/* Subroutine */ int ctrsen_(char *job, char *compq, logical *select, integer
*n, complex *t, integer *ldt, complex *q, integer *ldq, complex *w,
integer *m, real *s, real *sep, complex *work, integer *lwork,
integer *info)
{
/* System generated locals */
integer q_dim1, q_offset, t_dim1, t_offset, i__1, i__2, i__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer k, n1, n2, nn, ks;
real est;
integer kase, ierr;
real scale;
extern logical lsame_(char *, char *);
integer isave[3], lwmin;
logical wantq, wants;
real rnorm;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
real rwork[1];
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), xerbla_(char *,
integer *);
logical wantbh;
extern /* Subroutine */ int ctrexc_(char *, integer *, complex *, integer
*, complex *, integer *, integer *, integer *, integer *);
logical wantsp;
extern /* Subroutine */ int ctrsyl_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, integer *, complex *,
integer *, real *, integer *);
logical lquery;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTRSEN reorders the Schur factorization of a complex matrix */
/* A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in */
/* the leading positions on the diagonal of the upper triangular matrix */
/* T, and the leading columns of Q form an orthonormal basis of the */
/* corresponding right invariant subspace. */
/* Optionally the routine computes the reciprocal condition numbers of */
/* the cluster of eigenvalues and/or the invariant subspace. */
/* Arguments */
/* ========= */
/* JOB (input) CHARACTER*1 */
/* Specifies whether condition numbers are required for the */
/* cluster of eigenvalues (S) or the invariant subspace (SEP): */
/* = 'N': none; */
/* = 'E': for eigenvalues only (S); */
/* = 'V': for invariant subspace only (SEP); */
/* = 'B': for both eigenvalues and invariant subspace (S and */
/* SEP). */
/* COMPQ (input) CHARACTER*1 */
/* = 'V': update the matrix Q of Schur vectors; */
/* = 'N': do not update Q. */
/* SELECT (input) LOGICAL array, dimension (N) */
/* SELECT specifies the eigenvalues in the selected cluster. To */
/* select the j-th eigenvalue, SELECT(j) must be set to .TRUE.. */
/* N (input) INTEGER */
/* The order of the matrix T. N >= 0. */
/* T (input/output) COMPLEX array, dimension (LDT,N) */
/* On entry, the upper triangular matrix T. */
/* On exit, T is overwritten by the reordered matrix T, with the */
/* selected eigenvalues as the leading diagonal elements. */
/* LDT (input) INTEGER */
/* The leading dimension of the array T. LDT >= max(1,N). */
/* Q (input/output) COMPLEX array, dimension (LDQ,N) */
/* On entry, if COMPQ = 'V', the matrix Q of Schur vectors. */
/* On exit, if COMPQ = 'V', Q has been postmultiplied by the */
/* unitary transformation matrix which reorders T; the leading M */
/* columns of Q form an orthonormal basis for the specified */
/* invariant subspace. */
/* If COMPQ = 'N', Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. */
/* LDQ >= 1; and if COMPQ = 'V', LDQ >= N. */
/* W (output) COMPLEX array, dimension (N) */
/* The reordered eigenvalues of T, in the same order as they */
/* appear on the diagonal of T. */
/* M (output) INTEGER */
/* The dimension of the specified invariant subspace. */
/* 0 <= M <= N. */
/* S (output) REAL */
/* 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) REAL */
/* If JOB = 'V' or 'B', SEP is the estimated reciprocal */
/* condition number of the specified invariant subspace. If */
/* M = 0 or N, SEP = norm(T). */
/* If JOB = 'N' or 'E', SEP is not referenced. */
/* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. */
/* If JOB = 'N', LWORK >= 1; */
/* if JOB = 'E', LWORK = max(1,M*(N-M)); */
/* if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)). */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* CTRSEN first collects the selected eigenvalues by computing a unitary */
/* transformation Z to move them to the top left corner of T. In other */
/* words, the selected eigenvalues are the eigenvalues of T11 in: */
/* Z'*T*Z = ( T11 T12 ) n1 */
/* ( 0 T22 ) n2 */
/* n1 n2 */
/* where N = n1+n2 and Z' means the conjugate transpose of Z. The first */
/* n1 columns of Z span the specified invariant subspace of T. */
/* If T has been obtained from the Schur factorization of a matrix */
/* A = Q*T*Q', then the reordered Schur factorization of A is given by */
/* A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the */
/* corresponding invariant subspace of A. */
/* The reciprocal condition number of the average of the eigenvalues of */
/* T11 may be returned in S. S lies between 0 (very badly conditioned) */
/* and 1 (very well conditioned). It is computed as follows. First we */
/* compute R so that */
/* P = ( I R ) n1 */
/* ( 0 0 ) n2 */
/* n1 n2 */
/* is the projector on the invariant subspace associated with T11. */
/* R is the solution of the Sylvester equation: */
/* T11*R - R*T22 = T12. */
/* Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote */
/* the two-norm of M. Then S is computed as the lower bound */
/* (1 + F-norm(R)**2)**(-1/2) */
/* on the reciprocal of 2-norm(P), the true reciprocal condition number. */
/* S cannot underestimate 1 / 2-norm(P) by more than a factor of */
/* sqrt(N). */
/* An approximate error bound for the computed average of the */
/* eigenvalues of T11 is */
/* EPS * norm(T) / S */
/* where EPS is the machine precision. */
/* The reciprocal condition number of the right invariant subspace */
/* spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. */
/* SEP is defined as the separation of T11 and T22: */
/* sep( T11, T22 ) = sigma-min( C ) */
/* where sigma-min(C) is the smallest singular value of the */
/* n1*n2-by-n1*n2 matrix */
/* C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) */
/* I(m) is an m by m identity matrix, and kprod denotes the Kronecker */
/* product. We estimate sigma-min(C) by the reciprocal of an estimate of */
/* the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) */
/* cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). */
/* When SEP is small, small changes in T can cause large changes in */
/* the invariant subspace. An approximate bound on the maximum angular */
/* error in the computed right invariant subspace is */
/* EPS * norm(T) / SEP */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters. */
/* Parameter adjustments */
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--w;
--work;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
wantq = lsame_(compq, "V");
/* Set M to the number of selected eigenvalues. */
*m = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (select[k]) {
++(*m);
}
/* L10: */
}
n1 = *m;
n2 = *n - *m;
nn = n1 * n2;
*info = 0;
lquery = *lwork == -1;
if (wantsp) {
/* Computing MAX */
i__1 = 1, i__2 = nn << 1;
lwmin = max(i__1,i__2);
} else if (lsame_(job, "N")) {
lwmin = 1;
} else if (lsame_(job, "E")) {
lwmin = max(1,nn);
}
if (! lsame_(job, "N") && ! wants && ! wantsp) {
*info = -1;
} else if (! lsame_(compq, "N") && ! wantq) {
*info = -2;
} else if (*n < 0) {
*info = -4;
} else if (*ldt < max(1,*n)) {
*info = -6;
} else if (*ldq < 1 || wantq && *ldq < *n) {
*info = -8;
} else if (*lwork < lwmin && ! lquery) {
*info = -14;
}
if (*info == 0) {
work[1].r = (real) lwmin, work[1].i = 0.f;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTRSEN", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*m == *n || *m == 0) {
if (wants) {
*s = 1.f;
}
if (wantsp) {
*sep = clange_("1", n, n, &t[t_offset], ldt, rwork);
}
goto L40;
}
/* Collect the selected eigenvalues at the top left corner of T. */
ks = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (select[k]) {
++ks;
/* Swap the K-th eigenvalue to position KS. */
if (k != ks) {
ctrexc_(compq, n, &t[t_offset], ldt, &q[q_offset], ldq, &k, &
ks, &ierr);
}
}
/* L20: */
}
if (wants) {
/* Solve the Sylvester equation for R: */
/* T11*R - R*T22 = scale*T12 */
clacpy_("F", &n1, &n2, &t[(n1 + 1) * t_dim1 + 1], ldt, &work[1], &n1);
ctrsyl_("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 = clange_("F", &n1, &n2, &work[1], &n1, rwork);
if (rnorm == 0.f) {
*s = 1.f;
} else {
*s = scale / (sqrt(scale * scale / rnorm + rnorm) * sqrt(rnorm));
}
}
if (wantsp) {
/* Estimate sep(T11,T22). */
est = 0.f;
kase = 0;
L30:
clacn2_(&nn, &work[nn + 1], &work[1], &est, &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve T11*R - R*T22 = scale*X. */
