
/* Computing MAX */
i__4 = 1;
i__5 = k - *kd; // , expr subst
i__3 = k;
for (i__ = max(i__4,i__5);
i__ <= i__3;
++i__)
{
i__4 = *kd + 1 + i__ - k + k * ab_dim1;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ab[i__4].r, abs(d__1)) + (d__2 = d_imag(&ab[*kd + 1 + i__ - k + k * ab_dim1]), abs(d__2))) * ((d__3 = x[i__5] .r, abs(d__3)) + (d__4 = d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L110: */
}
rwork[k] += s;
/* L120: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
s = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[ k + j * x_dim1]), abs(d__2));
/* Computing MAX */
i__3 = 1;
i__4 = k - *kd; // , expr subst
i__5 = k - 1;
for (i__ = max(i__3,i__4);
i__ <= i__5;
++i__)
{
i__3 = *kd + 1 + i__ - k + k * ab_dim1;
i__4 = i__ + j * x_dim1;
s += ((d__1 = ab[i__3].r, abs(d__1)) + (d__2 = d_imag(&ab[*kd + 1 + i__ - k + k * ab_dim1]), abs(d__2))) * ((d__3 = x[i__4] .r, abs(d__3)) + (d__4 = d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L130: */
}
rwork[k] += s;
/* L140: */
}
}
}
else
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.;
/* Computing MIN */
i__3 = *n;
i__4 = k + *kd; // , expr subst
i__5 = min(i__3,i__4);
for (i__ = k;
i__ <= i__5;
++i__)
{
i__3 = i__ + 1 - k + k * ab_dim1;
i__4 = i__ + j * x_dim1;
s += ((d__1 = ab[i__3].r, abs(d__1)) + (d__2 = d_imag(&ab[i__ + 1 - k + k * ab_dim1]), abs(d__2))) * ((d__3 = x[i__4].r, abs( d__3)) + (d__4 = d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L150: */
}
rwork[k] += s;
/* L160: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__5 = k + j * x_dim1;
s = (d__1 = x[i__5].r, abs(d__1)) + (d__2 = d_imag(&x[ k + j * x_dim1]), abs(d__2));
/* Computing MIN */
i__3 = *n;
i__4 = k + *kd; // , expr subst
i__5 = min(i__3,i__4);
for (i__ = k + 1;
i__ <= i__5;
++i__)
{
i__3 = i__ + 1 - k + k * ab_dim1;
i__4 = i__ + j * x_dim1;
s += ((d__1 = ab[i__3].r, abs(d__1)) + (d__2 = d_imag(&ab[i__ + 1 - k + k * ab_dim1]), abs(d__2))) * ((d__3 = x[i__4].r, abs( d__3)) + (d__4 = d_imag(&x[i__ + j * x_dim1]), abs(d__4)));
/* L170: */
}
rwork[k] += s;
/* L180: */
}
}
}
}
s = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
/* Computing MAX */
i__5 = i__;
d__3 = s;
d__4 = ((d__1 = work[i__5].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2))) / rwork[i__]; // , expr subst
s = max(d__3,d__4);
}
else
{
/* Computing MAX */
i__5 = i__;
d__3 = s;
d__4 = ((d__1 = work[i__5].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__] + safe1); // , expr subst
s = max(d__3,d__4);
}
/* L190: */
}
berr[j] = s;
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(op(A)))* */
/* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(op(A)) is the inverse of op(A) */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(op(A))*abs(X) + abs(B) is less than SAFE2. */
/* Use ZLACN2 to estimate the infinity-norm of the matrix */
/* inv(op(A)) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
i__5 = i__;
rwork[i__] = (d__1 = work[i__5].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__] ;
}
else
{
i__5 = i__;
rwork[i__] = (d__1 = work[i__5].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__] + safe1;
}
/* L200: */
}
kase = 0;
L210:
zlacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Multiply by diag(W)*inv(op(A)**H). */
ztbsv_(uplo, transt, diag, n, kd, &ab[ab_offset], ldab, &work[ 1], &c__1);
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__5 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__3] * work[i__4].r;
z__1.i = rwork[i__3] * work[i__4].i; // , expr subst
work[i__5].r = z__1.r;
work[i__5].i = z__1.i; // , expr subst
/* L220: */
}
}
else
{
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__5 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__3] * work[i__4].r;
z__1.i = rwork[i__3] * work[i__4].i; // , expr subst
work[i__5].r = z__1.r;
work[i__5].i = z__1.i; // , expr subst
/* L230: */
}
ztbsv_(uplo, transn, diag, n, kd, &ab[ab_offset], ldab, &work[ 1], &c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
/* Computing MAX */
i__5 = i__ + j * x_dim1;
d__3 = lstres;
d__4 = (d__1 = x[i__5].r, abs(d__1)) + (d__2 = d_imag(&x[i__ + j * x_dim1]), abs(d__2)); // , expr subst
lstres = max(d__3,d__4);
/* L240: */
}
if (lstres != 0.)
