int normAm(double alpha, int m, double* est, double* dwrk, int* iwrk, expo_type* expo)
{
int tcol = 1; // Number of columns used by DLACN1
char trans = 'N';
double* mvwrk = dwrk;
double* v = mvwrk + 2*expo->dim;
double* x = v + expo->dim;
double* xold = x + expo->dim;
double* wrk = xold + expo->dim;
double* h = wrk + tcol;
int* ind = iwrk;
int* indh = ind + expo->dim;
int kase = 0;
int info = 0;
int iseed[] = { 153, 1673, 2, 3567 };
int isave[] = { 0, 0, 0 };
int mv = 0;
*est = 0.0;
// dlacn1_(&expo->dim,&tcol,v,x,&expo->dim,xold,&expo->dim,wrk,h,ind,indh,est,&kase,iseed,&info);
dlacn2_(&expo->dim,v,x,iwrk,est,&kase,isave);
while (kase != 0) {
if (kase == 1) trans = 'N';
else if (kase == 2) trans = 'T';
afun_power(trans,alpha,m,x,expo,mvwrk);
// dlacn1_(&expo->dim,&tcol,v,x,&expo->dim,xold,&expo->dim,wrk,h,ind,indh,est,&kase,iseed,&info);
dlacn2_(&expo->dim,v,x,iwrk,est,&kase,isave);
mv += m*tcol;
}
return mv;
}
/* Subroutine */
int dpbcon_(char *uplo, integer *n, integer *kd, doublereal * ab, integer *ldab, doublereal *anorm, doublereal *rcond, doublereal * work, integer *iwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1;
doublereal d__1;
/* Local variables */
integer ix, kase;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */
int drscl_(integer *, doublereal *, doublereal *, integer *);
logical upper;
extern /* Subroutine */
int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal scalel;
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */
int dlatbs_(char *, char *, char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *);
doublereal scaleu;
extern /* Subroutine */
int xerbla_(char *, integer *);
doublereal ainvnm;
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 .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--work;
--iwork;
/* 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_("DPBCON", &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:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (upper)
{
/* Multiply by inv(U**T). */
dlatbs_("Upper", "Transpose", "Non-unit", normin, n, kd, &ab[ ab_offset], ldab, &work[1], &scalel, &work[(*n << 1) + 1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
dlatbs_("Upper", "No transpose", "Non-unit", normin, n, kd, &ab[ ab_offset], ldab, &work[1], &scaleu, &work[(*n << 1) + 1], info);
}
else
{
/* Multiply by inv(L). */
dlatbs_("Lower", "No transpose", "Non-unit", normin, n, kd, &ab[ ab_offset], ldab, &work[1], &scalel, &work[(*n << 1) + 1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L**T). */
dlatbs_("Lower", "Transpose", "Non-unit", normin, n, kd, &ab[ ab_offset], ldab, &work[1], &scaleu, &work[(*n << 1) + 1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.)
{
ix = idamax_(n, &work[1], &c__1);
if (scale < (d__1 = work[ix], f2c_abs(d__1)) * smlnum || scale == 0.)
{
goto L20;
}
drscl_(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 DPBCON */
}
/* Subroutine */ int dgtrfs_(char *trans, integer *n, integer *nrhs,
doublereal *dl, doublereal *d__, doublereal *du, doublereal *dlf,
doublereal *df, doublereal *duf, doublereal *du2, integer *ipiv,
doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal *
ferr, doublereal *berr, doublereal *work, integer *iwork, integer *
info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2;
doublereal d__1, d__2, d__3, d__4;
/* Local variables */
integer i__, j;
doublereal s;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), daxpy_(integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *);
integer count;
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */ int dlagtm_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, doublereal *, doublereal *, integer *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
logical notran;
char transn[1];
extern /* Subroutine */ int dgttrs_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, integer *, integer *);
char transt[1];
doublereal lstres;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DGTRFS 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 = 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) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) subdiagonal elements of A. */
/* D (input) DOUBLE PRECISION array, dimension (N) */
/* The diagonal elements of A. */
/* DU (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) superdiagonal elements of A. */
/* DLF (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) multipliers that define the matrix L from the */
/* LU factorization of A as computed by DGTTRF. */
/* DF (input) DOUBLE PRECISION array, dimension (N) */
/* The n diagonal elements of the upper triangular matrix U from */
/* the LU factorization of A. */
/* DUF (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) elements of the first superdiagonal of U. */
/* DU2 (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by DGTTRS. */
/* 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* 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;
--iwork;
/* 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_("DGTRFS", &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 = 'T';
} else {
*(unsigned char *)transn = 'T';
*(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. */
dcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dlagtm_(trans, n, &c__1, &c_b18, &dl[1], &d__[1], &du[1], &x[j *
x_dim1 + 1], ldx, &c_b19, &work[*n + 1], n);
/* Compute abs(op(A))*abs(x) + abs(b) for use in the backward */
/* error bound. */
if (notran) {
if (*n == 1) {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2));
} else {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2)) + (d__3 = du[1] *
x[j * x_dim1 + 2], abs(d__3));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1)) + (
d__2 = dl[i__ - 1] * x[i__ - 1 + j * x_dim1], abs(
d__2)) + (d__3 = d__[i__] * x[i__ + j * x_dim1],
abs(d__3)) + (d__4 = du[i__] * x[i__ + 1 + j *
x_dim1], abs(d__4));
/* L30: */
}
work[*n] = (d__1 = b[*n + j * b_dim1], abs(d__1)) + (d__2 =
dl[*n - 1] * x[*n - 1 + j * x_dim1], abs(d__2)) + (
d__3 = d__[*n] * x[*n + j * x_dim1], abs(d__3));
}
} else {
if (*n == 1) {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2));
} else {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2)) + (d__3 = dl[1] *
x[j * x_dim1 + 2], abs(d__3));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1)) + (
d__2 = du[i__ - 1] * x[i__ - 1 + j * x_dim1], abs(
d__2)) + (d__3 = d__[i__] * x[i__ + j * x_dim1],
abs(d__3)) + (d__4 = dl[i__] * x[i__ + 1 + j *
x_dim1], abs(d__4));
/* L40: */
}
work[*n] = (d__1 = b[*n + j * b_dim1], abs(d__1)) + (d__2 =
du[*n - 1] * x[*n - 1 + j * x_dim1], abs(d__2)) + (
d__3 = d__[*n] * x[*n + j * x_dim1], abs(d__3));
}
}
/* 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 (work[i__] > safe2) {
/* Computing MAX */
d__2 = s, d__3 = (d__1 = work[*n + i__], abs(d__1)) / work[
i__];
s = max(d__2,d__3);
} else {
/* Computing MAX */
d__2 = s, d__3 = ((d__1 = work[*n + i__], abs(d__1)) + safe1)
/ (work[i__] + safe1);
s = max(d__2,d__3);
}
/* L50: */
}
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. */
dgttrs_(trans, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &ipiv[
1], &work[*n + 1], n, info);
daxpy_(n, &c_b19, &work[*n + 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 DLACN2 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 (work[i__] > safe2) {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__];
} else {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__] + safe1;
}
/* L60: */
}
kase = 0;
L70:
dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**T). */
dgttrs_(transt, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &
ipiv[1], &work[*n + 1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L80: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L90: */
}
dgttrs_(transn, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &
ipiv[1], &work[*n + 1], n, info);
}
goto L70;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = lstres, d__3 = (d__1 = x[i__ + j * x_dim1], abs(d__1));
lstres = max(d__2,d__3);
/* L100: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L110: */
}
return 0;
/* End of DGTRFS */
} /* dgtrfs_ */
/* ===================================================================== */
doublereal dla_syrcond_(char *uplo, integer *n, doublereal *a, integer *lda, doublereal *af, integer *ldaf, integer *ipiv, integer *cmode, doublereal *c__, integer *info, doublereal *work, integer *iwork)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2;
doublereal ret_val, d__1;
/* Local variables */
integer i__, j;
logical up;
doublereal tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */
int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */
int xerbla_(char *, integer *);
doublereal ainvnm;
char normin[1];
doublereal smlnum;
extern /* Subroutine */
int dsytrs_(char *, integer *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, integer *);
/* -- 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 .. */
/* .. */
/* .. 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;
--iwork;
/* Function Body */
ret_val = 0.;
*info = 0;
if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*n))
{
*info = -4;
}
else if (*ldaf < max(1,*n))
{
*info = -6;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("DLA_SYRCOND", &i__1);
return ret_val;
}
if (*n == 0)
{
ret_val = 1.;
return ret_val;
}
up = FALSE_;
if (lsame_(uplo, "U"))
{
up = TRUE_;
}
/* Compute the equilibration matrix R such that */
/* inv(R)*A*C has unit 1-norm. */
if (up)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.;
if (*cmode == 1)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[j + i__ * a_dim1] * c__[j], abs(d__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[i__ + j * a_dim1] * c__[j], abs(d__1));
}
}
else if (*cmode == 0)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[j + i__ * a_dim1], abs(d__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[i__ + j * a_dim1], abs(d__1));
}
}
else
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[j + i__ * a_dim1] / c__[j], abs(d__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[i__ + j * a_dim1] / c__[j], abs(d__1));
}
}
work[(*n << 1) + i__] = tmp;
}
}
else
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.;
if (*cmode == 1)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[i__ + j * a_dim1] * c__[j], abs(d__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[j + i__ * a_dim1] * c__[j], abs(d__1));
}
}
else if (*cmode == 0)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[i__ + j * a_dim1], abs(d__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[j + i__ * a_dim1], abs(d__1));
}
}
else
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[i__ + j * a_dim1] / c__[j], abs(d__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (d__1 = a[j + i__ * a_dim1] / c__[j], abs(d__1));
}
}
work[(*n << 1) + i__] = tmp;
}
}
/* Estimate the norm of inv(op(A)). */
smlnum = dlamch_("Safe minimum");
ainvnm = 0.;
*(unsigned char *)normin = 'N';
kase = 0;
L10:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (kase == 2)
{
/* Multiply by R. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] *= work[(*n << 1) + i__];
}
if (up)
{
dsytrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
else
{
dsytrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
/* Multiply by inv(C). */
if (*cmode == 1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] /= c__[i__];
}
}
else if (*cmode == -1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] *= c__[i__];
}
}
}
else
{
/* Multiply by inv(C**T). */
if (*cmode == 1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] /= c__[i__];
}
}
else if (*cmode == -1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] *= c__[i__];
}
}
if (up)
{
dsytrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
else
{
dsytrs_("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__)
{
work[i__] *= work[(*n << 1) + i__];
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.)