ctrsyl_("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. */
ctrsyl_("C", "C", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 +
1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &
ierr);
}
goto L30;
}
*sep = scale / est;
}
L40:
/* Copy reordered eigenvalues to W. */
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = k;
i__3 = k + k * t_dim1;
w[i__2].r = t[i__3].r, w[i__2].i = t[i__3].i;
/* L50: */
}
work[1].r = (real) lwmin, work[1].i = 0.f;
return 0;
/* End of CTRSEN */
} /* ctrsen_ */
/* Subroutine */ int ctrsna_(char *job, char *howmny, logical *select,
integer *n, complex *t, integer *ldt, complex *vl, integer *ldvl,
complex *vr, integer *ldvr, real *s, real *sep, integer *mm, integer *
m, complex *work, integer *ldwork, real *rwork, integer *info)
{
/* System generated locals */
integer t_dim1, t_offset, vl_dim1, vl_offset, vr_dim1, vr_offset,
work_dim1, work_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2;
complex q__1;
/* Builtin functions */
double c_abs(complex *), r_imag(complex *);
/* Local variables */
integer i__, j, k, ks, ix;
real eps, est;
integer kase, ierr;
complex prod;
real lnrm, rnrm, scale;
extern /* Complex */ VOID cdotc_(complex *, integer *, complex *, integer
*, complex *, integer *);
extern logical lsame_(char *, char *);
integer isave[3];
complex dummy[1];
logical wants;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
real xnorm;
extern doublereal scnrm2_(integer *, complex *, integer *);
extern /* Subroutine */ int slabad_(real *, real *);
extern integer icamax_(integer *, complex *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), xerbla_(char *,
integer *);
real bignum;
logical wantbh;
extern /* Subroutine */ int clatrs_(char *, char *, char *, char *,
integer *, complex *, integer *, complex *, real *, real *,
integer *), csrscl_(integer *,
real *, complex *, integer *), ctrexc_(char *, integer *, complex
*, integer *, complex *, integer *, integer *, integer *, integer
*);
logical somcon;
char normin[1];
real smlnum;
logical wantsp;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTRSNA estimates reciprocal condition numbers for specified */
/* eigenvalues and/or right eigenvectors of a complex upper triangular */
/* matrix T (or of any matrix Q*T*Q**H with Q unitary). */
/* Arguments */
/* ========= */
/* JOB (input) CHARACTER*1 */
/* Specifies whether condition numbers are required for */
/* eigenvalues (S) or eigenvectors (SEP): */
/* = 'E': for eigenvalues only (S); */
/* = 'V': for eigenvectors only (SEP); */
/* = 'B': for both eigenvalues and eigenvectors (S and SEP). */
/* HOWMNY (input) CHARACTER*1 */
/* = 'A': compute condition numbers for all eigenpairs; */
/* = 'S': compute condition numbers for selected eigenpairs */
/* specified by the array SELECT. */
/* SELECT (input) LOGICAL array, dimension (N) */
/* If HOWMNY = 'S', SELECT specifies the eigenpairs for which */
/* condition numbers are required. To select condition numbers */
/* for the j-th eigenpair, SELECT(j) must be set to .TRUE.. */
/* If HOWMNY = 'A', SELECT is not referenced. */
/* N (input) INTEGER */
/* The order of the matrix T. N >= 0. */
/* T (input) COMPLEX array, dimension (LDT,N) */
/* The upper triangular matrix T. */
/* LDT (input) INTEGER */
/* The leading dimension of the array T. LDT >= max(1,N). */
/* VL (input) COMPLEX array, dimension (LDVL,M) */
/* If JOB = 'E' or 'B', VL must contain left eigenvectors of T */
/* (or of any Q*T*Q**H with Q unitary), corresponding to the */
/* eigenpairs specified by HOWMNY and SELECT. The eigenvectors */
/* must be stored in consecutive columns of VL, as returned by */
/* CHSEIN or CTREVC. */
/* If JOB = 'V', VL is not referenced. */
/* LDVL (input) INTEGER */
/* The leading dimension of the array VL. */
/* LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. */
/* VR (input) COMPLEX array, dimension (LDVR,M) */
/* If JOB = 'E' or 'B', VR must contain right eigenvectors of T */
/* (or of any Q*T*Q**H with Q unitary), corresponding to the */
/* eigenpairs specified by HOWMNY and SELECT. The eigenvectors */
/* must be stored in consecutive columns of VR, as returned by */
/* CHSEIN or CTREVC. */
/* If JOB = 'V', VR is not referenced. */
/* LDVR (input) INTEGER */
/* The leading dimension of the array VR. */
/* LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. */
/* S (output) REAL array, dimension (MM) */
/* If JOB = 'E' or 'B', the reciprocal condition numbers of the */
/* selected eigenvalues, stored in consecutive elements of the */
/* array. Thus S(j), SEP(j), and the j-th columns of VL and VR */
/* all correspond to the same eigenpair (but not in general the */
/* j-th eigenpair, unless all eigenpairs are selected). */
/* If JOB = 'V', S is not referenced. */
/* SEP (output) REAL array, dimension (MM) */
/* If JOB = 'V' or 'B', the estimated reciprocal condition */
/* numbers of the selected eigenvectors, stored in consecutive */
/* elements of the array. */
/* If JOB = 'E', SEP is not referenced. */
/* MM (input) INTEGER */
/* The number of elements in the arrays S (if JOB = 'E' or 'B') */
/* and/or SEP (if JOB = 'V' or 'B'). MM >= M. */
/* M (output) INTEGER */
/* The number of elements of the arrays S and/or SEP actually */
/* used to store the estimated condition numbers. */
/* If HOWMNY = 'A', M is set to N. */
/* WORK (workspace) COMPLEX array, dimension (LDWORK,N+6) */
/* If JOB = 'E', WORK is not referenced. */
/* LDWORK (input) INTEGER */
/* The leading dimension of the array WORK. */
/* LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. */
/* RWORK (workspace) REAL array, dimension (N) */
/* If JOB = 'E', RWORK is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* The reciprocal of the condition number of an eigenvalue lambda is */
/* defined as */
/* S(lambda) = |v'*u| / (norm(u)*norm(v)) */
/* where u and v are the right and left eigenvectors of T corresponding */
/* to lambda; v' denotes the conjugate transpose of v, and norm(u) */
/* denotes the Euclidean norm. These reciprocal condition numbers always */
/* lie between zero (very badly conditioned) and one (very well */
/* conditioned). If n = 1, S(lambda) is defined to be 1. */
/* An approximate error bound for a computed eigenvalue W(i) is given by */
/* EPS * norm(T) / S(i) */
/* where EPS is the machine precision. */
/* The reciprocal of the condition number of the right eigenvector u */
/* corresponding to lambda is defined as follows. Suppose */
/* T = ( lambda c ) */
/* ( 0 T22 ) */
/* Then the reciprocal condition number is */
/* SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) */
/* where sigma-min denotes the smallest singular value. We approximate */
/* the smallest singular value by the reciprocal of an estimate of the */
/* one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is */
/* defined to be abs(T(1,1)). */
/* An approximate error bound for a computed right eigenvector VR(i) */
/* is given by */
/* EPS * norm(T) / SEP(i) */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/* Parameter adjustments */
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--s;
--sep;
work_dim1 = *ldwork;
work_offset = 1 + work_dim1;
work -= work_offset;
--rwork;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
somcon = lsame_(howmny, "S");
/* Set M to the number of eigenpairs for which condition numbers are */
/* to be computed. */
if (somcon) {
*m = 0;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (select[j]) {
++(*m);
}
/* L10: */
}
} else {
*m = *n;
}
*info = 0;
if (! wants && ! wantsp) {
*info = -1;
} else if (! lsame_(howmny, "A") && ! somcon) {
*info = -2;
} else if (*n < 0) {
*info = -4;
} else if (*ldt < max(1,*n)) {
*info = -6;
} else if (*ldvl < 1 || wants && *ldvl < *n) {
*info = -8;
} else if (*ldvr < 1 || wants && *ldvr < *n) {
*info = -10;
} else if (*mm < *m) {
*info = -13;
} else if (*ldwork < 1 || wantsp && *ldwork < *n) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTRSNA", &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.f;
}
if (wantsp) {
sep[1] = c_abs(&t[t_dim1 + 1]);
}
return 0;
}
/* Get machine constants */
eps = slamch_("P");
smlnum = slamch_("S") / eps;
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
ks = 1;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (somcon) {
if (! select[k]) {
goto L50;
}
}
if (wants) {
/* Compute the reciprocal condition number of the k-th */