{
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of ZTBRFS */
}
/* Subroutine */ int zsprfs_(char *uplo, integer *n, integer *nrhs,
doublecomplex *ap, doublecomplex *afp, integer *ipiv, doublecomplex *
b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *ferr,
doublereal *berr, doublecomplex *work, doublereal *rwork, integer *
info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3, d__4;
doublecomplex z__1;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer i__, j, k;
doublereal s;
integer ik, kk;
doublereal xk;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3], count;
logical upper;
extern /* Subroutine */ int zcopy_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *), zaxpy_(integer *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, integer *), zspmv_(
char *, integer *, doublecomplex *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *), zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal lstres;
extern /* Subroutine */ int zsptrs_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZSPRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is symmetric indefinite */
/* and packed, and provides error bounds and backward error estimates */
/* for the solution. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2) */
/* The upper or lower triangle of the symmetric matrix A, packed */
/* columnwise in a linear array. The j-th column of A is stored */
/* in the array AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. */
/* AFP (input) COMPLEX*16 array, dimension (N*(N+1)/2) */
/* The factored form of the matrix A. AFP contains the block */
/* diagonal matrix D and the multipliers used to obtain the */
/* factor U or L from the factorization A = U*D*U**T or */
/* A = L*D*L**T as computed by ZSPTRF, stored as a packed */
/* triangular matrix. */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by ZSPTRF. */
/* B (input) COMPLEX*16 array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by ZSPTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The estimated forward error bound for each solution vector */
/* X(j) (the j-th column of the solution matrix X). */
/* If XTRUE is the true solution corresponding to X(j), FERR(j) */
/* is an estimated upper bound for the magnitude of the largest */
/* element in (X(j) - XTRUE) divided by the magnitude of the */
/* largest element in X(j). The estimate is as reliable as */
/* the estimate for RCOND, and is almost always a slight */
/* overestimate of the true error. */
/* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Internal Parameters */
/* =================== */
/* ITMAX is the maximum number of steps of iterative refinement. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
--afp;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*ldb < max(1,*n)) {
*info = -8;
} else if (*ldx < max(1,*n)) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZSPRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.;
berr[j] = 0.;
/* L10: */
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
count = 1;
lstres = 3.;
L20:
/* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - A * X */
zcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
z__1.r = -1., z__1.i = -0.;
zspmv_(uplo, n, &z__1, &ap[1], &x[j * x_dim1 + 1], &c__1, &c_b1, &
work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 = d_imag(&b[
i__ + j * b_dim1]), abs(d__2));
/* L30: */
}
/* Compute abs(A)*abs(X) + abs(B). */
kk = 1;
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[k + j *
x_dim1]), abs(d__2));
ik = kk;
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = ik;
rwork[i__] += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 =
d_imag(&ap[ik]), abs(d__2))) * xk;
i__4 = ik;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 = d_imag(&ap[
ik]), abs(d__2))) * ((d__3 = x[i__5].r, abs(d__3))
+ (d__4 = d_imag(&x[i__ + j * x_dim1]), abs(d__4)
));
++ik;
/* L40: */
}
i__3 = kk + k - 1;
rwork[k] = rwork[k] + ((d__1 = ap[i__3].r, abs(d__1)) + (d__2
= d_imag(&ap[kk + k - 1]), abs(d__2))) * xk + s;
kk += k;
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[k + j *
x_dim1]), abs(d__2));
i__3 = kk;
rwork[k] += ((d__1 = ap[i__3].r, abs(d__1)) + (d__2 = d_imag(&
ap[kk]), abs(d__2))) * xk;
ik = kk + 1;
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = ik;
rwork[i__] += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 =
d_imag(&ap[ik]), abs(d__2))) * xk;
i__4 = ik;
i__5 = i__ + j * x_dim1;
s += ((d__1 = ap[i__4].r, abs(d__1)) + (d__2 = d_imag(&ap[
ik]), abs(d__2))) * ((d__3 = x[i__5].r, abs(d__3))
+ (d__4 = d_imag(&x[i__ + j * x_dim1]), abs(d__4)
));
++ik;
/* L60: */
}
rwork[k] += s;
kk += *n - k + 1;
/* L70: */
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2))) / rwork[i__];
s = max(d__3,d__4);
} else {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__]
+ safe1);
s = max(d__3,d__4);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2. <= lstres && count <= 5) {
/* Update solution and try again. */
zsptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
zaxpy_(n, &c_b1, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(A))* */
/* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(A) is the inverse of A */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(A)*abs(X) + abs(B) is less than SAFE2. */
/* Use ZLACN2 to estimate the infinity-norm of the matrix */
/* inv(A) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
;
} else {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
+ safe1;
}
/* L90: */
}
kase = 0;
L100:
zlacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
zsptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
/* L110: */
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
/* L120: */
}
zsptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = i__ + j * x_dim1;
d__3 = lstres, d__4 = (d__1 = x[i__3].r, abs(d__1)) + (d__2 =
d_imag(&x[i__ + j * x_dim1]), abs(d__2));
lstres = max(d__3,d__4);
/* L130: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of ZSPRFS */
} /* zsprfs_ */
/* Subroutine */
int ztrsna_(char *job, char *howmny, logical *select, integer *n, doublecomplex *t, integer *ldt, doublecomplex *vl, integer *ldvl, doublecomplex *vr, integer *ldvr, doublereal *s, doublereal *sep, integer *mm, integer *m, doublecomplex *work, integer *ldwork, doublereal *rwork, integer *info)
{
/* System generated locals */
integer t_dim1, t_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, work_dim1, work_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2;
doublecomplex z__1;
/* Builtin functions */
double z_abs(doublecomplex *), d_imag(doublecomplex *);
/* Local variables */
integer i__, j, k, ks, ix;
doublereal eps, est;
integer kase, ierr;
doublecomplex prod;
doublereal lnrm, rnrm, scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Double Complex */
VOID zdotc_f2c_(doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *);
doublecomplex dummy[1];