{
ret_val = 1. / ainvnm;
}
return ret_val;
}
/* Subroutine */ int dgbcon_(char *norm, integer *n, integer *kl, integer *ku,
doublereal *ab, integer *ldab, integer *ipiv, doublereal *anorm,
doublereal *rcond, doublereal *work, integer *iwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1, i__2, i__3;
doublereal d__1;
/* Local variables */
integer j;
doublereal t;
integer kd, lm, jp, ix, kase;
extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *,
integer *);
integer kase1;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int drscl_(integer *, doublereal *, doublereal *,
integer *);
logical lnoti;
extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *), dlacn2_(integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *,
integer *);
extern doublereal dlamch_(char *);
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */ int dlatbs_(char *, char *, char *, char *,
integer *, integer *, doublereal *, integer *, doublereal *,
doublereal *, doublereal *, integer *), xerbla_(char *, integer *);
doublereal ainvnm;
logical onenrm;
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 DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DGBCON estimates the reciprocal of the condition number of a real */
/* general band matrix A, in either the 1-norm or the infinity-norm, */
/* using the LU factorization computed by DGBTRF. */
/* 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) DOUBLE PRECISION array, dimension (LDAB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by DGBTRF. 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--ipiv;
--work;
--iwork;
/* 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_("DGBCON", &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:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[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];
t = work[jp];
if (jp != j) {
work[jp] = work[j];
work[j] = t;
}
d__1 = -t;
daxpy_(&lm, &d__1, &ab[kd + 1 + j * ab_dim1], &c__1, &
work[j + 1], &c__1);
/* L20: */
}
}
/* Multiply by inv(U). */
i__1 = *kl + *ku;
dlatbs_("Upper", "No transpose", "Non-unit", normin, n, &i__1, &
ab[ab_offset], ldab, &work[1], &scale, &work[(*n << 1) +
1], info);
} else {
/* Multiply by inv(U'). */
i__1 = *kl + *ku;
dlatbs_("Upper", "Transpose", "Non-unit", normin, n, &i__1, &ab[
ab_offset], ldab, &work[1], &scale, &work[(*n << 1) + 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);
work[j] -= ddot_(&lm, &ab[kd + 1 + j * ab_dim1], &c__1, &
work[j + 1], &c__1);
jp = ipiv[j];
if (jp != j) {
t = work[jp];
work[jp] = work[j];
work[j] = t;
}
/* L30: */
}
}
}
/* Divide X by 1/SCALE if doing so will not cause overflow. */
*(unsigned char *)normin = 'Y';
if (scale != 1.) {
ix = idamax_(n, &work[1], &c__1);
if (scale < (d__1 = work[ix], abs(d__1)) * smlnum || scale == 0.)
{
goto L40;
}
drscl_(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 DGBCON */
} /* dgbcon_ */
/* Subroutine */
int dtrsna_(char *job, char *howmny, logical *select, integer *n, doublereal *t, integer *ldt, doublereal *vl, integer * ldvl, doublereal *vr, integer *ldvr, doublereal *s, doublereal *sep, integer *mm, integer *m, doublereal *work, integer *ldwork, integer * iwork, 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;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j, k, n2;
doublereal cs;
integer nn, ks;
doublereal sn, mu, eps, est;
integer kase;
doublereal cond;
extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, integer *);
logical pair;
integer ierr;
doublereal dumm, prod;
integer ifst;
doublereal lnrm;
integer ilst;
doublereal rnrm;
extern doublereal dnrm2_(integer *, doublereal *, integer *);
doublereal prod1, prod2, scale, delta;
extern logical lsame_(char *, char *);
integer isave[3];
logical wants;
doublereal dummy[1];
extern /* Subroutine */
int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *);
extern doublereal dlapy2_(doublereal *, doublereal *);
extern /* Subroutine */
int dlabad_(doublereal *, doublereal *);
extern doublereal dlamch_(char *);
extern /* Subroutine */
int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *);
doublereal bignum;
logical wantbh;
extern /* Subroutine */
int dlaqtr_(logical *, logical *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *), dtrexc_(char *, integer * , doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *);
logical somcon;
doublereal smlnum;
logical wantsp;
/* -- 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;
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;
--iwork;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
somcon = lsame_(howmny, "S");
*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
{
/* Set M to the number of eigenpairs for which condition numbers */
/* are required, and test MM. */
if (somcon)
{
*m = 0;
pair = FALSE_;
i__1 = *n;
for (k = 1;
k <= i__1;
++k)
{
if (pair)
{
pair = FALSE_;
}
else
{
if (k < *n)
{
if (t[k + 1 + k * t_dim1] == 0.)
{
if (select[k])
{
++(*m);
}
}
else
{
pair = TRUE_;
if (select[k] || select[k + 1])
{
*m += 2;
}
}
}
else
{
if (select[*n])
{
++(*m);
}
}
}
/* L10: */
}
}
else
{
*m = *n;
}
if (*mm < *m)
{
*info = -13;
}
else if (*ldwork < 1 || wantsp && *ldwork < *n)
{
*info = -16;
}
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("DTRSNA", &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] = (d__1 = t[t_dim1 + 1], abs(d__1));
}
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S") / eps;
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
ks = 0;
pair = FALSE_;
i__1 = *n;
for (k = 1;
k <= i__1;
++k)
{
/* Determine whether T(k,k) begins a 1-by-1 or 2-by-2 block. */
if (pair)
{
pair = FALSE_;
goto L60;
}
else
{
if (k < *n)
{
pair = t[k + 1 + k * t_dim1] != 0.;
}
}
/* Determine whether condition numbers are required for the k-th */
/* eigenpair. */
if (somcon)
{
if (pair)
{
if (! select[k] && ! select[k + 1])
{
goto L60;
}
}
else
{
if (! select[k])
{
goto L60;
}
}
}
++ks;
if (wants)
{
/* Compute the reciprocal condition number of the k-th */
/* eigenvalue. */
if (! pair)
{
/* Real eigenvalue. */
prod = ddot_(n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks * vl_dim1 + 1], &c__1);
rnrm = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
lnrm = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
s[ks] = abs(prod) / (rnrm * lnrm);
}
else
{
/* Complex eigenvalue. */
prod1 = ddot_(n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks * vl_dim1 + 1], &c__1);
prod1 += ddot_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1);
prod2 = ddot_(n, &vl[ks * vl_dim1 + 1], &c__1, &vr[(ks + 1) * vr_dim1 + 1], &c__1);
prod2 -= ddot_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1, &vr[ks * vr_dim1 + 1], &c__1);
d__1 = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
d__2 = dnrm2_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1);
rnrm = dlapy2_(&d__1, &d__2);
d__1 = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
d__2 = dnrm2_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1);
lnrm = dlapy2_(&d__1, &d__2);
cond = dlapy2_(&prod1, &prod2) / (rnrm * lnrm);
s[ks] = cond;
s[ks + 1] = cond;
}
}
if (wantsp)
{
/* Estimate the reciprocal condition number of the k-th */
/* eigenvector. */
/* Copy the matrix T to the array WORK and swap the diagonal */
/* block beginning at T(k,k) to the (1,1) position. */
dlacpy_("Full", n, n, &t[t_offset], ldt, &work[work_offset], ldwork);
ifst = k;
ilst = 1;
dtrexc_("No Q", n, &work[work_offset], ldwork, dummy, &c__1, & ifst, &ilst, &work[(*n + 1) * work_dim1 + 1], &ierr);
if (ierr == 1 || ierr == 2)
{
/* Could not swap because blocks not well separated */
scale = 1.;
est = bignum;
}
else
{
/* Reordering successful */
if (work[work_dim1 + 2] == 0.)