/* eigenvalue. */
cdotc_(&q__1, n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks * vl_dim1 +
1], &c__1);
prod.r = q__1.r, prod.i = q__1.i;
rnrm = scnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
lnrm = scnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
s[ks] = c_abs(&prod) / (rnrm * lnrm);
}
if (wantsp) {
/* Estimate the reciprocal condition number of the k-th */
/* eigenvector. */
/* Copy the matrix T to the array WORK and swap the k-th */
/* diagonal element to the (1,1) position. */
clacpy_("Full", n, n, &t[t_offset], ldt, &work[work_offset],
ldwork);
ctrexc_("No Q", n, &work[work_offset], ldwork, dummy, &c__1, &k, &
c__1, &ierr);
/* Form C = T22 - lambda*I in WORK(2:N,2:N). */
i__2 = *n;
for (i__ = 2; i__ <= i__2; ++i__) {
i__3 = i__ + i__ * work_dim1;
i__4 = i__ + i__ * work_dim1;
i__5 = work_dim1 + 1;
q__1.r = work[i__4].r - work[i__5].r, q__1.i = work[i__4].i -
work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L20: */
}
/* Estimate a lower bound for the 1-norm of inv(C'). The 1st */
/* and (N+1)th columns of WORK are used to store work vectors. */
sep[ks] = 0.f;
est = 0.f;
kase = 0;
*(unsigned char *)normin = 'N';
L30:
i__2 = *n - 1;
clacn2_(&i__2, &work[(*n + 1) * work_dim1 + 1], &work[work_offset]
, &est, &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve C'*x = scale*b */
i__2 = *n - 1;
clatrs_("Upper", "Conjugate transpose", "Nonunit", normin,
&i__2, &work[(work_dim1 << 1) + 2], ldwork, &
work[work_offset], &scale, &rwork[1], &ierr);
} else {
/* Solve C*x = scale*b */
i__2 = *n - 1;
clatrs_("Upper", "No transpose", "Nonunit", normin, &i__2,
&work[(work_dim1 << 1) + 2], ldwork, &work[
work_offset], &scale, &rwork[1], &ierr);
}
*(unsigned char *)normin = 'Y';
if (scale != 1.f) {
/* Multiply by 1/SCALE if doing so will not cause */
/* overflow. */
i__2 = *n - 1;
ix = icamax_(&i__2, &work[work_offset], &c__1);
i__2 = ix + work_dim1;
xnorm = (r__1 = work[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&work[ix + work_dim1]), dabs(r__2));
if (scale < xnorm * smlnum || scale == 0.f) {
goto L40;
}
csrscl_(n, &scale, &work[work_offset], &c__1);
}
goto L30;
}
sep[ks] = 1.f / dmax(est,smlnum);
}
L40:
++ks;
L50:
;
}
return 0;
/* End of CTRSNA */
} /* ctrsna_ */
/* Subroutine */ int cgbcon_(char *norm, integer *n, integer *kl, integer *ku,
complex *ab, integer *ldab, integer *ipiv, real *anorm, real *rcond,
complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1, i__2, i__3;
real r__1, r__2;
complex q__1, q__2;
/* Local variables */
integer j;
complex t;
integer kd, lm, jp, ix, kase, kase1;
real scale;
integer isave[3];
logical lnoti;
real ainvnm;
logical onenrm;
char normin[1];
real smlnum;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* CGBCON estimates the reciprocal of the condition number of a complex */
/* general band matrix A, in either the 1-norm or the infinity-norm, */
/* using the LU factorization computed by CGBTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KL (input) INTEGER */
/* The number of subdiagonals within the band of A. KL >= 0. */
/* KU (input) INTEGER */
/* The number of superdiagonals within the band of A. KU >= 0. */
/* AB (input) COMPLEX array, dimension (LDAB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by CGBTRF. 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) REAL */
/* 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) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL 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;
--ipiv;
--work;
--rwork;
/* Function Body */
*info = 0;
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kl < 0) {
*info = -3;
} else if (*ku < 0) {
*info = -4;
} else if (*ldab < (*kl << 1) + *ku + 1) {
*info = -6;
} else if (*anorm < 0.f) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGBCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm == 0.f) {
return 0;
}
smlnum = slamch_("Safe minimum");
/* Estimate the norm of inv(A). */
ainvnm = 0.f;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kd = *kl + *ku + 1;
lnoti = *kl > 0;
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(L). */
if (lnoti) {
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
i__2 = *kl, i__3 = *n - j;
lm = min(i__2,i__3);
jp = ipiv[j];
i__2 = jp;
t.r = work[i__2].r, t.i = work[i__2].i;
if (jp != j) {
i__2 = jp;
i__3 = j;
work[i__2].r = work[i__3].r, work[i__2].i = work[i__3]
.i;
i__2 = j;
work[i__2].r = t.r, work[i__2].i = t.i;
}
q__1.r = -t.r, q__1.i = -t.i;
caxpy_(&lm, &q__1, &ab[kd + 1 + j * ab_dim1], &c__1, &
work[j + 1], &c__1);
}
}
/* Multiply by inv(U). */
i__1 = *kl + *ku;
clatbs_("Upper", "No transpose", "Non-unit", normin, n, &i__1, &
ab[ab_offset], ldab, &work[1], &scale, &rwork[1], info);
} else {
/* Multiply by inv(U'). */
i__1 = *kl + *ku;
clatbs_("Upper", "Conjugate transpose", "Non-unit", normin, n, &
i__1, &ab[ab_offset], ldab, &work[1], &scale, &rwork[1],
info);
/* Multiply by inv(L'). */
if (lnoti) {
for (j = *n - 1; j >= 1; --j) {
/* Computing MIN */
i__1 = *kl, i__2 = *n - j;
lm = min(i__1,i__2);
i__1 = j;
i__2 = j;
cdotc_(&q__2, &lm, &ab[kd + 1 + j * ab_dim1], &c__1, &
work[j + 1], &c__1);
q__1.r = work[i__2].r - q__2.r, q__1.i = work[i__2].i -
q__2.i;
work[i__1].r = q__1.r, work[i__1].i = q__1.i;
jp = ipiv[j];
if (jp != j) {
i__1 = jp;
t.r = work[i__1].r, t.i = work[i__1].i;
i__1 = jp;
i__2 = j;
work[i__1].r = work[i__2].r, work[i__1].i = work[i__2]
.i;
i__1 = j;
work[i__1].r = t.r, work[i__1].i = t.i;
}
}
}
}
/* Divide X by 1/SCALE if doing so will not cause overflow. */
*(unsigned char *)normin = 'Y';
if (scale != 1.f) {
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((r__1 = work[i__1].r, dabs(r__1)) + (r__2 = r_imag(&
work[ix]), dabs(r__2))) * smlnum || scale == 0.f) {
goto L40;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
L40:
return 0;
/* End of CGBCON */
} /* cgbcon_ */
/* Subroutine */ int cgbrfs_(char *trans, integer *n, integer *kl, integer *
ku, integer *nrhs, complex *ab, integer *ldab, complex *afb, integer *
ldafb, integer *ipiv, complex *b, integer *ldb, complex *x, integer *
ldx, real *ferr, real *berr, complex *work, real *rwork, integer *
info)
{
/* System generated locals */
integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset,
x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5, i__6, i__7;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j, k;
real s;
integer kk;
real xk;
integer nz;
real eps;
integer kase;
real safe1, safe2;
extern /* Subroutine */ int cgbmv_(char *, integer *, integer *, integer *
, integer *, complex *, complex *, integer *, complex *, integer *
, complex *, complex *, integer *);
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *), caxpy_(integer *, complex *, complex *,
integer *, complex *, integer *);
integer count;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
extern doublereal slamch_(char *);
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *), cgbtrs_(
char *, integer *, integer *, integer *, integer *, complex *,
integer *, integer *, complex *, integer *, integer *);
logical notran;
char transn[1], transt[1];
real lstres;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CGBRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is banded, and provides */
/* error bounds and backward error estimates for the solution. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate transpose) */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KL (input) INTEGER */
/* The number of subdiagonals within the band of A. KL >= 0. */
/* KU (input) INTEGER */
/* The number of superdiagonals within the band of A. KU >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* AB (input) COMPLEX array, dimension (LDAB,N) */
/* The original band matrix A, stored in rows 1 to KL+KU+1. */
/* The j-th column of A is stored in the j-th column of the */
/* array AB as follows: */
/* AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KL+KU+1. */
/* AFB (input) COMPLEX array, dimension (LDAFB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by CGBTRF. 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 CGBTRF; for 1<=i<=N, row i of the */
/* matrix was interchanged with row IPIV(i). */
/* B (input) COMPLEX array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) COMPLEX array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by CGBTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) REAL 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) REAL array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Internal Parameters */