logical wants;
doublereal xnorm;
extern /* Subroutine */
int zlacn2_(integer *, doublecomplex *, doublecomplex *, doublereal *, integer *, integer *), dlabad_( doublereal *, doublereal *);
extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_( char *);
extern /* Subroutine */
int xerbla_(char *, integer *);
doublereal bignum;
logical wantbh;
extern integer izamax_(integer *, doublecomplex *, integer *);
logical somcon;
extern /* Subroutine */
int zdrscl_(integer *, doublereal *, doublecomplex *, integer *);
char normin[1];
extern /* Subroutine */
int zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *);
doublereal smlnum;
logical wantsp;
extern /* Subroutine */
int zlatrs_(char *, char *, char *, char *, integer *, doublecomplex *, integer *, doublecomplex *, doublereal *, doublereal *, integer *), ztrexc_(char *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *, integer *, integer *);
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/* Parameter adjustments */
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--s;
--sep;
work_dim1 = *ldwork;
work_offset = 1 + work_dim1;
work -= work_offset;
--rwork;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
somcon = lsame_(howmny, "S");
/* Set M to the number of eigenpairs for which condition numbers are */
/* to be computed. */
if (somcon)
{
*m = 0;
i__1 = *n;
for (j = 1;
j <= i__1;
++j)
{
if (select[j])
{
++(*m);
}
/* L10: */
}
}
else
{
*m = *n;
}
*info = 0;
if (! wants && ! wantsp)
{
*info = -1;
}
else if (! lsame_(howmny, "A") && ! somcon)
{
*info = -2;
}
else if (*n < 0)
{
*info = -4;
}
else if (*ldt < max(1,*n))
{
*info = -6;
}
else if (*ldvl < 1 || wants && *ldvl < *n)
{
*info = -8;
}
else if (*ldvr < 1 || wants && *ldvr < *n)
{
*info = -10;
}
else if (*mm < *m)
{
*info = -13;
}
else if (*ldwork < 1 || wantsp && *ldwork < *n)
{
*info = -16;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZTRSNA", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0)
{
return 0;
}
if (*n == 1)
{
if (somcon)
{
if (! select[1])
{
return 0;
}
}
if (wants)
{
s[1] = 1.;
}
if (wantsp)
{
sep[1] = z_abs(&t[t_dim1 + 1]);
}
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S") / eps;
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
ks = 1;
i__1 = *n;
for (k = 1;
k <= i__1;
++k)
{
if (somcon)
{
if (! select[k])
{
goto L50;
}
}
if (wants)
{
/* Compute the reciprocal condition number of the k-th */
/* eigenvalue. */
zdotc_f2c_(&z__1, n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks * vl_dim1 + 1], &c__1);
prod.r = z__1.r;
prod.i = z__1.i; // , expr subst
rnrm = dznrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
lnrm = dznrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
s[ks] = z_abs(&prod) / (rnrm * lnrm);
}
if (wantsp)
{
/* Estimate the reciprocal condition number of the k-th */
/* eigenvector. */
/* Copy the matrix T to the array WORK and swap the k-th */
/* diagonal element to the (1,1) position. */
zlacpy_("Full", n, n, &t[t_offset], ldt, &work[work_offset], ldwork);
ztrexc_("No Q", n, &work[work_offset], ldwork, dummy, &c__1, &k, & c__1, &ierr);
/* Form C = T22 - lambda*I in WORK(2:N,2:N). */
i__2 = *n;
for (i__ = 2;
i__ <= i__2;
++i__)
{
i__3 = i__ + i__ * work_dim1;
i__4 = i__ + i__ * work_dim1;
i__5 = work_dim1 + 1;
z__1.r = work[i__4].r - work[i__5].r;
z__1.i = work[i__4].i - work[i__5].i; // , expr subst
work[i__3].r = z__1.r;
work[i__3].i = z__1.i; // , expr subst
/* L20: */
}
/* Estimate a lower bound for the 1-norm of inv(C**H). The 1st */
/* and (N+1)th columns of WORK are used to store work vectors. */
sep[ks] = 0.;
est = 0.;
kase = 0;
*(unsigned char *)normin = 'N';
L30:
i__2 = *n - 1;
zlacn2_(&i__2, &work[(*n + 1) * work_dim1 + 1], &work[work_offset] , &est, &kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Solve C**H*x = scale*b */
i__2 = *n - 1;
zlatrs_("Upper", "Conjugate transpose", "Nonunit", normin, &i__2, &work[(work_dim1 << 1) + 2], ldwork, & work[work_offset], &scale, &rwork[1], &ierr);
}
else
{
/* Solve C*x = scale*b */
i__2 = *n - 1;
zlatrs_("Upper", "No transpose", "Nonunit", normin, &i__2, &work[(work_dim1 << 1) + 2], ldwork, &work[ work_offset], &scale, &rwork[1], &ierr);
}
*(unsigned char *)normin = 'Y';
if (scale != 1.)
{
/* Multiply by 1/SCALE if doing so will not cause */
/* overflow. */
i__2 = *n - 1;
ix = izamax_(&i__2, &work[work_offset], &c__1);
i__2 = ix + work_dim1;
xnorm = (d__1 = work[i__2].r, f2c_abs(d__1)) + (d__2 = d_imag( &work[ix + work_dim1]), f2c_abs(d__2));
if (scale < xnorm * smlnum || scale == 0.)
{
goto L40;
}
zdrscl_(n, &scale, &work[work_offset], &c__1);
}
goto L30;
}
sep[ks] = 1. / max(est,smlnum);
}
L40:
++ks;
L50:
;
}
return 0;
/* End of ZTRSNA */
}
/* Subroutine */
int zgecon_(char *norm, integer *n, doublecomplex *a, integer *lda, doublereal *anorm, doublereal *rcond, doublecomplex * work, doublereal *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
doublereal d__1, d__2;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
doublereal sl;
integer ix;
doublereal su;
integer kase, kase1;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */
int zlacn2_(integer *, doublecomplex *, doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */
int xerbla_(char *, integer *);
doublereal ainvnm;
extern integer izamax_(integer *, doublecomplex *, integer *);
logical onenrm;
extern /* Subroutine */
int zdrscl_(integer *, doublereal *, doublecomplex *, integer *);
char normin[1];
doublereal smlnum;
extern /* Subroutine */
int zlatrs_(char *, char *, char *, char *, integer *, doublecomplex *, integer *, doublecomplex *, doublereal *, doublereal *, integer *);
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
if (! onenrm && ! lsame_(norm, "I"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*n))
{
*info = -4;
}
else if (*anorm < 0.)
{
*info = -5;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZGECON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0)
{
*rcond = 1.;
return 0;
}
else if (*anorm == 0.)
{
return 0;
}
smlnum = dlamch_("Safe minimum");
/* Estimate the norm of inv(A). */
ainvnm = 0.;
*(unsigned char *)normin = 'N';
if (onenrm)
{
kase1 = 1;
}
else
{
kase1 = 2;
}
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (kase == kase1)
{
/* Multiply by inv(L). */
zlatrs_("Lower", "No transpose", "Unit", normin, n, &a[a_offset], lda, &work[1], &sl, &rwork[1], info);
/* Multiply by inv(U). */
zlatrs_("Upper", "No transpose", "Non-unit", normin, n, &a[ a_offset], lda, &work[1], &su, &rwork[*n + 1], info);
}
else
{
/* Multiply by inv(U**H). */
zlatrs_("Upper", "Conjugate transpose", "Non-unit", normin, n, &a[ a_offset], lda, &work[1], &su, &rwork[*n + 1], info);
/* Multiply by inv(L**H). */
zlatrs_("Lower", "Conjugate transpose", "Unit", normin, n, &a[ a_offset], lda, &work[1], &sl, &rwork[1], info);
}
/* Divide X by 1/(SL*SU) if doing so will not cause overflow. */
scale = sl * su;
*(unsigned char *)normin = 'Y';
if (scale != 1.)