{
/* Form C = T22 - lambda*I in WORK(2:N,2:N). */
i__2 = *n;
for (i__ = 2;
i__ <= i__2;
++i__)
{
work[i__ + i__ * work_dim1] -= work[work_dim1 + 1];
/* L20: */
}
n2 = 1;
nn = *n - 1;
}
else
{
/* Triangularize the 2 by 2 block by unitary */
/* transformation U = [ cs i*ss ] */
/* [ i*ss cs ]. */
/* such that the (1,1) position of WORK is complex */
/* eigenvalue lambda with positive imaginary part. (2,2) */
/* position of WORK is the complex eigenvalue lambda */
/* with negative imaginary part. */
mu = sqrt((d__1 = work[(work_dim1 << 1) + 1], abs(d__1))) * sqrt((d__2 = work[work_dim1 + 2], abs(d__2)));
delta = dlapy2_(&mu, &work[work_dim1 + 2]);
cs = mu / delta;
sn = -work[work_dim1 + 2] / delta;
/* Form */
/* C**T = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ] */
/* [ mu ] */
/* [ .. ] */
/* [ .. ] */
/* [ mu ] */
/* where C**T is transpose of matrix C, */
/* and RWORK is stored starting in the N+1-st column of */
/* WORK. */
i__2 = *n;
for (j = 3;
j <= i__2;
++j)
{
work[j * work_dim1 + 2] = cs * work[j * work_dim1 + 2] ;
work[j + j * work_dim1] -= work[work_dim1 + 1];
/* L30: */
}
work[(work_dim1 << 1) + 2] = 0.;
work[(*n + 1) * work_dim1 + 1] = mu * 2.;
i__2 = *n - 1;
for (i__ = 2;
i__ <= i__2;
++i__)
{
work[i__ + (*n + 1) * work_dim1] = sn * work[(i__ + 1) * work_dim1 + 1];
/* L40: */
}
n2 = 2;
nn = *n - 1 << 1;
}
/* Estimate norm(inv(C**T)) */
est = 0.;
kase = 0;
L50:
dlacn2_(&nn, &work[(*n + 2) * work_dim1 + 1], &work[(*n + 4) * work_dim1 + 1], &iwork[1], &est, &kase, isave);
if (kase != 0)
{
if (kase == 1)
{
if (n2 == 1)
{
/* Real eigenvalue: solve C**T*x = scale*c. */
i__2 = *n - 1;
dlaqtr_(&c_true, &c_true, &i__2, &work[(work_dim1 << 1) + 2], ldwork, dummy, &dumm, &scale, &work[(*n + 4) * work_dim1 + 1], &work[(* n + 6) * work_dim1 + 1], &ierr);
}
else
{
/* Complex eigenvalue: solve */
/* C**T*(p+iq) = scale*(c+id) in real arithmetic. */
i__2 = *n - 1;
dlaqtr_(&c_true, &c_false, &i__2, &work[( work_dim1 << 1) + 2], ldwork, &work[(*n + 1) * work_dim1 + 1], &mu, &scale, &work[(* n + 4) * work_dim1 + 1], &work[(*n + 6) * work_dim1 + 1], &ierr);
}
}
else
{
if (n2 == 1)
{
/* Real eigenvalue: solve C*x = scale*c. */
i__2 = *n - 1;
dlaqtr_(&c_false, &c_true, &i__2, &work[( work_dim1 << 1) + 2], ldwork, dummy, & dumm, &scale, &work[(*n + 4) * work_dim1 + 1], &work[(*n + 6) * work_dim1 + 1], & ierr);
}
else
{
/* Complex eigenvalue: solve */
/* C*(p+iq) = scale*(c+id) in real arithmetic. */
i__2 = *n - 1;
dlaqtr_(&c_false, &c_false, &i__2, &work[( work_dim1 << 1) + 2], ldwork, &work[(*n + 1) * work_dim1 + 1], &mu, &scale, &work[(* n + 4) * work_dim1 + 1], &work[(*n + 6) * work_dim1 + 1], &ierr);
}
}
goto L50;
}
}
sep[ks] = scale / max(est,smlnum);
if (pair)
{
sep[ks + 1] = sep[ks];
}
}
if (pair)
{
++ks;
}
L60:
;
}
return 0;
/* End of DTRSNA */
}
/* Subroutine */
int dtbrfs_(char *uplo, char *trans, char *diag, integer *n, integer *kd, integer *nrhs, doublereal *ab, integer *ldab, doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal *ferr, doublereal *berr, doublereal *work, integer *iwork, 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;
/* 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];
extern /* Subroutine */
int dtbmv_(char *, char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), dcopy_(integer *, doublereal *, integer * , doublereal *, integer *), dtbsv_(char *, char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), daxpy_(integer *, doublereal * , doublereal *, integer *, doublereal *, integer *);
logical upper;
extern /* Subroutine */
int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */
int xerbla_(char *, integer *);
logical notran;
char 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 .. */
/* .. */
/* .. 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;
--iwork;
/* 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_("DTBRFS", &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 *)transt = 'T';
}
else
{
*(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 or A**T, depending on TRANS. */
dcopy_(n, &x[j * x_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dtbmv_(uplo, trans, diag, n, kd, &ab[ab_offset], ldab, &work[*n + 1], &c__1);
daxpy_(n, &c_b19, &b[j * b_dim1 + 1], &c__1, &work[*n + 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__)
{
work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1));
/* L20: */
}
if (notran)
{
/* Compute abs(A)*abs(X) + abs(B). */
if (upper)
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
/* 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__)
{
work[i__] += (d__1 = ab[*kd + 1 + i__ - k + k * ab_dim1], abs(d__1)) * xk;
/* L30: */
}
/* L40: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
/* 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__)
{
work[i__] += (d__1 = ab[*kd + 1 + i__ - k + k * ab_dim1], abs(d__1)) * xk;
/* L50: */
}
work[k] += xk;
/* L60: */
}
}
}
else
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
/* 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__)
{
work[i__] += (d__1 = ab[i__ + 1 - k + k * ab_dim1] , abs(d__1)) * xk;
/* L70: */
}
/* L80: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
/* 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__)
{
work[i__] += (d__1 = ab[i__ + 1 - k + k * ab_dim1] , abs(d__1)) * xk;
/* L90: */
}
work[k] += xk;
/* L100: */
}
}
}
}
else
{
/* Compute abs(A**T)*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__)
{
s += (d__1 = ab[*kd + 1 + i__ - k + k * ab_dim1], abs(d__1)) * (d__2 = x[i__ + j * x_dim1], abs(d__2));
/* L110: */
}
work[k] += s;
/* L120: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = (d__1 = x[k + j * x_dim1], abs(d__1));
/* 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__)
{
s += (d__1 = ab[*kd + 1 + i__ - k + k * ab_dim1], abs(d__1)) * (d__2 = x[i__ + j * x_dim1], abs(d__2));
/* L130: */
}
work[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__)
{
s += (d__1 = ab[i__ + 1 - k + k * ab_dim1], abs( d__1)) * (d__2 = x[i__ + j * x_dim1], abs( d__2));
/* L150: */
}
work[k] += s;
/* L160: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = (d__1 = x[k + j * x_dim1], abs(d__1));
/* 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__)
{
s += (d__1 = ab[i__ + 1 - k + k * ab_dim1], abs( d__1)) * (d__2 = x[i__ + j * x_dim1], abs( d__2));
/* L170: */
}
work[k] += s;
/* L180: */
}
}
}
}
s = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (work[i__] > safe2)
{
/* Computing MAX */
d__2 = s;
d__3 = (d__1 = work[*n + i__], abs(d__1)) / work[ i__]; // , expr subst
s = max(d__2,d__3);
}
else
{
/* Computing MAX */
d__2 = s;
d__3 = ((d__1 = work[*n + i__], abs(d__1)) + safe1) / (work[i__] + safe1); // , expr subst
s = max(d__2,d__3);
}
/* 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 DLACN2 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 (work[i__] > safe2)
{
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps * work[i__];
}
else
{
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps * work[i__] + safe1;
}
/* L200: */
}
kase = 0;
L210:
dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], & kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Multiply by diag(W)*inv(op(A)**T). */
dtbsv_(uplo, transt, diag, n, kd, &ab[ab_offset], ldab, &work[ *n + 1], &c__1);
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
work[*n + i__] = work[i__] * work[*n + i__];
/* L220: */
}
}
else
{
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
work[*n + i__] = work[i__] * work[*n + i__];
/* L230: */
}
dtbsv_(uplo, trans, diag, n, kd, &ab[ab_offset], ldab, &work[* n + 1], &c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
/* Computing MAX */
d__2 = lstres;
d__3 = (d__1 = x[i__ + j * x_dim1], abs(d__1)); // , expr subst
lstres = max(d__2,d__3);
/* L240: */
}
if (lstres != 0.)