/* =================== */
/* ITMAX is the maximum number of steps of iterative refinement. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
notran = lsame_(trans, "N");
if (! notran && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kl < 0) {
*info = -3;
} else if (*ku < 0) {
*info = -4;
} else if (*nrhs < 0) {
*info = -5;
} else if (*ldab < *kl + *ku + 1) {
*info = -7;
} else if (*ldafb < (*kl << 1) + *ku + 1) {
*info = -9;
} else if (*ldb < max(1,*n)) {
*info = -12;
} else if (*ldx < max(1,*n)) {
*info = -14;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGBRFS", &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.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
if (notran) {
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
} else {
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
/* Computing MIN */
i__1 = *kl + *ku + 2, i__2 = *n + 1;
nz = min(i__1,i__2);
eps = slamch_("Epsilon");
safmin = slamch_("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.f;
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. */
ccopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
q__1.r = -1.f, q__1.i = -0.f;
cgbmv_(trans, n, n, kl, ku, &q__1, &ab[ab_offset], ldab, &x[j *
x_dim1 + 1], &c__1, &c_b1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&b[
i__ + j * b_dim1]), dabs(r__2));
/* L30: */
}
/* Compute abs(op(A))*abs(X) + abs(B). */
if (notran) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
kk = *ku + 1 - k;
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x[k + j
* x_dim1]), dabs(r__2));
/* Computing MAX */
i__3 = 1, i__4 = k - *ku;
/* Computing MIN */
i__6 = *n, i__7 = k + *kl;
i__5 = min(i__6,i__7);
for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) {
i__3 = kk + i__ + k * ab_dim1;
rwork[i__] += ((r__1 = ab[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&ab[kk + i__ + k * ab_dim1]), dabs(r__2)))
* xk;
/* L40: */
}
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
kk = *ku + 1 - k;
/* Computing MAX */
i__5 = 1, i__3 = k - *ku;
/* Computing MIN */
i__6 = *n, i__7 = k + *kl;
i__4 = min(i__6,i__7);
for (i__ = max(i__5,i__3); i__ <= i__4; ++i__) {
i__5 = kk + i__ + k * ab_dim1;
i__3 = i__ + j * x_dim1;
s += ((r__1 = ab[i__5].r, dabs(r__1)) + (r__2 = r_imag(&
ab[kk + i__ + k * ab_dim1]), dabs(r__2))) * ((
r__3 = x[i__3].r, dabs(r__3)) + (r__4 = r_imag(&x[
i__ + j * x_dim1]), dabs(r__4)));
/* L60: */
}
rwork[k] += s;
/* L70: */
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__4 = i__;
r__3 = s, r__4 = ((r__1 = work[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2))) / rwork[i__];
s = dmax(r__3,r__4);
} else {
/* Computing MAX */
i__4 = i__;
r__3 = s, r__4 = ((r__1 = work[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__]
+ safe1);
s = dmax(r__3,r__4);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2.f <= lstres && count <= 5) {
/* Update solution and try again. */
cgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1]
, &work[1], n, info);
caxpy_(n, &c_b1, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(op(A)))* */
/* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(op(A)) is the inverse of op(A) */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(op(A))*abs(X) + abs(B) is less than SAFE2. */
/* Use CLACN2 to estimate the infinity-norm of the matrix */
/* inv(op(A)) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__4 = i__;
rwork[i__] = (r__1 = work[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__];
} else {
i__4 = i__;
rwork[i__] = (r__1 = work[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
clacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**H). */
cgbtrs_(transt, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &
ipiv[1], &work[1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__4 = i__;
i__5 = i__;
i__3 = i__;
q__1.r = rwork[i__5] * work[i__3].r, q__1.i = rwork[i__5]
* work[i__3].i;
work[i__4].r = q__1.r, work[i__4].i = q__1.i;
/* L110: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__4 = i__;
i__5 = i__;
i__3 = i__;
q__1.r = rwork[i__5] * work[i__3].r, q__1.i = rwork[i__5]
* work[i__3].i;
work[i__4].r = q__1.r, work[i__4].i = q__1.i;
/* L120: */
}
cgbtrs_(transn, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &
ipiv[1], &work[1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__4 = i__ + j * x_dim1;
r__3 = lstres, r__4 = (r__1 = x[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__2));
lstres = dmax(r__3,r__4);
/* L130: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of CGBRFS */
} /* cgbrfs_ */
/* Subroutine */ int cgtcon_(char *norm, integer *n, complex *dl, complex *
d__, complex *du, complex *du2, integer *ipiv, real *anorm, real *
rcond, complex *work, integer *info)
{
/* System generated locals */
integer i__1, i__2;
/* Local variables */
integer i__, kase, kase1;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *), xerbla_(char *, integer *);
real ainvnm;
logical onenrm;
extern /* Subroutine */ int cgttrs_(char *, integer *, integer *, complex
*, complex *, complex *, complex *, integer *, complex *, integer
*, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CGTCON estimates the reciprocal of the condition number of a complex */
/* tridiagonal matrix A using the LU factorization as computed by */
/* CGTTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* DL (input) COMPLEX array, dimension (N-1) */
/* The (n-1) multipliers that define the matrix L from the */
/* LU factorization of A as computed by CGTTRF. */
/* D (input) COMPLEX array, dimension (N) */
/* The n diagonal elements of the upper triangular matrix U from */
/* the LU factorization of A. */
/* DU (input) COMPLEX array, dimension (N-1) */
/* The (n-1) elements of the first superdiagonal of U. */
/* DU2 (input) COMPLEX 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) REAL */
/* 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) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments. */
/* Parameter adjustments */
--work;
--ipiv;
--du2;
--du;
--d__;
--dl;
/* Function Body */
*info = 0;
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*anorm < 0.f) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGTCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm == 0.f) {
return 0;
}
/* Check that D(1:N) is non-zero. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
if (d__[i__2].r == 0.f && d__[i__2].i == 0.f) {
return 0;
}
/* L10: */
}
ainvnm = 0.f;
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L20:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(U)*inv(L). */
cgttrs_("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'). */
cgttrs_("Conjugate transpose", n, &c__1, &dl[1], &d__[1], &du[1],
&du2[1], &ipiv[1], &work[1], n, info);
}
goto L20;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
return 0;
/* End of CGTCON */
} /* cgtcon_ */
/* Subroutine */ int cppcon_(char *uplo, integer *n, complex *ap, real *anorm,
real *rcond, complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer i__1;
real r__1, r__2;
/* Local variables */
integer ix, kase;
real scale;
integer isave[3];
logical upper;
real scalel;
real scaleu;
real ainvnm;
char normin[1];
real smlnum;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* CPPCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a complex Hermitian positive definite packed matrix using */
/* the Cholesky factorization A = U**H*U or A = L*L**H computed by */
/* CPPTRF. */
/* 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) COMPLEX array, dimension (N*(N+1)/2) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**H*U or A = L*L**H, 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) REAL */
/* The 1-norm (or infinity-norm) of the Hermitian matrix A. */
/* RCOND (output) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
--rwork;
--work;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*anorm < 0.f) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPPCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm == 0.f) {
return 0;
}
smlnum = slamch_("Safe minimum");
/* Estimate the 1-norm of the inverse. */
kase = 0;
*(unsigned char *)normin = 'N';
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (upper) {
/* Multiply by inv(U'). */
clatps_("Upper", "Conjugate transpose", "Non-unit", normin, n, &