{
ix = izamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((d__1 = work[i__1].r, f2c_abs(d__1)) + (d__2 = d_imag(& work[ix]), f2c_abs(d__2))) * smlnum || scale == 0.)
{
goto L20;
}
zdrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.)
{
*rcond = 1. / ainvnm / *anorm;
}
L20:
return 0;
/* End of ZGECON */
}
int ztgsen_(int *ijob, int *wantq, int *wantz,
int *select, int *n, doublecomplex *a, int *lda,
doublecomplex *b, int *ldb, doublecomplex *alpha, doublecomplex *
beta, doublecomplex *q, int *ldq, doublecomplex *z__, int *
ldz, int *m, double *pl, double *pr, double *dif,
doublecomplex *work, int *lwork, int *iwork, int *liwork,
int *info)
{
/* System generated locals */
int a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1,
z_offset, i__1, i__2, i__3;
doublecomplex z__1, z__2;
/* Builtin functions */
double sqrt(double), z_abs(doublecomplex *);
void d_cnjg(doublecomplex *, doublecomplex *);
/* Local variables */
int i__, k, n1, n2, ks, mn2, ijb, kase, ierr;
double dsum;
int swap;
doublecomplex temp1, temp2;
int isave[3];
extern int zscal_(int *, doublecomplex *,
doublecomplex *, int *);
int wantd;
int lwmin;
int wantp;
extern int zlacn2_(int *, doublecomplex *,
doublecomplex *, double *, int *, int *);
int wantd1, wantd2;
extern double dlamch_(char *);
double dscale, rdscal, safmin;
extern int xerbla_(char *, int *);
int liwmin;
extern int zlacpy_(char *, int *, int *,
doublecomplex *, int *, doublecomplex *, int *),
ztgexc_(int *, int *, int *, doublecomplex *, int
*, doublecomplex *, int *, doublecomplex *, int *,
doublecomplex *, int *, int *, int *, int *),
zlassq_(int *, doublecomplex *, int *, double *,
double *);
int lquery;
extern int ztgsyl_(char *, int *, int *, int
*, doublecomplex *, int *, doublecomplex *, int *,
doublecomplex *, int *, doublecomplex *, int *,
doublecomplex *, int *, doublecomplex *, int *,
double *, double *, doublecomplex *, int *, int *,
int *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* January 2007 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZTGSEN reorders the generalized Schur decomposition of a complex */
/* matrix pair (A, B) (in terms of an unitary equivalence trans- */
/* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */
/* appears in the leading diagonal blocks of the pair (A,B). The leading */
/* columns of Q and Z form unitary bases of the corresponding left and */
/* right eigenspaces (deflating subspaces). (A, B) must be in */
/* generalized Schur canonical form, that is, A and B are both upper */
/* triangular. */
/* ZTGSEN also computes the generalized eigenvalues */
/* w(j)= ALPHA(j) / BETA(j) */
/* of the reordered matrix pair (A, B). */
/* Optionally, the routine computes estimates of reciprocal condition */
/* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */
/* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */
/* between the matrix pairs (A11, B11) and (A22,B22) that correspond to */
/* the selected cluster and the eigenvalues outside the cluster, resp., */
/* and norms of "projections" onto left and right eigenspaces w.r.t. */
/* the selected cluster in the (1,1)-block. */
/* Arguments */
/* ========= */
/* IJOB (input) int */
/* Specifies whether condition numbers are required for the */
/* cluster of eigenvalues (PL and PR) or the deflating subspaces */
/* (Difu and Difl): */
/* =0: Only reorder w.r.t. SELECT. No extras. */
/* =1: Reciprocal of norms of "projections" onto left and right */
/* eigenspaces w.r.t. the selected cluster (PL and PR). */
/* =2: Upper bounds on Difu and Difl. F-norm-based estimate */
/* (DIF(1:2)). */
/* =3: Estimate of Difu and Difl. 1-norm-based estimate */
/* (DIF(1:2)). */
/* About 5 times as expensive as IJOB = 2. */
/* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */
/* version to get it all. */
/* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */
/* WANTQ (input) LOGICAL */
/* .TRUE. : update the left transformation matrix Q; */
/* .FALSE.: do not update Q. */
/* WANTZ (input) LOGICAL */
/* .TRUE. : update the right transformation matrix Z; */
/* .FALSE.: do not update Z. */
/* SELECT (input) LOGICAL array, dimension (N) */
/* SELECT specifies the eigenvalues in the selected cluster. To */
/* select an eigenvalue w(j), SELECT(j) must be set to */
/* .TRUE.. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* A (input/output) COMPLEX*16 array, dimension(LDA,N) */
/* On entry, the upper triangular matrix A, in generalized */
/* Schur canonical form. */
/* On exit, A is overwritten by the reordered matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,N). */
/* B (input/output) COMPLEX*16 array, dimension(LDB,N) */
/* On entry, the upper triangular matrix B, in generalized */
/* Schur canonical form. */
/* On exit, B is overwritten by the reordered matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= MAX(1,N). */
/* ALPHA (output) COMPLEX*16 array, dimension (N) */
/* BETA (output) COMPLEX*16 array, dimension (N) */
/* The diagonal elements of A and B, respectively, */
/* when the pair (A,B) has been reduced to generalized Schur */
/* form. ALPHA(i)/BETA(i) i=1,...,N are the generalized */
/* eigenvalues. */
/* Q (input/output) COMPLEX*16 array, dimension (LDQ,N) */
/* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */
/* On exit, Q has been postmultiplied by the left unitary */
/* transformation matrix which reorder (A, B); The leading M */
/* columns of Q form orthonormal bases for the specified pair of */
/* left eigenspaces (deflating subspaces). */
/* If WANTQ = .FALSE., Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= 1. */
/* If WANTQ = .TRUE., LDQ >= N. */
/* Z (input/output) COMPLEX*16 array, dimension (LDZ,N) */
/* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */
/* On exit, Z has been postmultiplied by the left unitary */
/* transformation matrix which reorder (A, B); The leading M */
/* columns of Z form orthonormal bases for the specified pair of */
/* left eigenspaces (deflating subspaces). */
/* If WANTZ = .FALSE., Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1. */
/* If WANTZ = .TRUE., LDZ >= N. */
/* M (output) INTEGER */
/* The dimension of the specified pair of left and right */
/* eigenspaces, (deflating subspaces) 0 <= M <= N. */
/* PL (output) DOUBLE PRECISION */
/* PR (output) DOUBLE PRECISION */
/* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */
/* reciprocal of the norm of "projections" onto left and right */
/* eigenspace with respect to the selected cluster. */
/* 0 < PL, PR <= 1. */
/* If M = 0 or M = N, PL = PR = 1. */
/* If IJOB = 0, 2 or 3 PL, PR are not referenced. */
/* DIF (output) DOUBLE PRECISION array, dimension (2). */
/* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */
/* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */
/* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */
/* estimates of Difu and Difl, computed using reversed */