{
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of DTBRFS */
}
/* Subroutine */ int dsyrfs_(char *uplo, integer *n, integer *nrhs,
doublereal *a, integer *lda, doublereal *af, integer *ldaf, integer *
ipiv, doublereal *b, integer *ldb, doublereal *x, integer *ldx,
doublereal *ferr, doublereal *berr, doublereal *work, integer *iwork,
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;
doublereal d__1, d__2, d__3;
/* 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];
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), daxpy_(integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *);
integer count;
logical upper;
extern /* Subroutine */ int dsymv_(char *, integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
doublereal *, integer *), dlacn2_(integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal lstres;
extern /* Subroutine */ int dsytrs_(char *, integer *, integer *,
doublereal *, integer *, integer *, doublereal *, integer *,
integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSYRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is symmetric indefinite, 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) DOUBLE PRECISION array, dimension (LDA,N) */
/* The symmetric 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) DOUBLE PRECISION array, dimension (LDAF,N) */
/* The factored form of the matrix A. AF 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 DSYTRF. */
/* 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 DSYTRF. */
/* B (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by DSYTRS. */
/* 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* 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;
--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;
--iwork;
/* 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 = -10;
} else if (*ldx < max(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSYRFS", &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 */
dcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dsymv_(uplo, n, &c_b12, &a[a_offset], lda, &x[j * x_dim1 + 1], &c__1,
&c_b14, &work[*n + 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__) {
work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1));
/* L30: */
}
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
work[i__] += (d__1 = a[i__ + k * a_dim1], abs(d__1)) * xk;
s += (d__1 = a[i__ + k * a_dim1], abs(d__1)) * (d__2 = x[
i__ + j * x_dim1], abs(d__2));
/* L40: */
}
work[k] = work[k] + (d__1 = a[k + k * a_dim1], abs(d__1)) *
xk + s;
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
work[k] += (d__1 = a[k + k * a_dim1], abs(d__1)) * xk;
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
work[i__] += (d__1 = a[i__ + k * a_dim1], abs(d__1)) * xk;
s += (d__1 = a[i__ + k * a_dim1], abs(d__1)) * (d__2 = x[
i__ + j * x_dim1], abs(d__2));
/* L60: */
}
work[k] += s;
/* L70: */
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
/* Computing MAX */
d__2 = s, d__3 = (d__1 = work[*n + i__], abs(d__1)) / work[
i__];
s = max(d__2,d__3);
} else {
/* Computing MAX */
d__2 = s, d__3 = ((d__1 = work[*n + i__], abs(d__1)) + safe1)
/ (work[i__] + safe1);
s = max(d__2,d__3);
}
/* 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. */
dsytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[*n
+ 1], n, info);
daxpy_(n, &c_b14, &work[*n + 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 DLACN2 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 (work[i__] > safe2) {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__];
} else {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
dsytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
*n + 1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L110: */
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L120: */
}
dsytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
*n + 1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = lstres, d__3 = (d__1 = x[i__ + j * x_dim1], abs(d__1));
lstres = max(d__2,d__3);
/* L130: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of DSYRFS */
} /* dsyrfs_ */
doublereal dla_gercond__(char *trans, integer *n, doublereal *a, integer *lda,
doublereal *af, integer *ldaf, integer *ipiv, integer *cmode,
doublereal *c__, integer *info, doublereal *work, integer *iwork,
ftnlen trans_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2;
doublereal ret_val, d__1;
/* Local variables */
integer i__, j;
doublereal tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *), xerbla_(char *,
integer *);
doublereal ainvnm;
extern /* Subroutine */ int dgetrs_(char *, integer *, integer *,
doublereal *, integer *, integer *, doublereal *, 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 */
/* ======= */
/* DLA_GERCOND estimates the Skeel condition number of op(A) * op2(C) */
/* where op2 is determined by CMODE as follows */
/* CMODE = 1 op2(C) = C */
/* CMODE = 0 op2(C) = I */
/* CMODE = -1 op2(C) = inv(C) */
/* The Skeel condition number cond(A) = norminf( |inv(A)||A| ) */
/* is computed by computing scaling factors R such that */
/* diag(R)*A*op2(C) is row equilibrated and computing the standard */
/* infinity-norm condition number. */
/* 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDAF,N) */
/* The factors L and U from the factorization */
/* A = P*L*U as computed by DGETRF. */
/* 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 DGETRF; row i of the matrix was interchanged */
/* with row IPIV(i). */
/* CMODE (input) INTEGER */
/* Determines op2(C) in the formula op(A) * op2(C) as follows: */
/* CMODE = 1 op2(C) = C */
/* CMODE = 0 op2(C) = I */
/* CMODE = -1 op2(C) = inv(C) */
/* C (input) DOUBLE PRECISION array, dimension (N) */
/* The vector C in the formula op(A) * op2(C). */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) DOUBLE PRECISION array, dimension (3*N). */
/* Workspace. */
/* IWORK (input) INTEGER array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. 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;
--iwork;
/* 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 (*lda < max(1,*n)) {
*info = -4;
} else if (*ldaf < max(1,*n)) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DLA_GERCOND", &i__1);
return ret_val;
}
if (*n == 0) {
ret_val = 1.;
return ret_val;
}
/* Compute the equilibration matrix R such that */
/* inv(R)*A*C has unit 1-norm. */
if (notrans) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*cmode == 1) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
tmp += (d__1 = a[i__ + j * a_dim1] * c__[j], abs(d__1));
}
} else if (*cmode == 0) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
tmp += (d__1 = a[i__ + j * a_dim1], abs(d__1));
}
} else {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
tmp += (d__1 = a[i__ + j * a_dim1] / c__[j], abs(d__1));
}
}
work[(*n << 1) + i__] = tmp;
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*cmode == 1) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
tmp += (d__1 = a[j + i__ * a_dim1] * c__[j], abs(d__1));
}
} else if (*cmode == 0) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
tmp += (d__1 = a[j + i__ * a_dim1], abs(d__1));
}
} else {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
tmp += (d__1 = a[j + i__ * a_dim1] / c__[j], abs(d__1));
}
}
work[(*n << 1) + i__] = tmp;
}
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.;
kase = 0;
L10:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] *= work[(*n << 1) + i__];
}
if (notrans) {
dgetrs_("No transpose", n, &c__1, &af[af_offset], ldaf, &ipiv[
1], &work[1], n, info);
} else {
dgetrs_("Transpose", n, &c__1, &af[af_offset], ldaf, &ipiv[1],
&work[1], n, info);
}
/* Multiply by inv(C). */
if (*cmode == 1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] /= c__[i__];
}
} else if (*cmode == -1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] *= c__[i__];
}
}
} else {
/* Multiply by inv(C'). */
if (*cmode == 1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] /= c__[i__];
}
} else if (*cmode == -1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] *= c__[i__];
}
}
if (notrans) {
dgetrs_("Transpose", n, &c__1, &af[af_offset], ldaf, &ipiv[1],
&work[1], n, info);
} else {
dgetrs_("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__) {
work[i__] *= work[(*n << 1) + i__];
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
ret_val = 1. / ainvnm;
}
return ret_val;
} /* dla_gercond__ */
/* Subroutine */ int dgecon_(char *norm, integer *n, doublereal *a, integer *
lda, doublereal *anorm, doublereal *rcond, doublereal *work, integer *
iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
doublereal d__1;
/* Local variables */
doublereal sl;
integer ix;
doublereal su;
integer kase, kase1;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int drscl_(integer *, doublereal *, doublereal *,
integer *), dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal ainvnm;
extern /* Subroutine */ int dlatrs_(char *, char *, char *, char *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
doublereal *, integer *);
logical onenrm;
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 DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DGECON estimates the reciprocal of the condition number of a general */
/* real matrix A, in either the 1-norm or the infinity-norm, using */
/* the LU factorization computed by DGETRF. */
/* 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) DOUBLE PRECISION array, dimension (LDA,N) */
/* The factors L and U from the factorization A = P*L*U */
/* as computed by DGETRF. */
/* 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) DOUBLE PRECISION array, dimension (4*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--iwork;
/* 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_("DGECON", &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:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(L). */
dlatrs_("Lower", "No transpose", "Unit", normin, n, &a[a_offset],
lda, &work[1], &sl, &work[(*n << 1) + 1], info);
/* Multiply by inv(U). */
dlatrs_("Upper", "No transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &su, &work[*n * 3 + 1], info);
} else {
/* Multiply by inv(U'). */
dlatrs_("Upper", "Transpose", "Non-unit", normin, n, &a[a_offset],
lda, &work[1], &su, &work[*n * 3 + 1], info);
/* Multiply by inv(L'). */
dlatrs_("Lower", "Transpose", "Unit", normin, n, &a[a_offset],
lda, &work[1], &sl, &work[(*n << 1) + 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 = idamax_(n, &work[1], &c__1);
if (scale < (d__1 = work[ix], abs(d__1)) * smlnum || scale == 0.)
{
goto L20;
}
drscl_(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 DGECON */
} /* dgecon_ */
/* Subroutine */ int dspcon_(char *uplo, integer *n, doublereal *ap, integer *
ipiv, doublereal *anorm, doublereal *rcond, doublereal *work, integer
*iwork, integer *info)
{
/* System generated locals */
integer i__1;
/* Local variables */
integer i__, ip, kase;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *), xerbla_(char *,
integer *);
doublereal ainvnm;
extern /* Subroutine */ int dsptrs_(char *, integer *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *);
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSPCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a real symmetric packed matrix A using the factorization */
/* A = U*D*U**T or A = L*D*L**T computed by DSPTRF. */
/* 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. */
/* AP (input) DOUBLE PRECISION 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 DSPTRF, 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 DSPTRF. */
/* 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) DOUBLE PRECISION array, dimension (2*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--iwork;
--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_("DSPCON", &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__) {
if (ipiv[i__] > 0 && ap[ip] == 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__) {
if (ipiv[i__] > 0 && ap[ip] == 0.) {
return 0;
}
ip = ip + *n - i__ + 1;
/* L20: */
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
/* Multiply by inv(L*D*L') or inv(U*D*U'). */
dsptrs_(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 DSPCON */
} /* dspcon_ */
/* Subroutine */ int dtgsen_(integer *ijob, logical *wantq, logical *wantz,
logical *select, integer *n, doublereal *a, integer *lda, doublereal *
b, integer *ldb, doublereal *alphar, doublereal *alphai, doublereal *
beta, doublereal *q, integer *ldq, doublereal *z__, integer *ldz,
integer *m, doublereal *pl, doublereal *pr, doublereal *dif,
doublereal *work, integer *lwork, integer *iwork, integer *liwork,
integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1,
z_offset, i__1, i__2;
doublereal d__1;
/* Local variables */
integer i__, k, n1, n2, kk, ks, mn2, ijb;
doublereal eps;
integer kase;
logical pair;
integer ierr;
doublereal dsum;
logical swap;
integer isave[3];
logical wantd;
integer lwmin;
logical wantp;
logical wantd1, wantd2;
doublereal dscale, rdscal;
integer liwmin;
doublereal smlnum;
logical lquery;