ap[1], &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
clatps_("Upper", "No transpose", "Non-unit", normin, n, &ap[1], &
work[1], &scaleu, &rwork[1], info);
} else {
/* Multiply by inv(L). */
clatps_("Lower", "No transpose", "Non-unit", normin, n, &ap[1], &
work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L'). */
clatps_("Lower", "Conjugate transpose", "Non-unit", normin, n, &
ap[1], &work[1], &scaleu, &rwork[1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.f) {
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((r__1 = work[i__1].r, dabs(r__1)) + (r__2 = r_imag(&
work[ix]), dabs(r__2))) * smlnum || scale == 0.f) {
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
L20:
return 0;
/* End of CPPCON */
} /* cppcon_ */
/* Subroutine */ int ctprfs_(char *uplo, char *trans, char *diag, integer *n,
integer *nrhs, complex *ap, complex *b, integer *ldb, complex *x,
integer *ldx, real *ferr, real *berr, complex *work, real *rwork,
integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j, k;
real s;
integer kc;
real xk;
integer nz;
real eps;
integer kase;
real safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *), caxpy_(integer *, complex *, complex *,
integer *, complex *, integer *), ctpmv_(char *, char *, char *,
integer *, complex *, complex *, integer *);
logical upper;
extern /* Subroutine */ int ctpsv_(char *, char *, char *, integer *,
complex *, complex *, integer *), clacn2_(
integer *, complex *, complex *, real *, integer *, integer *);
extern doublereal slamch_(char *);
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
logical notran;
char transn[1], transt[1];
logical nounit;
real lstres;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTPRFS 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 CTPTRS or some other */
/* means before entering this routine. CTPRFS does not do iterative */
/* refinement because doing so cannot improve the backward error. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': A is upper triangular; */
/* = 'L': A is lower triangular. */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate transpose) */
/* DIAG (input) CHARACTER*1 */
/* = 'N': A is non-unit triangular; */
/* = 'U': A is unit triangular. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* AP (input) COMPLEX array, dimension (N*(N+1)/2) */
/* The upper or lower triangular matrix A, packed columnwise in */
/* a linear array. The j-th column of A is stored in the array */
/* AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */
/* If DIAG = 'U', the diagonal elements of A are not referenced */
/* and are assumed to be 1. */
/* B (input) COMPLEX array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input) COMPLEX array, dimension (LDX,NRHS) */
/* The solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) REAL 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) REAL array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
notran = lsame_(trans, "N");
nounit = lsame_(diag, "N");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "T") && !
lsame_(trans, "C")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*nrhs < 0) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -8;
} else if (*ldx < max(1,*n)) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTPRFS", &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.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
if (notran) {
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
} else {
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
/* Compute residual R = B - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
ccopy_(n, &x[j * x_dim1 + 1], &c__1, &work[1], &c__1);
ctpmv_(uplo, trans, diag, n, &ap[1], &work[1], &c__1);
q__1.r = -1.f, q__1.i = -0.f;
caxpy_(n, &q__1, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&b[
i__ + j * b_dim1]), dabs(r__2));
/* L20: */
}
if (notran) {
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
i__3 = k;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&ap[kc + i__ - 1]), dabs(
r__2))) * xk;
/* L30: */
}
kc += k;
/* L40: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&ap[kc + i__ - 1]), dabs(
r__2))) * xk;
/* L50: */
}
rwork[k] += xk;
kc += k;
/* L60: */
}
}
} else {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
i__3 = *n;
for (i__ = k; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&ap[kc + i__ - k]), dabs(
r__2))) * xk;
/* L70: */
}
kc = kc + *n - k + 1;
/* L80: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&ap[kc + i__ - k]), dabs(
r__2))) * xk;
/* L90: */
}
rwork[k] += xk;
kc = kc + *n - k + 1;
/* L100: */
}
}
}
} else {
/* Compute abs(A**H)*abs(X) + abs(B). */
if (upper) {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = k;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&ap[kc + i__ - 1]), dabs(r__2))) *
((r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__4)))
;
/* L110: */
}
rwork[k] += s;
kc += k;
/* L120: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
s = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - 1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&ap[kc + i__ - 1]), dabs(r__2))) *
((r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__4)))
;
/* L130: */
}
rwork[k] += s;
kc += k;
/* L140: */
}
}
} else {
kc = 1;
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = *n;
for (i__ = k; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
i__5 = i__ + j * x_dim1;
s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&ap[kc + i__ - k]), dabs(r__2))) *
((r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__4)))
;
/* L150: */
}
rwork[k] += s;
kc = kc + *n - k + 1;
/* L160: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
s = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = kc + i__ - k;
i__5 = i__ + j * x_dim1;
s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&ap[kc + i__ - k]), dabs(r__2))) *
((r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__4)))
;
/* L170: */
}
rwork[k] += s;
kc = kc + *n - k + 1;
/* L180: */
}
}
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2))) / rwork[i__];
s = dmax(r__3,r__4);
} else {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__]
+ safe1);
s = dmax(r__3,r__4);
}
/* L190: */
}
berr[j] = s;
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(op(A)))* */
/* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(op(A)) is the inverse of op(A) */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(op(A))*abs(X) + abs(B) is less than SAFE2. */
/* Use CLACN2 to estimate the infinity-norm of the matrix */
/* inv(op(A)) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__];
} else {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__] + safe1;
}
/* L200: */
}
kase = 0;
L210:
clacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**H). */
ctpsv_(uplo, transt, diag, n, &ap[1], &work[1], &c__1);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L220: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L230: */
}
ctpsv_(uplo, transn, diag, n, &ap[1], &work[1], &c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = i__ + j * x_dim1;
r__3 = lstres, r__4 = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__2));
lstres = dmax(r__3,r__4);
/* L240: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of CTPRFS */
} /* ctprfs_ */
/* Subroutine */ int ctgsen_(integer *ijob, logical *wantq, logical *wantz,
logical *select, integer *n, complex *a, integer *lda, complex *b,
integer *ldb, complex *alpha, complex *beta, complex *q, integer *ldq,
complex *z__, integer *ldz, integer *m, real *pl, real *pr, real *
dif, complex *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, i__3;
complex q__1, q__2;
/* Builtin functions */
double sqrt(doublereal), c_abs(complex *);
void r_cnjg(complex *, complex *);
/* Local variables */
integer i__, k, n1, n2, ks, mn2, ijb, kase, ierr;
real dsum;
logical swap;
complex temp1, temp2;
extern /* Subroutine */ int cscal_(integer *, complex *, complex *,
integer *);
integer isave[3];
logical wantd;
integer lwmin;
logical wantp;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
logical wantd1, wantd2;
real dscale;
extern doublereal slamch_(char *);
real rdscal;
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *);
real safmin;
extern /* Subroutine */ int ctgexc_(logical *, logical *, integer *,
complex *, integer *, complex *, integer *, complex *, integer *,
complex *, integer *, integer *, integer *, integer *), xerbla_(