/* communication with ZLACN2. */
/* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */
/* If IJOB = 0 or 1, DIF is not referenced. */
/* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) */
/* IF IJOB = 0, WORK is not referenced. Otherwise, */
/* on exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= 1 */
/* If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) */
/* If IJOB = 3 or 5, LWORK >= 4*M*(N-M) */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */
/* IF IJOB = 0, IWORK is not referenced. Otherwise, */
/* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. LIWORK >= 1. */
/* If IJOB = 1, 2 or 4, LIWORK >= N+2; */
/* If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); */
/* If LIWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the optimal size of the IWORK array, */
/* returns this value as the first entry of the IWORK array, and */
/* no error message related to LIWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* =0: Successful exit. */
/* <0: If INFO = -i, the i-th argument had an illegal value. */
/* =1: Reordering of (A, B) failed because the transformed */
/* matrix pair (A, B) would be too far from generalized */
/* Schur form; the problem is very ill-conditioned. */
/* (A, B) may have been partially reordered. */
/* If requested, 0 is returned in DIF(*), PL and PR. */
/* Further Details */
/* =============== */
/* ZTGSEN first collects the selected eigenvalues by computing unitary */
/* U and W that move them to the top left corner of (A, B). In other */
/* words, the selected eigenvalues are the eigenvalues of (A11, B11) in */
/* U'*(A, B)*W = (A11 A12) (B11 B12) n1 */
/* ( 0 A22),( 0 B22) n2 */
/* n1 n2 n1 n2 */
/* where N = n1+n2 and U' means the conjugate transpose of U. The first */
/* n1 columns of U and W span the specified pair of left and right */
/* eigenspaces (deflating subspaces) of (A, B). */
/* If (A, B) has been obtained from the generalized float Schur */
/* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */
/* reordered generalized Schur form of (C, D) is given by */
/* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', */
/* and the first n1 columns of Q*U and Z*W span the corresponding */
/* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */
/* Note that if the selected eigenvalue is sufficiently ill-conditioned, */
/* then its value may differ significantly from its value before */
/* reordering. */
/* The reciprocal condition numbers of the left and right eigenspaces */
/* spanned by the first n1 columns of U and W (or Q*U and Z*W) may */
/* be returned in DIF(1:2), corresponding to Difu and Difl, resp. */
/* The Difu and Difl are defined as: */
/* Difu[(A11, B11), (A22, B22)] = sigma-MIN( Zu ) */
/* and */
/* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */
/* where sigma-MIN(Zu) is the smallest singular value of the */
/* (2*n1*n2)-by-(2*n1*n2) matrix */
/* Zu = [ kron(In2, A11) -kron(A22', In1) ] */
/* [ kron(In2, B11) -kron(B22', In1) ]. */
/* Here, Inx is the identity matrix of size nx and A22' is the */
/* transpose of A22. kron(X, Y) is the Kronecker product between */
/* the matrices X and Y. */
/* When DIF(2) is small, small changes in (A, B) can cause large changes */
/* in the deflating subspace. An approximate (asymptotic) bound on the */
/* maximum angular error in the computed deflating subspaces is */
/* EPS * norm((A, B)) / DIF(2), */
/* where EPS is the machine precision. */
/* The reciprocal norm of the projectors on the left and right */
/* eigenspaces associated with (A11, B11) may be returned in PL and PR. */
/* They are computed as follows. First we compute L and R so that */
/* P*(A, B)*Q is block diagonal, where */
/* P = ( I -L ) n1 Q = ( I R ) n1 */
/* ( 0 I ) n2 and ( 0 I ) n2 */
/* n1 n2 n1 n2 */
/* and (L, R) is the solution to the generalized Sylvester equation */
/* A11*R - L*A22 = -A12 */
/* B11*R - L*B22 = -B12 */
/* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */
/* An approximate (asymptotic) bound on the average absolute error of */
/* the selected eigenvalues is */
/* EPS * norm((A, B)) / PL. */
/* There are also global error bounds which valid for perturbations up */
/* to a certain restriction: A lower bound (x) on the smallest */
/* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */
/* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */
/* (i.e. (A + E, B + F), is */
/* x = MIN(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*MAX(1/PL,1/PR)). */
/* An approximate bound on x can be computed from DIF(1:2), PL and PR. */
/* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */
/* (L', R') and unperturbed (L, R) left and right deflating subspaces */
/* associated with the selected cluster in the (1,1)-blocks can be */
/* bounded as */
/* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */
/* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */
/* See LAPACK User's Guide section 4.11 or the following references */
/* for more information. */
/* Note that if the default method for computing the Frobenius-norm- */
/* based estimate DIF is not wanted (see ZLATDF), then the parameter */
/* IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF */
/* (IJOB = 2 will be used)). See ZTGSYL for more details. */
/* Based on contributions by */
/* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */
/* Umea University, S-901 87 Umea, Sweden. */
/* References */
/* ========== */
/* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */
/* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */
/* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */
/* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */
/* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */
/* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */
/* Estimation: Theory, Algorithms and Software, Report */
/* UMINF - 94.04, Department of Computing Science, Umea University, */
/* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. */
/* To appear in Numerical Algorithms, 1996. */
/* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */
/* for Solving the Generalized Sylvester Equation and Estimating the */
/* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */
/* Department of Computing Science, Umea University, S-901 87 Umea, */
/* Sweden, December 1993, Revised April 1994, Also as LAPACK working */
/* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */
/* 1996. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/* Parameter adjustments */
--select;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--alpha;
--beta;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--dif;
--work;
--iwork;
/* Function Body */
*info = 0;
lquery = *lwork == -1 || *liwork == -1;
if (*ijob < 0 || *ijob > 5) {
*info = -1;
} else if (*n < 0) {
*info = -5;
} else if (*lda < MAX(1,*n)) {
*info = -7;
} else if (*ldb < MAX(1,*n)) {
*info = -9;