/* -- LAPACK routine (version 3.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* January 2007 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* DTGSEN reorders the generalized real Schur decomposition of a real */
/* matrix pair (A, B) (in terms of an orthonormal equivalence trans- */
/* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */
/* appears in the leading diagonal blocks of the upper quasi-triangular */
/* matrix A and the upper triangular B. The leading columns of Q and */
/* Z form orthonormal bases of the corresponding left and right eigen- */
/* spaces (deflating subspaces). (A, B) must be in generalized real */
/* Schur canonical form (as returned by DGGES), i.e. A is block upper */
/* triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper */
/* triangular. */
/* DTGSEN also computes the generalized eigenvalues */
/* w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j) */
/* of the reordered matrix pair (A, B). */
/* Optionally, DTGSEN computes the 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) INTEGER */
/* 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 a real eigenvalue w(j), SELECT(j) must be set to */
/* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, */
/* either SELECT(j) or SELECT(j+1) or both must be set to */
/* .TRUE.; a complex conjugate pair of eigenvalues must be */
/* either both included in the cluster or both excluded. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* A (input/output) DOUBLE PRECISION array, dimension(LDA,N) */
/* On entry, the upper quasi-triangular matrix A, with (A, B) in */
/* generalized real 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) DOUBLE PRECISION array, dimension(LDB,N) */
/* On entry, the upper triangular matrix B, with (A, B) in */
/* generalized real 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). */
/* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */
/* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */
/* BETA (output) DOUBLE PRECISION array, dimension (N) */
/* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i */
/* form (S,T) that would result if the 2-by-2 diagonal blocks of */
/* the real generalized Schur form of (A,B) were further reduced */
/* to triangular form using complex unitary transformations. */
/* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if */
/* positive, then the j-th and (j+1)-st eigenvalues are a */
/* complex conjugate pair, with ALPHAI(j+1) negative. */
/* Q (input/output) DOUBLE PRECISION 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 orthogonal */
/* 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; */
/* and if WANTQ = .TRUE., LDQ >= N. */
/* Z (input/output) DOUBLE PRECISION 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 orthogonal */
/* 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 eigen- */
/* spaces (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 */
/* eigenspaces 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 and 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. */
/* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */
/* If IJOB = 0 or 1, DIF is not referenced. */
/* WORK (workspace/output) DOUBLE PRECISION 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. LWORK >= 4*N+16. */
/* If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). */
/* If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 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+6. */
/* If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). */
/* 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 */
/* =============== */
/* DTGSEN first collects the selected eigenvalues by computing */
/* orthogonal 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 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 real Schur */
/* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */
/* reordered generalized real 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 DLATDF), then the parameter */
/* IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF */
/* (IJOB = 2 will be used)). See DTGSYL 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. */
/* ===================================================================== */
/* 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;
--alphar;
--alphai;
--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 = -14;
} else if (*ldz < 1 || *wantz && *ldz < *n) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DTGSEN", &i__1);
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S") / eps;
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;
pair = FALSE_;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (pair) {
pair = FALSE_;
} else {
if (k < *n) {
if (a[k + 1 + k * a_dim1] == 0.) {
if (select[k]) {
++(*m);
}
} else {
pair = TRUE_;
if (select[k] || select[k + 1]) {
*m += 2;
}
}
} else {
if (select[*n]) {
++(*m);
}
}
}
}
if (*ijob == 1 || *ijob == 2 || *ijob == 4) {
/* Computing MAX */
i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), i__2 = (*m <<
1) * (*n - *m);
lwmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n + 6;
liwmin = max(i__1,i__2);
} else if (*ijob == 3 || *ijob == 5) {
/* Computing MAX */
i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), 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 + 6;
liwmin = max(i__1,i__2);
} else {
/* Computing MAX */
i__1 = 1, i__2 = (*n << 2) + 16;
lwmin = max(i__1,i__2);
liwmin = 1;
}
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -22;
} else if (*liwork < liwmin && ! lquery) {
*info = -24;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DTGSEN", &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__) {
dlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum);
dlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum);
}
dif[1] = dscale * sqrt(dsum);
dif[2] = dif[1];
}
goto L60;
}
/* Collect the selected blocks at the top-left corner of (A, B). */
ks = 0;
pair = FALSE_;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (pair) {
pair = FALSE_;
} else {
swap = select[k];
if (k < *n) {
if (a[k + 1 + k * a_dim1] != 0.) {
pair = TRUE_;
swap = swap || select[k + 1];
}
}
if (swap) {
++ks;
/* Swap the K-th block to position KS. */
/* Perform the reordering of diagonal blocks in (A, B) */
/* by orthogonal transformation matrices and update */
/* Q and Z accordingly (if requested): */
kk = k;
if (k != ks) {
dtgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset],
ldb, &q[q_offset], ldq, &z__[z_offset], ldz, &kk,
&ks, &work[1], lwork, &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 L60;
}
if (pair) {
++ks;
}
}
}
}
if (wantp) {
/* Solve generalized Sylvester equation for R and L */
/* and compute PL and PR. */
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 0;
dlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1);
dlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 +
1], &n1);
i__1 = *lwork - (n1 << 1) * n2;
dtgsyl_("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;
dlassq_(&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;
dlassq_(&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 of 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;
dtgsyl_("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 << 1) * n2 + 1], &i__1, &iwork[1], &
ierr);
/* Frobenius norm-based Difl-estimate. */
i__1 = *lwork - (n1 << 1) * n2;
dtgsyl_("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 << 1) * n2 + 1], &i__1, &iwork[1], &
ierr);
} else {
/* Compute 1-norm-based estimates of Difu and Difl using */
/* reversed communication with DLACN2. 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:
dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[1], &kase,
isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation. */
i__1 = *lwork - (n1 << 1) * n2;
dtgsyl_("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 << 1) * n2 +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
dtgsyl_("T", &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 << 1) * n2 +
1], &i__1, &iwork[1], &ierr);
}
goto L40;
}
dif[1] = dscale / dif[1];
/* 1-norm-based estimate of Difl. */
L50:
dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[2], &kase,
isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation. */
i__1 = *lwork - (n1 << 1) * n2;
dtgsyl_("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 << 1) * n2 +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
dtgsyl_("T", &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 << 1) * n2 +
1], &i__1, &iwork[1], &ierr);
}
goto L50;
}
dif[2] = dscale / dif[2];
}
}
L60:
/* Compute generalized eigenvalues of reordered pair (A, B) and */
/* normalize the generalized Schur form. */
pair = FALSE_;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (pair) {
pair = FALSE_;
} else {
if (k < *n) {
if (a[k + 1 + k * a_dim1] != 0.) {
pair = TRUE_;
}
}
if (pair) {
/* Compute the eigenvalue(s) at position K. */
work[1] = a[k + k * a_dim1];
work[2] = a[k + 1 + k * a_dim1];
work[3] = a[k + (k + 1) * a_dim1];
work[4] = a[k + 1 + (k + 1) * a_dim1];
work[5] = b[k + k * b_dim1];
work[6] = b[k + 1 + k * b_dim1];
work[7] = b[k + (k + 1) * b_dim1];
work[8] = b[k + 1 + (k + 1) * b_dim1];
d__1 = smlnum * eps;
dlag2_(&work[1], &c__2, &work[5], &c__2, &d__1, &beta[k], &
beta[k + 1], &alphar[k], &alphar[k + 1], &alphai[k]);
alphai[k + 1] = -alphai[k];
} else {
if (d_sign(&c_b28, &b[k + k * b_dim1]) < 0.) {
/* If B(K,K) is negative, make it positive */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
a[k + i__ * a_dim1] = -a[k + i__ * a_dim1];
b[k + i__ * b_dim1] = -b[k + i__ * b_dim1];
if (*wantq) {
q[i__ + k * q_dim1] = -q[i__ + k * q_dim1];
}
}
}
alphar[k] = a[k + k * a_dim1];
alphai[k] = 0.;
beta[k] = b[k + k * b_dim1];
}
}
}
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
return 0;
/* End of DTGSEN */
} /* dtgsen_ */
/* Subroutine */ int dgtcon_(char *norm, integer *n, doublereal *dl,
doublereal *d__, doublereal *du, doublereal *du2, integer *ipiv,
doublereal *anorm, doublereal *rcond, doublereal *work, integer *
iwork, integer *info)
{
/* System generated locals */
integer i__1;
/* Local variables */
integer i__, kase, kase1;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *), xerbla_(char *,
integer *);
doublereal ainvnm;
logical onenrm;
extern /* Subroutine */ int dgttrs_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DGTCON estimates the reciprocal of the condition number of a real */
/* tridiagonal matrix A using the LU factorization as computed by */
/* DGTTRF. */
/* 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) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) multipliers that define the matrix L from the */
/* LU factorization of A as computed by DGTTRF. */
/* D (input) DOUBLE PRECISION array, dimension (N) */
/* The n diagonal elements of the upper triangular matrix U from */
/* the LU factorization of A. */
/* DU (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) elements of the first superdiagonal of U. */
/* DU2 (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (2*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments. */
/* Parameter adjustments */
--iwork;
--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_("DGTCON", &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__) {
if (d__[i__] == 0.) {
return 0;
}
/* L10: */
}
ainvnm = 0.;
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L20:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(U)*inv(L). */
dgttrs_("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'). */
dgttrs_("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 DGTCON */
} /* dgtcon_ */
int dgerfs_(char *trans, int *n, int *nrhs,
double *a, int *lda, double *af, int *ldaf, int *
ipiv, double *b, int *ldb, double *x, int *ldx,
double *ferr, double *berr, double *work, int *iwork,
int *info)
{
/* System generated locals */
int a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2, i__3;
double d__1, d__2, d__3;
/* Local variables */
int i__, j, k;
double s, xk;
int nz;
double eps;
int kase;
double safe1, safe2;
extern int lsame_(char *, char *);
extern int dgemv_(char *, int *, int *,
double *, double *, int *, double *, int *,
double *, double *, int *);
int isave[3];
extern int dcopy_(int *, double *, int *,
double *, int *), daxpy_(int *, double *,
double *, int *, double *, int *);
int count;
extern int dlacn2_(int *, double *, double *,
int *, double *, int *, int *);
extern double dlamch_(char *);
double safmin;
extern int xerbla_(char *, int *), dgetrs_(
char *, int *, int *, double *, int *, int *,
double *, int *, int *);
int notran;
char transt[1];
double lstres;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DGERFS improves the computed solution to a system of linear */
/* equations 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 = 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 matrices B and X. NRHS >= 0. */
/* A (input) DOUBLE PRECISION array, dimension (LDA,N) */
/* The original N-by-N matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,N). */
/* AF (input) DOUBLE PRECISION array, dimension (LDAF,N) */
/* The factors L and U from the factorization A = P*L*U */