char *, integer *), classq_(integer *, complex *, integer
*, real *, real *);
integer liwmin;
extern /* Subroutine */ int ctgsyl_(char *, integer *, integer *, integer
*, complex *, integer *, complex *, integer *, complex *, integer
*, complex *, integer *, complex *, integer *, complex *, integer
*, real *, real *, complex *, integer *, integer *, integer *);
logical lquery;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* January 2007 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTGSEN reorders the generalized Schur decomposition of a complex */
/* matrix pair (A, B) (in terms of an unitary equivalence trans- */
/* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */
/* appears in the leading diagonal blocks of the pair (A,B). The leading */
/* columns of Q and Z form unitary bases of the corresponding left and */
/* right eigenspaces (deflating subspaces). (A, B) must be in */
/* generalized Schur canonical form, that is, A and B are both upper */
/* triangular. */
/* CTGSEN also computes the generalized eigenvalues */
/* w(j)= ALPHA(j) / BETA(j) */
/* of the reordered matrix pair (A, B). */
/* Optionally, the routine computes estimates of reciprocal condition */
/* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */
/* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */
/* between the matrix pairs (A11, B11) and (A22,B22) that correspond to */
/* the selected cluster and the eigenvalues outside the cluster, resp., */
/* and norms of "projections" onto left and right eigenspaces w.r.t. */
/* the selected cluster in the (1,1)-block. */
/* Arguments */
/* ========= */
/* IJOB (input) 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 an eigenvalue w(j), SELECT(j) must be set to */
/* .TRUE.. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* A (input/output) COMPLEX array, dimension(LDA,N) */
/* On entry, the upper triangular matrix A, in generalized */
/* Schur canonical form. */
/* On exit, A is overwritten by the reordered matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* B (input/output) COMPLEX array, dimension(LDB,N) */
/* On entry, the upper triangular matrix B, in generalized */
/* Schur canonical form. */
/* On exit, B is overwritten by the reordered matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* ALPHA (output) COMPLEX array, dimension (N) */
/* BETA (output) COMPLEX array, dimension (N) */
/* The diagonal elements of A and B, respectively, */
/* when the pair (A,B) has been reduced to generalized Schur */
/* form. ALPHA(i)/BETA(i) i=1,...,N are the generalized */
/* eigenvalues. */
/* Q (input/output) COMPLEX array, dimension (LDQ,N) */
/* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */
/* On exit, Q has been postmultiplied by the left unitary */
/* transformation matrix which reorder (A, B); The leading M */
/* columns of Q form orthonormal bases for the specified pair of */
/* left eigenspaces (deflating subspaces). */
/* If WANTQ = .FALSE., Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= 1. */
/* If WANTQ = .TRUE., LDQ >= N. */
/* Z (input/output) COMPLEX array, dimension (LDZ,N) */
/* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */
/* On exit, Z has been postmultiplied by the left unitary */
/* transformation matrix which reorder (A, B); The leading M */
/* columns of Z form orthonormal bases for the specified pair of */
/* left eigenspaces (deflating subspaces). */
/* If WANTZ = .FALSE., Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1. */
/* If WANTZ = .TRUE., LDZ >= N. */
/* M (output) INTEGER */
/* The dimension of the specified pair of left and right */
/* eigenspaces, (deflating subspaces) 0 <= M <= N. */
/* PL (output) REAL */
/* PR (output) REAL */
/* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */
/* reciprocal of the norm of "projections" onto left and right */
/* eigenspace with respect to the selected cluster. */
/* 0 < PL, PR <= 1. */
/* If M = 0 or M = N, PL = PR = 1. */
/* If IJOB = 0, 2 or 3 PL, PR are not referenced. */
/* DIF (output) REAL array, dimension (2). */
/* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */
/* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */
/* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */
/* estimates of Difu and Difl, computed using reversed */
/* communication with CLACN2. */
/* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */
/* If IJOB = 0 or 1, DIF is not referenced. */
/* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) */
/* IF IJOB = 0, WORK is not referenced. Otherwise, */
/* on exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= 1 */
/* If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) */
/* If IJOB = 3 or 5, LWORK >= 4*M*(N-M) */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */
/* IF IJOB = 0, IWORK is not referenced. Otherwise, */
/* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. LIWORK >= 1. */
/* If IJOB = 1, 2 or 4, LIWORK >= N+2; */
/* If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); */
/* If LIWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the optimal size of the IWORK array, */
/* returns this value as the first entry of the IWORK array, and */
/* no error message related to LIWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* =0: Successful exit. */
/* <0: If INFO = -i, the i-th argument had an illegal value. */
/* =1: Reordering of (A, B) failed because the transformed */
/* matrix pair (A, B) would be too far from generalized */
/* Schur form; the problem is very ill-conditioned. */
/* (A, B) may have been partially reordered. */
/* If requested, 0 is returned in DIF(*), PL and PR. */
/* Further Details */
/* =============== */
/* CTGSEN first collects the selected eigenvalues by computing unitary */
/* U and W that move them to the top left corner of (A, B). In other */
/* words, the selected eigenvalues are the eigenvalues of (A11, B11) in */
/* U'*(A, B)*W = (A11 A12) (B11 B12) n1 */
/* ( 0 A22),( 0 B22) n2 */
/* n1 n2 n1 n2 */
/* where N = n1+n2 and U' means the conjugate transpose of U. The first */
/* n1 columns of U and W span the specified pair of left and right */
/* eigenspaces (deflating subspaces) of (A, B). */
/* If (A, B) has been obtained from the generalized real Schur */
/* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */
/* reordered generalized Schur form of (C, D) is given by */
/* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', */
/* and the first n1 columns of Q*U and Z*W span the corresponding */
/* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */
/* Note that if the selected eigenvalue is sufficiently ill-conditioned, */
/* then its value may differ significantly from its value before */
/* reordering. */
/* The reciprocal condition numbers of the left and right eigenspaces */
/* spanned by the first n1 columns of U and W (or Q*U and Z*W) may */
/* be returned in DIF(1:2), corresponding to Difu and Difl, resp. */
/* The Difu and Difl are defined as: */
/* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) */
/* and */
/* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */
/* where sigma-min(Zu) is the smallest singular value of the */
/* (2*n1*n2)-by-(2*n1*n2) matrix */
/* Zu = [ kron(In2, A11) -kron(A22', In1) ] */
/* [ kron(In2, B11) -kron(B22', In1) ]. */
/* Here, Inx is the identity matrix of size nx and A22' is the */
/* transpose of A22. kron(X, Y) is the Kronecker product between */
/* the matrices X and Y. */
/* When DIF(2) is small, small changes in (A, B) can cause large changes */
/* in the deflating subspace. An approximate (asymptotic) bound on the */
/* maximum angular error in the computed deflating subspaces is */
/* EPS * norm((A, B)) / DIF(2), */
/* where EPS is the machine precision. */
/* The reciprocal norm of the projectors on the left and right */
/* eigenspaces associated with (A11, B11) may be returned in PL and PR. */
/* They are computed as follows. First we compute L and R so that */
/* P*(A, B)*Q is block diagonal, where */
/* P = ( I -L ) n1 Q = ( I R ) n1 */
/* ( 0 I ) n2 and ( 0 I ) n2 */
/* n1 n2 n1 n2 */
/* and (L, R) is the solution to the generalized Sylvester equation */
/* A11*R - L*A22 = -A12 */
/* B11*R - L*B22 = -B12 */
/* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */
/* An approximate (asymptotic) bound on the average absolute error of */
/* the selected eigenvalues is */
/* EPS * norm((A, B)) / PL. */