} else if (*ldq < 1 || *wantq && *ldq < *n) {
*info = -13;
} else if (*ldz < 1 || *wantz && *ldz < *n) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTGSEN", &i__1);
return 0;
}
ierr = 0;
wantp = *ijob == 1 || *ijob >= 4;
wantd1 = *ijob == 2 || *ijob == 4;
wantd2 = *ijob == 3 || *ijob == 5;
wantd = wantd1 || wantd2;
/* Set M to the dimension of the specified pair of deflating */
/* subspaces. */
*m = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = k;
i__3 = k + k * a_dim1;
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = k + k * b_dim1;
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
if (k < *n) {
if (select[k]) {
++(*m);
}
} else {
if (select[*n]) {
++(*m);
}
}
/* L10: */
}
if (*ijob == 1 || *ijob == 2 || *ijob == 4) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m);
lwmin = MAX(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n + 2;
liwmin = MAX(i__1,i__2);
} else if (*ijob == 3 || *ijob == 5) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 2) * (*n - *m);
lwmin = MAX(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = MAX(i__1,i__2), i__2 =
*n + 2;
liwmin = MAX(i__1,i__2);
} else {
lwmin = 1;
liwmin = 1;
}
work[1].r = (double) lwmin, work[1].i = 0.;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -21;
} else if (*liwork < liwmin && ! lquery) {
*info = -23;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTGSEN", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible. */
if (*m == *n || *m == 0) {
if (wantp) {
*pl = 1.;
*pr = 1.;
}
if (wantd) {
dscale = 0.;
dsum = 1.;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
zlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum);
zlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum);
/* L20: */
}
dif[1] = dscale * sqrt(dsum);
dif[2] = dif[1];
}
goto L70;
}
/* Get machine constant */
safmin = dlamch_("S");
/* Collect the selected blocks at the top-left corner of (A, B). */
ks = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
swap = select[k];
if (swap) {
++ks;
/* Swap the K-th block to position KS. Compute unitary Q */
/* and Z that will swap adjacent diagonal blocks in (A, B). */
if (k != ks) {
ztgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb,
&q[q_offset], ldq, &z__[z_offset], ldz, &k, &ks, &
ierr);
}
if (ierr > 0) {
/* Swap is rejected: exit. */
*info = 1;
if (wantp) {
*pl = 0.;
*pr = 0.;
}
if (wantd) {
dif[1] = 0.;
dif[2] = 0.;
}
goto L70;
}
}
/* L30: */
}
if (wantp) {
/* Solve generalized Sylvester equation for R and L: */
/* A11 * R - L * A22 = A12 */
/* B11 * R - L * B22 = B12 */
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
zlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1);
zlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 +
1], &n1);
ijb = 0;
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1]
, lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ *
b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &
work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr);
/* Estimate the reciprocal of norms of "projections" onto */
/* left and right eigenspaces */
rdscal = 0.;
dsum = 1.;
i__1 = n1 * n2;
zlassq_(&i__1, &work[1], &c__1, &rdscal, &dsum);
*pl = rdscal * sqrt(dsum);
if (*pl == 0.) {
*pl = 1.;
} else {
*pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl));
}
rdscal = 0.;
dsum = 1.;
i__1 = n1 * n2;
zlassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum);
*pr = rdscal * sqrt(dsum);
if (*pr == 0.) {
*pr = 1.;
} else {
*pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr));
}
}
if (wantd) {
/* Compute estimates Difu and Difl. */
if (wantd1) {
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 3;
/* Frobenius norm-based Difu estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ *
a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ +
i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &
dif[1], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
ierr);
/* Frobenius norm-based Difl estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[
a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1],
ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale,
&dif[2], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
ierr);
} else {
/* Compute 1-norm-based estimates of Difu and Difl using */
/* reversed communication with ZLACN2. In each step a */
/* generalized Sylvester equation or a transposed variant */
/* is solved. */
kase = 0;
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 0;
mn2 = (n1 << 1) * n2;
/* 1-norm-based estimate of Difu. */
L40:
zlacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[1], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("C", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
}
goto L40;
}
dif[1] = dscale / dif[1];
/* 1-norm-based estimate of Difl. */
L50:
zlacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[2], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
&a[a_offset], lda, &work[1], &n2, &b[i__ + i__ *
b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 +
1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("C", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
&a[a_offset], lda, &work[1], &n2, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
}
goto L50;
}
dif[2] = dscale / dif[2];
}
}
/* If B(K,K) is complex, make it float and positive (normalization */
/* of the generalized Schur form) and Store the generalized */
/* eigenvalues of reordered pair (A, B) */
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
dscale = z_abs(&b[k + k * b_dim1]);
if (dscale > safmin) {
i__2 = k + k * b_dim1;
z__2.r = b[i__2].r / dscale, z__2.i = b[i__2].i / dscale;
d_cnjg(&z__1, &z__2);
temp1.r = z__1.r, temp1.i = z__1.i;
i__2 = k + k * b_dim1;
z__1.r = b[i__2].r / dscale, z__1.i = b[i__2].i / dscale;
temp2.r = z__1.r, temp2.i = z__1.i;
i__2 = k + k * b_dim1;
b[i__2].r = dscale, b[i__2].i = 0.;
i__2 = *n - k;
zscal_(&i__2, &temp1, &b[k + (k + 1) * b_dim1], ldb);
i__2 = *n - k + 1;
zscal_(&i__2, &temp1, &a[k + k * a_dim1], lda);
if (*wantq) {
zscal_(n, &temp2, &q[k * q_dim1 + 1], &c__1);
}
} else {
i__2 = k + k * b_dim1;
b[i__2].r = 0., b[i__2].i = 0.;
}
i__2 = k;
i__3 = k + k * a_dim1;
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = k + k * b_dim1;
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
/* L60: */
}
L70:
work[1].r = (double) lwmin, work[1].i = 0.;
iwork[1] = liwmin;
return 0;
/* End of ZTGSEN */
} /* ztgsen_ */
doublereal zla_hercond_x__(char *uplo, integer *n, doublecomplex *a, integer *
lda, doublecomplex *af, integer *ldaf, integer *ipiv, doublecomplex *
x, integer *info, doublecomplex *work, doublereal *rwork, ftnlen
uplo_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2, i__3, i__4;
doublereal ret_val, d__1, d__2;
doublecomplex z__1, z__2;
/* Local variables */
integer i__, j;
logical up;
doublereal tmp;
integer kase;
integer isave[3];
doublereal anorm;
doublereal ainvnm;
/* -- LAPACK routine (version 3.2.1) -- */
/* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/* -- Jason Riedy of Univ. of California Berkeley. -- */
/* -- April 2009 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley and NAG Ltd. -- */
/* Purpose */
/* ======= */
/* ZLA_HERCOND_X computes the infinity norm condition number of */
/* op(A) * diag(X) where X is a COMPLEX*16 vector. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* A (input) COMPLEX*16 array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) COMPLEX*16 array, dimension (LDAF,N) */
/* The block diagonal matrix D and the multipliers used to */
/* obtain the factor U or L as computed by ZHETRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by CHETRF. */
/* X (input) COMPLEX*16 array, dimension (N) */
/* The vector X in the formula op(A) * diag(X). */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) COMPLEX*16 array, dimension (2*N). */
/* Workspace. */
/* RWORK (input) DOUBLE PRECISION array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
--x;
--work;
--rwork;
/* Function Body */
ret_val = 0.;
*info = 0;
if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZLA_HERCOND_X", &i__1);
return ret_val;
}
up = FALSE_;
if (lsame_(uplo, "U")) {
up = TRUE_;
}
/* Compute norm of op(A)*op2(C). */
anorm = 0.;
if (up) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
i__4 = j;
z__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
z__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
z__1.r = z__2.r, z__1.i = z__2.i;
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1),
abs(d__2));
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
i__4 = j;
z__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
z__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
z__1.r = z__2.r, z__1.i = z__2.i;
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1),
abs(d__2));
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
i__4 = j;
z__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
z__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
z__1.r = z__2.r, z__1.i = z__2.i;
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1),
abs(d__2));
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
i__4 = j;
z__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
z__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
z__1.r = z__2.r, z__1.i = z__2.i;
tmp += (d__1 = z__1.r, abs(d__1)) + (d__2 = d_imag(&z__1),
abs(d__2));
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0) {
ret_val = 1.;
return ret_val;
} else if (anorm == 0.) {
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.;
kase = 0;
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
if (up) {
zhetrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
} else {
zhetrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
}
/* Multiply by inv(X). */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
z_div(&z__1, &work[i__], &x[i__]);
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
} else {
/* Multiply by inv(X'). */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
z_div(&z__1, &work[i__], &x[i__]);
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
if (up) {
zhetrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
} else {
zhetrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
z__1.r = rwork[i__4] * work[i__3].r, z__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
ret_val = 1. / ainvnm;
}
return ret_val;
} /* zla_hercond_x__ */
/* Subroutine */ int zsycon_(char *uplo, integer *n, doublecomplex *a,
integer *lda, integer *ipiv, doublereal *anorm, doublereal *rcond,
doublecomplex *work, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
/* Local variables */
integer i__, kase;
integer isave[3];
logical upper;
doublereal ainvnm;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* ZSYCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a complex symmetric matrix A using the factorization */
/* A = U*D*U**T or A = L*D*L**T computed by ZSYTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* Specifies whether the details of the factorization are stored */
/* as an upper or lower triangular matrix. */
/* = 'U': Upper triangular, form is A = U*D*U**T; */
/* = 'L': Lower triangular, form is A = L*D*L**T. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input) COMPLEX*16 array, dimension (LDA,N) */
/* The block diagonal matrix D and the multipliers used to */
/* obtain the factor U or L as computed by ZSYTRF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by ZSYTRF. */
/* ANORM (input) DOUBLE PRECISION */
/* The 1-norm of the original matrix A. */
/* RCOND (output) DOUBLE PRECISION */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
--work;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*anorm < 0.) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZSYCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0) {
*rcond = 1.;
return 0;
} else if (*anorm <= 0.) {
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper) {
/* Upper triangular storage: examine D from bottom to top */
for (i__ = *n; i__ >= 1; --i__) {
i__1 = i__ + i__ * a_dim1;
if (ipiv[i__] > 0 && (a[i__1].r == 0. && a[i__1].i == 0.)) {
return 0;
}
}
} else {
/* Lower triangular storage: examine D from top to bottom. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__ + i__ * a_dim1;
if (ipiv[i__] > 0 && (a[i__2].r == 0. && a[i__2].i == 0.)) {
return 0;
}
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
/* Multiply by inv(L*D*L') or inv(U*D*U'). */
zsytrs_(uplo, n, &c__1, &a[a_offset], lda, &ipiv[1], &work[1], n,
info);
goto L30;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
*rcond = 1. / ainvnm / *anorm;
}
return 0;
/* End of ZSYCON */
} /* zsycon_ */
/* Subroutine */ int zporfs_(char *uplo, integer *n, integer *nrhs,
doublecomplex *a, integer *lda, doublecomplex *af, integer *ldaf,
doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx,
doublereal *ferr, doublereal *berr, doublecomplex *work, doublereal *
rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3, d__4;
doublecomplex z__1;
/* Local variables */
integer i__, j, k;
doublereal s, xk;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
integer isave[3], count;
logical upper;
doublereal safmin;
doublereal lstres;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* ZPORFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is Hermitian positive definite, */
/* and provides error bounds and backward error estimates for the */
/* solution. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* A (input) COMPLEX*16 array, dimension (LDA,N) */
/* The Hermitian matrix A. If UPLO = 'U', the leading N-by-N */
/* upper triangular part of A contains the upper triangular part */
/* of the matrix A, and the strictly lower triangular part of A */
/* is not referenced. If UPLO = 'L', the leading N-by-N lower */
/* triangular part of A contains the lower triangular part of */