/* as computed by DGETRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= MAX(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from DGETRF; for 1<=i<=N, row i of the */
/* matrix was interchanged with row IPIV(i). */
/* B (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by DGETRS. */
/* 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* 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;
--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;
--iwork;
/* 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 (*lda < MAX(1,*n)) {
*info = -5;
} else if (*ldaf < MAX(1,*n)) {
*info = -7;
} else if (*ldb < MAX(1,*n)) {
*info = -10;
} else if (*ldx < MAX(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGERFS", &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 *)transt = 'T';
} else {
*(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) {
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. */
dcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dgemv_(trans, n, n, &c_b15, &a[a_offset], lda, &x[j * x_dim1 + 1], &
c__1, &c_b17, &work[*n + 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__) {
work[i__] = (d__1 = b[i__ + j * b_dim1], ABS(d__1));
/* L30: */
}
/* Compute ABS(op(A))*ABS(X) + ABS(B). */
if (notran) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
xk = (d__1 = x[k + j * x_dim1], ABS(d__1));
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
work[i__] += (d__1 = a[i__ + k * a_dim1], ABS(d__1)) * xk;
/* L40: */
}
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
s += (d__1 = a[i__ + k * a_dim1], ABS(d__1)) * (d__2 = x[
i__ + j * x_dim1], ABS(d__2));
/* L60: */
}
work[k] += s;
/* L70: */
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
/* Computing MAX */
d__2 = s, d__3 = (d__1 = work[*n + i__], ABS(d__1)) / work[
i__];
s = MAX(d__2,d__3);
} else {
/* Computing MAX */
d__2 = s, d__3 = ((d__1 = work[*n + i__], ABS(d__1)) + safe1)
/ (work[i__] + safe1);
s = MAX(d__2,d__3);
}
/* 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. */
dgetrs_(trans, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[*n
+ 1], n, info);
daxpy_(n, &c_b17, &work[*n + 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 DLACN2 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 (work[i__] > safe2) {
work[i__] = (d__1 = work[*n + i__], ABS(d__1)) + nz * eps *
work[i__];
} else {
work[i__] = (d__1 = work[*n + i__], ABS(d__1)) + nz * eps *
work[i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**T). */
dgetrs_(transt, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &
work[*n + 1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L110: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L120: */
}
dgetrs_(trans, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &
work[*n + 1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = lstres, d__3 = (d__1 = x[i__ + j * x_dim1], ABS(d__1));
lstres = MAX(d__2,d__3);
/* L130: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of DGERFS */
} /* dgerfs_ */
/* Subroutine */
int dsprfs_(char *uplo, integer *n, integer *nrhs, doublereal *ap, doublereal *afp, integer *ipiv, doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal *ferr, doublereal *berr, doublereal *work, integer *iwork, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3;
doublereal d__1, d__2, d__3;
/* 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];
extern /* Subroutine */
int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), daxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *);
integer count;
extern /* Subroutine */
int dspmv_(char *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *);
logical upper;
extern /* Subroutine */
int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */
int xerbla_(char *, integer *);
doublereal lstres;
extern /* Subroutine */
int dsptrs_(char *, integer *, integer *, doublereal *, integer *, doublereal *, 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 .. */
/* .. */
/* .. 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;
--iwork;
/* 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_("DSPRFS", &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 */
dcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dspmv_(uplo, n, &c_b12, &ap[1], &x[j * x_dim1 + 1], &c__1, &c_b14, & work[*n + 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__)
{
work[i__] = (d__1 = b[i__ + j * b_dim1], f2c_abs(d__1));
/* 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.;
xk = (d__1 = x[k + j * x_dim1], f2c_abs(d__1));
ik = kk;
i__3 = k - 1;
for (i__ = 1;
i__ <= i__3;
++i__)
{
work[i__] += (d__1 = ap[ik], f2c_abs(d__1)) * xk;
s += (d__1 = ap[ik], f2c_abs(d__1)) * (d__2 = x[i__ + j * x_dim1], f2c_abs(d__2));
++ik;
/* L40: */
}
work[k] = work[k] + (d__1 = ap[kk + k - 1], f2c_abs(d__1)) * xk + s;
kk += k;
/* L50: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.;
xk = (d__1 = x[k + j * x_dim1], f2c_abs(d__1));
work[k] += (d__1 = ap[kk], f2c_abs(d__1)) * xk;
ik = kk + 1;
i__3 = *n;
for (i__ = k + 1;
i__ <= i__3;
++i__)
{
work[i__] += (d__1 = ap[ik], f2c_abs(d__1)) * xk;
s += (d__1 = ap[ik], f2c_abs(d__1)) * (d__2 = x[i__ + j * x_dim1], f2c_abs(d__2));
++ik;
/* L60: */
}
work[k] += s;
kk += *n - k + 1;
/* L70: */
}
}
s = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (work[i__] > safe2)
{
/* Computing MAX */
d__2 = s;
d__3 = (d__1 = work[*n + i__], f2c_abs(d__1)) / work[ i__]; // , expr subst
s = max(d__2,d__3);
}
else
{
/* Computing MAX */
d__2 = s;
d__3 = ((d__1 = work[*n + i__], f2c_abs(d__1)) + safe1) / (work[i__] + safe1); // , expr subst
s = max(d__2,d__3);
}
/* 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. */
dsptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[*n + 1], n, info);
daxpy_(n, &c_b14, &work[*n + 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 DLACN2 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 (work[i__] > safe2)
{
work[i__] = (d__1 = work[*n + i__], f2c_abs(d__1)) + nz * eps * work[i__];
}
else
{
work[i__] = (d__1 = work[*n + i__], f2c_abs(d__1)) + nz * eps * work[i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], & kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Multiply by diag(W)*inv(A**T). */
dsptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[*n + 1], n, info);
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
work[*n + i__] = work[i__] * work[*n + i__];
/* L110: */
}
}
else if (kase == 2)
{
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
work[*n + i__] = work[i__] * work[*n + i__];
/* L120: */
}
dsptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[*n + 1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
/* Computing MAX */
d__2 = lstres;
d__3 = (d__1 = x[i__ + j * x_dim1], f2c_abs(d__1)); // , expr subst
lstres = max(d__2,d__3);
/* L130: */
}
if (lstres != 0.)
{
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of DSPRFS */
}
/* Subroutine */ int dtrcon_(char *norm, char *uplo, char *diag, integer *n,
doublereal *a, integer *lda, doublereal *rcond, doublereal *work,
integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
doublereal d__1;
/* Local variables */
integer ix, kase, kase1;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int drscl_(integer *, doublereal *, doublereal *,
integer *);
doublereal anorm;
logical upper;
doublereal xnorm;
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */ int xerbla_(char *, integer *);
extern doublereal dlantr_(char *, char *, char *, integer *, integer *,
doublereal *, integer *, doublereal *);
doublereal ainvnm;
extern /* Subroutine */ int dlatrs_(char *, char *, char *, char *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
doublereal *, integer *);
logical onenrm;
char normin[1];
doublereal smlnum;
logical nounit;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DTRCON estimates the reciprocal of the condition number of a */
/* 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. */
/* A (input) DOUBLE PRECISION array, dimension (LDA,N) */
/* The triangular matrix A. If UPLO = 'U', the leading N-by-N */
/* upper triangular part of the array A contains the upper */
/* triangular matrix, and the strictly lower triangular part of */
/* A is not referenced. If UPLO = 'L', the leading N-by-N lower */
/* triangular part of the array A contains the lower triangular */
/* matrix, and the strictly upper triangular part of A is not */
/* referenced. If DIAG = 'U', the diagonal elements of A are */
/* also not referenced and are assumed to be 1. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--iwork;
/* 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 (*lda < max(1,*n)) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DTRCON", &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 = dlantr_(norm, uplo, diag, n, n, &a[a_offset], lda, &work[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:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(A). */
dlatrs_(uplo, "No transpose", diag, normin, n, &a[a_offset],
lda, &work[1], &scale, &work[(*n << 1) + 1], info);
} else {
/* Multiply by inv(A'). */
dlatrs_(uplo, "Transpose", diag, normin, n, &a[a_offset], lda,
&work[1], &scale, &work[(*n << 1) + 1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.) {
ix = idamax_(n, &work[1], &c__1);
xnorm = (d__1 = work[ix], abs(d__1));
if (scale < xnorm * smlnum || scale == 0.) {
goto L20;
}
drscl_(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 DTRCON */
} /* dtrcon_ */
/* Subroutine */ int dtbcon_(char *norm, char *uplo, char *diag, integer *n,
integer *kd, doublereal *ab, integer *ldab, doublereal *rcond,
doublereal *work, integer *iwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1;
doublereal d__1;
/* 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 DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* DTBCON 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--work;
--iwork;
/* 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_("DTBCON", &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 = dlantb_(norm, uplo, diag, n, kd, &ab[ab_offset], ldab, &work[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:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(A). */
dlatbs_(uplo, "No transpose", diag, normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scale, &work[(*n << 1) +
1], info)
;
} else {
/* Multiply by inv(A'). */
dlatbs_(uplo, "Transpose", diag, normin, n, kd, &ab[ab_offset]
, ldab, &work[1], &scale, &work[(*n << 1) + 1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.) {
ix = idamax_(n, &work[1], &c__1);
xnorm = (d__1 = work[ix], abs(d__1));
if (scale < xnorm * smlnum || scale == 0.) {
goto L20;
}
drscl_(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 DTBCON */
} /* dtbcon_ */
/* Subroutine */ int dppcon_(char *uplo, integer *n, doublereal *ap,
doublereal *anorm, doublereal *rcond, doublereal *work, integer *
iwork, integer *info)
{
/* System generated locals */
integer i__1;
doublereal d__1;
/* Local variables */
integer ix, kase;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int drscl_(integer *, doublereal *, doublereal *,
integer *);
logical upper;
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal scalel;
extern integer idamax_(integer *, doublereal *, integer *);
doublereal scaleu;
extern /* Subroutine */ int xerbla_(char *, integer *), dlatps_(
char *, char *, char *, char *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *);
doublereal ainvnm;
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 DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DPPCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a real symmetric positive definite packed matrix using */
/* the Cholesky factorization A = U**T*U or A = L*L**T computed by */
/* DPPTRF. */
/* 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 triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**T*U or A = L*L**T, packed columnwise in a linear */
/* array. The j-th column of U or L is stored in the array AP */
/* as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n. */
/* ANORM (input) DOUBLE PRECISION */
/* The 1-norm (or infinity-norm) of the symmetric 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--iwork;
--work;
--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 = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DPPCON", &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:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (upper) {
/* Multiply by inv(U'). */
dlatps_("Upper", "Transpose", "Non-unit", normin, n, &ap[1], &
work[1], &scalel, &work[(*n << 1) + 1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
dlatps_("Upper", "No transpose", "Non-unit", normin, n, &ap[1], &
work[1], &scaleu, &work[(*n << 1) + 1], info);
} else {
/* Multiply by inv(L). */
dlatps_("Lower", "No transpose", "Non-unit", normin, n, &ap[1], &
work[1], &scalel, &work[(*n << 1) + 1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L'). */
dlatps_("Lower", "Transpose", "Non-unit", normin, n, &ap[1], &
work[1], &scaleu, &work[(*n << 1) + 1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.) {
ix = idamax_(n, &work[1], &c__1);
if (scale < (d__1 = work[ix], abs(d__1)) * smlnum || scale == 0.)