/* There are also global error bounds which valid for perturbations up */
/* to a certain restriction: A lower bound (x) on the smallest */
/* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */
/* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */
/* (i.e. (A + E, B + F), is */
/* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). */
/* An approximate bound on x can be computed from DIF(1:2), PL and PR. */
/* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */
/* (L', R') and unperturbed (L, R) left and right deflating subspaces */
/* associated with the selected cluster in the (1,1)-blocks can be */
/* bounded as */
/* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */
/* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */
/* See LAPACK User's Guide section 4.11 or the following references */
/* for more information. */
/* Note that if the default method for computing the Frobenius-norm- */
/* based estimate DIF is not wanted (see CLATDF), then the parameter */
/* IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF */
/* (IJOB = 2 will be used)). See CTGSYL for more details. */
/* Based on contributions by */
/* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */
/* Umea University, S-901 87 Umea, Sweden. */
/* References */
/* ========== */
/* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */
/* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */
/* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */
/* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */
/* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */
/* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */
/* Estimation: Theory, Algorithms and Software, Report */
/* UMINF - 94.04, Department of Computing Science, Umea University, */
/* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. */
/* To appear in Numerical Algorithms, 1996. */
/* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */
/* for Solving the Generalized Sylvester Equation and Estimating the */
/* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */
/* Department of Computing Science, Umea University, S-901 87 Umea, */
/* Sweden, December 1993, Revised April 1994, Also as LAPACK working */
/* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */
/* 1996. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/* Parameter adjustments */
--select;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--alpha;
--beta;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--dif;
--work;
--iwork;
/* Function Body */
*info = 0;
lquery = *lwork == -1 || *liwork == -1;
if (*ijob < 0 || *ijob > 5) {
*info = -1;
} else if (*n < 0) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldq < 1 || *wantq && *ldq < *n) {
*info = -13;
} else if (*ldz < 1 || *wantz && *ldz < *n) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTGSEN", &i__1);
return 0;
}
ierr = 0;
wantp = *ijob == 1 || *ijob >= 4;
wantd1 = *ijob == 2 || *ijob == 4;
wantd2 = *ijob == 3 || *ijob == 5;
wantd = wantd1 || wantd2;
/* Set M to the dimension of the specified pair of deflating */
/* subspaces. */
*m = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = k;
i__3 = k + k * a_dim1;
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = k + k * b_dim1;
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
if (k < *n) {
if (select[k]) {
++(*m);
}
} else {
if (select[*n]) {
++(*m);
}
}
/* L10: */
}
if (*ijob == 1 || *ijob == 2 || *ijob == 4) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m);
lwmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n + 2;
liwmin = max(i__1,i__2);
} else if (*ijob == 3 || *ijob == 5) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 2) * (*n - *m);
lwmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = max(i__1,i__2), i__2 =
*n + 2;
liwmin = max(i__1,i__2);
} else {
lwmin = 1;
liwmin = 1;
}
work[1].r = (real) lwmin, work[1].i = 0.f;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -21;
} else if (*liwork < liwmin && ! lquery) {
*info = -23;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTGSEN", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible. */
if (*m == *n || *m == 0) {
if (wantp) {
*pl = 1.f;
*pr = 1.f;
}
if (wantd) {
dscale = 0.f;
dsum = 1.f;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
classq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum);
classq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum);
/* L20: */
}
dif[1] = dscale * sqrt(dsum);
dif[2] = dif[1];
}
goto L70;
}
/* Get machine constant */
safmin = slamch_("S");
/* Collect the selected blocks at the top-left corner of (A, B). */
ks = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
swap = select[k];
if (swap) {
++ks;
/* Swap the K-th block to position KS. Compute unitary Q */
/* and Z that will swap adjacent diagonal blocks in (A, B). */
if (k != ks) {
ctgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb,
&q[q_offset], ldq, &z__[z_offset], ldz, &k, &ks, &
ierr);
}
if (ierr > 0) {
/* Swap is rejected: exit. */
*info = 1;
if (wantp) {
*pl = 0.f;
*pr = 0.f;
}
if (wantd) {
dif[1] = 0.f;
dif[2] = 0.f;
}
goto L70;
}
}
/* L30: */
}
if (wantp) {
/* Solve generalized Sylvester equation for R and L: */
/* A11 * R - L * A22 = A12 */
/* B11 * R - L * B22 = B12 */
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
clacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1);
clacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 +
1], &n1);
ijb = 0;
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("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.f;
dsum = 1.f;
i__1 = n1 * n2;
classq_(&i__1, &work[1], &c__1, &rdscal, &dsum);
*pl = rdscal * sqrt(dsum);
if (*pl == 0.f) {
*pl = 1.f;
} else {
*pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl));
}
rdscal = 0.f;
dsum = 1.f;
i__1 = n1 * n2;
classq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum);
*pr = rdscal * sqrt(dsum);
if (*pr == 0.f) {
*pr = 1.f;
} else {
*pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr));
}
}
if (wantd) {
/* Compute estimates Difu and Difl. */
if (wantd1) {
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 3;
/* Frobenius norm-based Difu estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ *
a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ +
i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &
dif[1], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
ierr);
/* Frobenius norm-based Difl estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[
a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1],
ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale,
&dif[2], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
ierr);
} else {
/* Compute 1-norm-based estimates of Difu and Difl using */
/* reversed communication with CLACN2. 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:
clacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[1], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("C", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
}
goto L40;
}
dif[1] = dscale / dif[1];
/* 1-norm-based estimate of Difl. */
L50:
clacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[2], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
&a[a_offset], lda, &work[1], &n2, &b[i__ + i__ *
b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 +
1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("C", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
&a[a_offset], lda, &work[1], &n2, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
}
goto L50;
}
dif[2] = dscale / dif[2];
}
}
/* If B(K,K) is complex, make it real and positive (normalization */
/* of the generalized Schur form) and Store the generalized */
/* eigenvalues of reordered pair (A, B) */
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
dscale = c_abs(&b[k + k * b_dim1]);
if (dscale > safmin) {
i__2 = k + k * b_dim1;
q__2.r = b[i__2].r / dscale, q__2.i = b[i__2].i / dscale;
r_cnjg(&q__1, &q__2);
temp1.r = q__1.r, temp1.i = q__1.i;
i__2 = k + k * b_dim1;
q__1.r = b[i__2].r / dscale, q__1.i = b[i__2].i / dscale;
temp2.r = q__1.r, temp2.i = q__1.i;
i__2 = k + k * b_dim1;
b[i__2].r = dscale, b[i__2].i = 0.f;
i__2 = *n - k;
cscal_(&i__2, &temp1, &b[k + (k + 1) * b_dim1], ldb);
i__2 = *n - k + 1;
cscal_(&i__2, &temp1, &a[k + k * a_dim1], lda);
if (*wantq) {
cscal_(n, &temp2, &q[k * q_dim1 + 1], &c__1);
}
} else {
i__2 = k + k * b_dim1;
b[i__2].r = 0.f, b[i__2].i = 0.f;