/* the matrix A, and the strictly upper triangular part of A is */
/* not referenced. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) COMPLEX*16 array, dimension (LDAF,N) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**H*U or A = L*L**H, as computed by ZPOTRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* B (input) COMPLEX*16 array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by ZPOTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The estimated forward error bound for each solution vector */
/* X(j) (the j-th column of the solution matrix X). */
/* If XTRUE is the true solution corresponding to X(j), FERR(j) */
/* is an estimated upper bound for the magnitude of the largest */
/* element in (X(j) - XTRUE) divided by the magnitude of the */
/* largest element in X(j). The estimate is as reliable as */
/* the estimate for RCOND, and is almost always a slight */
/* overestimate of the true error. */
/* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Internal Parameters */
/* =================== */
/* ITMAX is the maximum number of steps of iterative refinement. */
/* ==================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldaf < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldx < max(1,*n)) {
*info = -11;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZPORFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.;
berr[j] = 0.;
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
count = 1;
lstres = 3.;
L20:
/* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - A * X */
zcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
z__1.r = -1., z__1.i = -0.;
zhemv_(uplo, n, &z__1, &a[a_offset], lda, &x[j * x_dim1 + 1], &c__1, &
c_b1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 = d_imag(&b[
i__ + j * b_dim1]), abs(d__2));
}
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[k + j *
x_dim1]), abs(d__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = i__ + k * a_dim1;
rwork[i__] += ((d__1 = a[i__4].r, abs(d__1)) + (d__2 =
d_imag(&a[i__ + k * a_dim1]), abs(d__2))) * xk;
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((d__1 = a[i__4].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + k * a_dim1]), abs(d__2))) * ((d__3 = x[i__5]
.r, abs(d__3)) + (d__4 = d_imag(&x[i__ + j *
x_dim1]), abs(d__4)));
}
i__3 = k + k * a_dim1;
rwork[k] = rwork[k] + (d__1 = a[i__3].r, abs(d__1)) * xk + s;
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
i__3 = k + j * x_dim1;
xk = (d__1 = x[i__3].r, abs(d__1)) + (d__2 = d_imag(&x[k + j *
x_dim1]), abs(d__2));
i__3 = k + k * a_dim1;
rwork[k] += (d__1 = a[i__3].r, abs(d__1)) * xk;
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = i__ + k * a_dim1;
rwork[i__] += ((d__1 = a[i__4].r, abs(d__1)) + (d__2 =
d_imag(&a[i__ + k * a_dim1]), abs(d__2))) * xk;
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((d__1 = a[i__4].r, abs(d__1)) + (d__2 = d_imag(&a[
i__ + k * a_dim1]), abs(d__2))) * ((d__3 = x[i__5]
.r, abs(d__3)) + (d__4 = d_imag(&x[i__ + j *
x_dim1]), abs(d__4)));
}
rwork[k] += s;
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2))) / rwork[i__];
s = max(d__3,d__4);
} else {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__]
+ safe1);
s = max(d__3,d__4);
}
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2. <= lstres && count <= 5) {
/* Update solution and try again. */
zpotrs_(uplo, n, &c__1, &af[af_offset], ldaf, &work[1], n, info);
zaxpy_(n, &c_b1, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(A))* */
/* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(A) is the inverse of A */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(A)*abs(X) + abs(B) is less than SAFE2. */
/* Use ZLACN2 to estimate the infinity-norm of the matrix */
/* inv(A) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
;
} else {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
+ safe1;
}
}
kase = 0;
L100:
zlacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
zpotrs_(uplo, n, &c__1, &af[af_offset], ldaf, &work[1], n,
info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
z__1.r = rwork[i__4] * work[i__5].r, z__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
}
zpotrs_(uplo, n, &c__1, &af[af_offset], ldaf, &work[1], n,
info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = i__ + j * x_dim1;
d__3 = lstres, d__4 = (d__1 = x[i__3].r, abs(d__1)) + (d__2 =
d_imag(&x[i__ + j * x_dim1]), abs(d__2));
lstres = max(d__3,d__4);
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
}
return 0;
/* End of ZPORFS */
} /* zporfs_ */
/* Subroutine */
int zpbcon_(char *uplo, integer *n, integer *kd, doublecomplex *ab, integer *ldab, doublereal *anorm, doublereal * rcond, doublecomplex *work, doublereal *rwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1;
doublereal d__1, d__2;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
integer ix, kase;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */
int zlacn2_(integer *, doublecomplex *, doublecomplex *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal scalel, scaleu;
extern /* Subroutine */
int xerbla_(char *, integer *);
doublereal ainvnm;
extern integer izamax_(integer *, doublecomplex *, integer *);
extern /* Subroutine */
int zlatbs_(char *, char *, char *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublereal *, doublereal *, integer *), zdrscl_(integer *, doublereal *, doublecomplex *, integer *);
char normin[1];
doublereal smlnum;
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*kd < 0)
{
*info = -3;
}
else if (*ldab < *kd + 1)
{
*info = -5;
}
else if (*anorm < 0.)
{
*info = -6;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZPBCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.;
if (*n == 0)
{
*rcond = 1.;
return 0;
}
else if (*anorm == 0.)
{
return 0;
}
smlnum = dlamch_("Safe minimum");
/* Estimate the 1-norm of the inverse. */
kase = 0;
*(unsigned char *)normin = 'N';
L10:
zlacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (upper)
{
/* Multiply by inv(U**H). */
zlatbs_("Upper", "Conjugate transpose", "Non-unit", normin, n, kd, &ab[ab_offset], ldab, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
zlatbs_("Upper", "No transpose", "Non-unit", normin, n, kd, &ab[ ab_offset], ldab, &work[1], &scaleu, &rwork[1], info);
}
else
{
/* Multiply by inv(L). */
zlatbs_("Lower", "No transpose", "Non-unit", normin, n, kd, &ab[ ab_offset], ldab, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L**H). */
zlatbs_("Lower", "Conjugate transpose", "Non-unit", normin, n, kd, &ab[ab_offset], ldab, &work[1], &scaleu, &rwork[1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.)
{
ix = izamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((d__1 = work[i__1].r, f2c_abs(d__1)) + (d__2 = d_imag(& work[ix]), f2c_abs(d__2))) * smlnum || scale == 0.)
{
goto L20;
}
zdrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.)
{
*rcond = 1. / ainvnm / *anorm;
}
L20:
return 0;
/* End of ZPBCON */
}