{
goto L20;
}
drscl_(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 DPPCON */
} /* dppcon_ */
int dtrsen_(char *job, char *compq, int *select, int
*n, double *t, int *ldt, double *q, int *ldq,
double *wr, double *wi, int *m, double *s, double
*sep, double *work, int *lwork, int *iwork, int *
liwork, int *info)
{
/* System generated locals */
int q_dim1, q_offset, t_dim1, t_offset, i__1, i__2;
double d__1, d__2;
/* Builtin functions */
double sqrt(double);
/* Local variables */
int k, n1, n2, kk, nn, ks;
double est;
int kase;
int pair;
int ierr;
int swap;
double scale;
extern int lsame_(char *, char *);
int isave[3], lwmin;
int wantq, wants;
double rnorm;
extern int dlacn2_(int *, double *, double *,
int *, double *, int *, int *);
extern double dlange_(char *, int *, int *, double *,
int *, double *);
extern int dlacpy_(char *, int *, int *,
double *, int *, double *, int *),
xerbla_(char *, int *);
int wantbh;
extern int dtrexc_(char *, int *, double *,
int *, double *, int *, int *, int *,
double *, int *);
int liwmin;
int wantsp, lquery;
extern int dtrsyl_(char *, char *, int *, int *,
int *, double *, int *, double *, int *,
double *, int *, double *, int *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DTRSEN reorders the float Schur factorization of a float matrix */
/* A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in */
/* the leading diagonal blocks of the upper quasi-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. */
/* T must be in Schur canonical form (as returned by DHSEQR), that is, */
/* block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each */
/* 2-by-2 diagonal block has its diagonal elemnts equal and its */
/* off-diagonal elements of opposite sign. */
/* 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 */
/* select a float eigenvalue w(j), SELECT(j) must be set to */
/* .TRUE.. To select a complex conjugate pair of eigenvalues */
/* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, */
/* either SELECT(j) or SELECT(j+1) or both must be set to */
/* .TRUE.; a complex conjugate pair of eigenvalues must be */
/* either both included in the cluster or both excluded. */
/* N (input) INTEGER */
/* The order of the matrix T. N >= 0. */
/* T (input/output) DOUBLE PRECISION array, dimension (LDT,N) */
/* On entry, the upper quasi-triangular matrix T, in Schur */
/* canonical form. */
/* On exit, T is overwritten by the reordered matrix T, again in */
/* Schur canonical form, with the selected eigenvalues in the */
/* leading diagonal blocks. */
/* LDT (input) INTEGER */
/* The leading dimension of the array T. LDT >= MAX(1,N). */
/* Q (input/output) DOUBLE PRECISION 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 */
/* orthogonal 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. */
/* WR (output) DOUBLE PRECISION array, dimension (N) */
/* WI (output) DOUBLE PRECISION array, dimension (N) */
/* The float and imaginary parts, respectively, of the reordered */
/* eigenvalues of T. The eigenvalues are stored in the same */
/* order as on the diagonal of T, with WR(i) = T(i,i) and, if */
/* T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and */
/* WI(i+1) = -WI(i). Note that if a complex eigenvalue is */
/* sufficiently ill-conditioned, then its value may differ */
/* significantly from its value before reordering. */
/* 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) DOUBLE PRECISION 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 >= MAX(1,N); */
/* 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. */
/* IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK)) */
/* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. */
/* If JOB = 'N' or 'E', LIWORK >= 1; */
/* if JOB = 'V' or 'B', LIWORK >= MAX(1,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 T failed because some eigenvalues are too */
/* close to separate (the problem is very ill-conditioned); */
/* T may have been partially reordered, and WR and WI */
/* contain the eigenvalues in the same order as in T; S and */
/* SEP (if requested) are set to zero. */
/* Further Details */
/* =============== */
/* DTRSEN first collects the selected eigenvalues by computing an */
/* orthogonal 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 transpose of Z. The first n1 columns */
/* of Z span the specified invariant subspace of T. */
/* If T has been obtained from the float Schur factorization of a matrix */
/* A = Q*T*Q', then the reordered float 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 */
/* ===================================================================== */
/* .. 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;
--wr;
--wi;
--work;
--iwork;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
wantq = lsame_(compq, "V");
*info = 0;
lquery = *lwork == -1;
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 {
/* Set M to the dimension of the specified invariant subspace, */
/* and test LWORK and LIWORK. */
*m = 0;
pair = FALSE;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (pair) {
pair = FALSE;
} else {
if (k < *n) {
if (t[k + 1 + k * t_dim1] == 0.) {
if (select[k]) {
++(*m);
}
} else {
pair = TRUE;
if (select[k] || select[k + 1]) {
*m += 2;
}
}
} else {
if (select[*n]) {
++(*m);
}
}
}
/* L10: */
}
n1 = *m;
n2 = *n - *m;
nn = n1 * n2;
if (wantsp) {
/* Computing MAX */
i__1 = 1, i__2 = nn << 1;
lwmin = MAX(i__1,i__2);
liwmin = MAX(1,nn);
} else if (lsame_(job, "N")) {
lwmin = MAX(1,*n);
liwmin = 1;
} else if (lsame_(job, "E")) {
lwmin = MAX(1,nn);
liwmin = 1;
}
if (*lwork < lwmin && ! lquery) {
*info = -15;
} else if (*liwork < liwmin && ! lquery) {
*info = -17;
}
}
if (*info == 0) {
work[1] = (double) lwmin;
iwork[1] = liwmin;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DTRSEN", &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 = dlange_("1", n, n, &t[t_offset], ldt, &work[1]);
}
goto L40;
}
/* Collect the selected blocks at the top-left corner of T. */
ks = 0;
pair = FALSE;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (pair) {
pair = FALSE;
} else {
swap = select[k];
if (k < *n) {
if (t[k + 1 + k * t_dim1] != 0.) {
pair = TRUE;
swap = swap || select[k + 1];
}
}
if (swap) {
++ks;
/* Swap the K-th block to position KS. */
ierr = 0;
kk = k;
if (k != ks) {
dtrexc_(compq, n, &t[t_offset], ldt, &q[q_offset], ldq, &
kk, &ks, &work[1], &ierr);
}
if (ierr == 1 || ierr == 2) {
/* Blocks too close to swap: exit. */
*info = 1;
if (wants) {
*s = 0.;
}
if (wantsp) {
*sep = 0.;
}
goto L40;
}
if (pair) {
++ks;
}
}
}
/* L20: */
}
if (wants) {
/* Solve Sylvester equation for R: */
/* T11*R - R*T22 = scale*T12 */
dlacpy_("F", &n1, &n2, &t[(n1 + 1) * t_dim1 + 1], ldt, &work[1], &n1);
dtrsyl_("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 = dlange_("F", &n1, &n2, &work[1], &n1, &work[1]);
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:
dlacn2_(&nn, &work[nn + 1], &work[1], &iwork[1], &est, &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve T11*R - R*T22 = scale*X. */
dtrsyl_("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. */
dtrsyl_("T", "T", &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:
/* Store the output eigenvalues in WR and WI. */
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
wr[k] = t[k + k * t_dim1];
wi[k] = 0.;
/* L50: */
}
i__1 = *n - 1;
for (k = 1; k <= i__1; ++k) {
if (t[k + 1 + k * t_dim1] != 0.) {
wi[k] = sqrt((d__1 = t[k + (k + 1) * t_dim1], ABS(d__1))) * sqrt((
d__2 = t[k + 1 + k * t_dim1], ABS(d__2)));
wi[k + 1] = -wi[k];
}
/* L60: */
}
work[1] = (double) lwmin;
iwork[1] = liwmin;
return 0;
/* End of DTRSEN */
} /* dtrsen_ */
doublereal dla_gbrcond__(char *trans, integer *n, integer *kl, integer *ku,
doublereal *ab, integer *ldab, doublereal *afb, integer *ldafb,
integer *ipiv, integer *cmode, doublereal *c__, integer *info,
doublereal *work, integer *iwork, 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;
/* Local variables */
integer i__, j, kd, ke;
doublereal tmp;
integer kase;
integer isave[3];
doublereal ainvnm;
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. -- */
/* Purpose */
/* ======= */
/* DLA_GERCOND Estimates the Skeel condition number of op(A) * op2(C) */
/* where op2 is determined by CMODE as follows */
/* CMODE = 1 op2(C) = C */
/* CMODE = 0 op2(C) = I */
/* CMODE = -1 op2(C) = inv(C) */
/* The Skeel condition number cond(A) = norminf( |inv(A)||A| ) */
/* is computed by computing scaling factors R such that */
/* diag(R)*A*op2(C) is row equilibrated and computing the standard */
/* infinity-norm condition number. */
/* 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDAFB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by DGBTRF. 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 DGBTRF; row i of the matrix was interchanged */
/* with row IPIV(i). */
/* CMODE (input) INTEGER */
/* Determines op2(C) in the formula op(A) * op2(C) as follows: */
/* CMODE = 1 op2(C) = C */
/* CMODE = 0 op2(C) = I */
/* CMODE = -1 op2(C) = inv(C) */
/* C (input) DOUBLE PRECISION array, dimension (N) */
/* The vector C in the formula op(A) * op2(C). */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) DOUBLE PRECISION array, dimension (5*N). */
/* Workspace. */
/* IWORK (input) INTEGER array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* 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;
--c__;
--work;
--iwork;
/* 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_("DLA_GBRCOND", &i__1);
return ret_val;
}
if (*n == 0) {
ret_val = 1.;
return ret_val;
}
/* Compute the equilibration matrix R such that */
/* inv(R)*A*C has unit 1-norm. */
kd = *ku + 1;
ke = *kl + 1;
if (notrans) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*cmode == 1) {
/* 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) {
tmp += (d__1 = ab[kd + i__ - j + j * ab_dim1] * c__[j],
abs(d__1));
}
} else if (*cmode == 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) {
tmp += (d__1 = ab[kd + i__ - j + j * ab_dim1], abs(d__1));
}
} else {
/* 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) {
tmp += (d__1 = ab[kd + i__ - j + j * ab_dim1] / c__[j],
abs(d__1));
}
}
work[(*n << 1) + i__] = tmp;
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.;
if (*cmode == 1) {
/* 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) {
tmp += (d__1 = ab[ke - i__ + j + i__ * ab_dim1] * c__[j],
abs(d__1));
}
} else if (*cmode == 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) {
tmp += (d__1 = ab[ke - i__ + j + i__ * ab_dim1], abs(d__1)
);
}
} else {
/* 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) {
tmp += (d__1 = ab[ke - i__ + j + i__ * ab_dim1] / c__[j],
abs(d__1));
}
}
work[(*n << 1) + i__] = tmp;
}
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.;
kase = 0;
L10:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] *= work[(*n << 1) + i__];
}
if (notrans) {
dgbtrs_("No transpose", n, kl, ku, &c__1, &afb[afb_offset],
ldafb, &ipiv[1], &work[1], n, info);
} else {
dgbtrs_("Transpose", n, kl, ku, &c__1, &afb[afb_offset],
ldafb, &ipiv[1], &work[1], n, info);
}
/* Multiply by inv(C). */
if (*cmode == 1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] /= c__[i__];
}
} else if (*cmode == -1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] *= c__[i__];
}
}
} else {
/* Multiply by inv(C'). */
if (*cmode == 1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] /= c__[i__];
}
} else if (*cmode == -1) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] *= c__[i__];
}
}
if (notrans) {
dgbtrs_("Transpose", n, kl, ku, &c__1, &afb[afb_offset],
ldafb, &ipiv[1], &work[1], n, info);
} else {
dgbtrs_("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__) {
work[i__] *= work[(*n << 1) + i__];
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.) {
ret_val = 1. / ainvnm;
}
return ret_val;
} /* dla_gbrcond__ */
/* Subroutine */ int dtprfs_(char *uplo, char *trans, char *diag, integer *n,
integer *nrhs, doublereal *ap, doublereal *b, integer *ldb,
doublereal *x, integer *ldx, doublereal *ferr, doublereal *berr,
doublereal *work, integer *iwork, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3;
doublereal d__1, d__2, d__3;
/* 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];
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), daxpy_(integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *), dtpmv_(char *,
char *, char *, integer *, doublereal *, doublereal *, integer *);
logical upper;
extern /* Subroutine */ int dtpsv_(char *, char *, char *, integer *,
doublereal *, doublereal *, integer *),
dlacn2_(integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
logical notran;
char transt[1];
logical nounit;
doublereal lstres;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DTPRFS 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 DTPTRS or some other */
/* means before entering this routine. DTPRFS 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 = 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) DOUBLE PRECISION 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)*(2*n-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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (3*N) */
/* IWORK (workspace) INTEGER 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 .. */
/* .. */
/* .. 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;
--iwork;
/* 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_("DTPRFS", &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 *)transt = 'T';
} else {
*(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 or A', depending on TRANS. */
dcopy_(n, &x[j * x_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dtpmv_(uplo, trans, diag, n, &ap[1], &work[*n + 1], &c__1);
daxpy_(n, &c_b19, &b[j * b_dim1 + 1], &c__1, &work[*n + 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__) {
work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1));
/* 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) {
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
i__3 = k;
for (i__ = 1; i__ <= i__3; ++i__) {
work[i__] += (d__1 = ap[kc + i__ - 1], abs(d__1))
* xk;
/* L30: */
}
kc += k;
/* L40: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
work[i__] += (d__1 = ap[kc + i__ - 1], abs(d__1))
* xk;
/* L50: */
}
work[k] += xk;
kc += k;
/* L60: */
}
}
} else {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
i__3 = *n;
for (i__ = k; i__ <= i__3; ++i__) {
work[i__] += (d__1 = ap[kc + i__ - k], abs(d__1))
* xk;
/* L70: */
}
kc = kc + *n - k + 1;
/* L80: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
work[i__] += (d__1 = ap[kc + i__ - k], abs(d__1))
* xk;
/* L90: */
}
work[k] += xk;
kc = kc + *n - k + 1;
/* L100: */
}
}
}
} else {
/* Compute abs(A')*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__) {
s += (d__1 = ap[kc + i__ - 1], abs(d__1)) * (d__2
= x[i__ + j * x_dim1], abs(d__2));
/* L110: */
}
work[k] += s;
kc += k;
/* L120: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = (d__1 = x[k + j * x_dim1], abs(d__1));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
s += (d__1 = ap[kc + i__ - 1], abs(d__1)) * (d__2
= x[i__ + j * x_dim1], abs(d__2));
/* L130: */
}
work[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__) {
s += (d__1 = ap[kc + i__ - k], abs(d__1)) * (d__2
= x[i__ + j * x_dim1], abs(d__2));
/* L150: */
}
work[k] += s;
kc = kc + *n - k + 1;
/* L160: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = (d__1 = x[k + j * x_dim1], abs(d__1));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
s += (d__1 = ap[kc + i__ - k], abs(d__1)) * (d__2
= x[i__ + j * x_dim1], abs(d__2));
/* L170: */
}
work[k] += s;
kc = kc + *n - k + 1;
/* L180: */
}
}
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
/* Computing MAX */
d__2 = s, d__3 = (d__1 = work[*n + i__], abs(d__1)) / work[
i__];
s = max(d__2,d__3);
} else {
/* Computing MAX */
d__2 = s, d__3 = ((d__1 = work[*n + i__], abs(d__1)) + safe1)
/ (work[i__] + safe1);
s = max(d__2,d__3);
}
/* 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 DLACN2 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 (work[i__] > safe2) {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__];
} else {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__] + safe1;
}
/* L200: */
}
kase = 0;
L210:
dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)'). */
dtpsv_(uplo, transt, diag, n, &ap[1], &work[*n + 1], &c__1);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L220: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L230: */
}
dtpsv_(uplo, trans, diag, n, &ap[1], &work[*n + 1], &c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = lstres, d__3 = (d__1 = x[i__ + j * x_dim1], abs(d__1));
lstres = max(d__2,d__3);
/* L240: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of DTPRFS */
} /* dtprfs_ */
/* Subroutine */
int dtrcon_(char *norm, char *uplo, char *diag, integer *n, doublereal *a, integer *lda, doublereal *rcond, doublereal *work, integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
doublereal d__1;
/* Local variables */
integer ix, kase, kase1;
doublereal scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */
int drscl_(integer *, doublereal *, doublereal *, integer *);
doublereal anorm;
logical upper;
doublereal xnorm;
extern /* Subroutine */
int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */
int xerbla_(char *, integer *);
extern doublereal dlantr_(char *, char *, char *, integer *, integer *, doublereal *, integer *, doublereal *);
doublereal ainvnm;
extern /* Subroutine */
int dlatrs_(char *, char *, char *, char *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *);
logical onenrm;
char normin[1];
doublereal smlnum;
logical nounit;
/* -- 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 .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--iwork;
/* 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 (*lda < max(1,*n))
{
*info = -6;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("DTRCON", &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 = dlantr_(norm, uplo, diag, n, n, &a[a_offset], lda, &work[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:
dlacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (kase == kase1)
{
/* Multiply by inv(A). */
dlatrs_(uplo, "No transpose", diag, normin, n, &a[a_offset], lda, &work[1], &scale, &work[(*n << 1) + 1], info);
}
else
{
/* Multiply by inv(A**T). */
dlatrs_(uplo, "Transpose", diag, normin, n, &a[a_offset], lda, &work[1], &scale, &work[(*n << 1) + 1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.)
{
ix = idamax_(n, &work[1], &c__1);
xnorm = (d__1 = work[ix], abs(d__1));
if (scale < xnorm * smlnum || scale == 0.)
{
goto L20;
}
drscl_(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 DTRCON */
}