}
i__2 = k;
i__3 = k + k * a_dim1;
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = k + k * b_dim1;
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
/* L60: */
}
L70:
work[1].r = (real) lwmin, work[1].i = 0.f;
iwork[1] = liwmin;
return 0;
/* End of CTGSEN */
} /* ctgsen_ */
/* Subroutine */
int cpocon_(char *uplo, integer *n, complex *a, integer *lda, real *anorm, real *rcond, complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
real r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer ix, kase;
real scale;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */
int clacn2_(integer *, complex *, complex *, real *, integer *, integer *);
extern integer icamax_(integer *, complex *, integer *);
real scalel;
extern real slamch_(char *);
real scaleu;
extern /* Subroutine */
int xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */
int clatrs_(char *, char *, char *, char *, integer *, complex *, integer *, complex *, real *, real *, integer *), csrscl_(integer *, real *, complex *, integer *);
char normin[1];
real smlnum;
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*n))
{
*info = -4;
}
else if (*anorm < 0.f)
{
*info = -5;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("CPOCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0)
{
*rcond = 1.f;
return 0;
}
else if (*anorm == 0.f)
{
return 0;
}
smlnum = slamch_("Safe minimum");
/* Estimate the 1-norm of inv(A). */
kase = 0;
*(unsigned char *)normin = 'N';
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (upper)
{
/* Multiply by inv(U**H). */
clatrs_("Upper", "Conjugate transpose", "Non-unit", normin, n, &a[ a_offset], lda, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
clatrs_("Upper", "No transpose", "Non-unit", normin, n, &a[ a_offset], lda, &work[1], &scaleu, &rwork[1], info);
}
else
{
/* Multiply by inv(L). */
clatrs_("Lower", "No transpose", "Non-unit", normin, n, &a[ a_offset], lda, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L**H). */
clatrs_("Lower", "Conjugate transpose", "Non-unit", normin, n, &a[ a_offset], lda, &work[1], &scaleu, &rwork[1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.f)
{
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((r__1 = work[i__1].r, abs(r__1)) + (r__2 = r_imag(& work[ix]), abs(r__2))) * smlnum || scale == 0.f)
{
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f)
{
*rcond = 1.f / ainvnm / *anorm;
}
L20:
return 0;
/* End of CPOCON */
}
int cpbcon_(char *uplo, int *n, int *kd, complex *ab,
int *ldab, float *anorm, float *rcond, complex *work, float *rwork,
int *info)
{
/* System generated locals */
int ab_dim1, ab_offset, i__1;
float r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
int ix, kase;
float scale;
extern int lsame_(char *, char *);
int isave[3];
int upper;
extern int clacn2_(int *, complex *, complex *, float
*, int *, int *);
extern int icamax_(int *, complex *, int *);
float scalel;
extern double slamch_(char *);
extern int clatbs_(char *, char *, char *, char *,
int *, int *, complex *, int *, complex *, float *,
float *, int *);
float scaleu;
extern int xerbla_(char *, int *);
float ainvnm;
extern int csrscl_(int *, float *, complex *, int
*);
char normin[1];
float smlnum;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CPBCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a complex Hermitian positive definite band matrix using */
/* the Cholesky factorization A = U**H*U or A = L*L**H computed by */
/* CPBTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangular factor stored in AB; */
/* = 'L': Lower triangular factor stored in AB. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals of the matrix A if UPLO = 'U', */
/* or the number of sub-diagonals if UPLO = 'L'. KD >= 0. */
/* AB (input) COMPLEX array, dimension (LDAB,N) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**H*U or A = L*L**H of the band matrix A, stored in the */
/* first KD+1 rows of the array. The j-th column of U or L is */
/* stored in the j-th column of the array AB as follows: */
/* if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for MAX(1,j-kd)<=i<=j; */
/* if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=MIN(n,j+kd). */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD+1. */
/* ANORM (input) REAL */
/* The 1-norm (or infinity-norm) of the Hermitian band matrix A. */
/* RCOND (output) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kd < 0) {
*info = -3;
} else if (*ldab < *kd + 1) {
*info = -5;
} else if (*anorm < 0.f) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPBCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm == 0.f) {
return 0;
}
smlnum = slamch_("Safe minimum");
/* Estimate the 1-norm of the inverse. */
kase = 0;
*(unsigned char *)normin = 'N';
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (upper) {
/* Multiply by inv(U'). */
clatbs_("Upper", "Conjugate transpose", "Non-unit", normin, n, kd,
&ab[ab_offset], ldab, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
clatbs_("Upper", "No transpose", "Non-unit", normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scaleu, &rwork[1], info);
} else {
/* Multiply by inv(L). */
clatbs_("Lower", "No transpose", "Non-unit", normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L'). */
clatbs_("Lower", "Conjugate transpose", "Non-unit", normin, n, kd,
&ab[ab_offset], ldab, &work[1], &scaleu, &rwork[1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.f) {
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((r__1 = work[i__1].r, ABS(r__1)) + (r__2 = r_imag(&
work[ix]), ABS(r__2))) * smlnum || scale == 0.f) {
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
L20:
return 0;
/* End of CPBCON */
} /* cpbcon_ */
/* Subroutine */
int ctrrfs_(char *uplo, char *trans, char *diag, integer *n, integer *nrhs, complex *a, integer *lda, complex *b, integer *ldb, complex *x, integer *ldx, real *ferr, real *berr, complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j, k;
real s, xk;
integer nz;
real eps;
integer kase;
real safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */
int ccopy_(integer *, complex *, integer *, complex *, integer *), caxpy_(integer *, complex *, complex *, integer *, complex *, integer *);
logical upper;
extern /* Subroutine */
int ctrmv_(char *, char *, char *, integer *, complex *, integer *, complex *, integer *), ctrsv_(char *, char *, char *, integer *, complex *, integer *, complex *, integer *), clacn2_( integer *, complex *, complex *, real *, integer *, integer *);
extern real slamch_(char *);
real safmin;
extern /* Subroutine */
int xerbla_(char *, integer *);
logical notran;
char transn[1], transt[1];
logical nounit;
real lstres;
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
notran = lsame_(trans, "N");
nounit = lsame_(diag, "N");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (! notran && ! lsame_(trans, "T") && ! lsame_(trans, "C"))
{
*info = -2;
}
else if (! nounit && ! lsame_(diag, "U"))
{
*info = -3;
}
else if (*n < 0)
{
*info = -4;
}
else if (*nrhs < 0)
{
*info = -5;
}
else if (*lda < max(1,*n))
{
*info = -7;
}
else if (*ldb < max(1,*n))
{
*info = -9;
}
else if (*ldx < max(1,*n))
{
*info = -11;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("CTRRFS", &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.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
if (notran)
{
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
}
else
{
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
/* Compute residual R = B - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
ccopy_(n, &x[j * x_dim1 + 1], &c__1, &work[1], &c__1);
ctrmv_(uplo, trans, diag, n, &a[a_offset], lda, &work[1], &c__1);
q__1.r = -1.f;
q__1.i = -0.f; // , expr subst
caxpy_(n, &q__1, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( f2c_abs(R(i)) / ( f2c_abs(op(A))*f2c_abs(X) + f2c_abs(B) )(i) ) */
/* where f2c_abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__ + j * b_dim1;
rwork[i__] = (r__1 = b[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&b[ i__ + j * b_dim1]), f2c_abs(r__2));
/* L20: */
}
if (notran)
{
/* Compute f2c_abs(A)*f2c_abs(X) + f2c_abs(B). */
if (upper)
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(& x[k + j * x_dim1]), f2c_abs(r__2));
i__3 = k;
for (i__ = 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, f2c_abs(r__1)) + ( r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs( r__2))) * xk;
/* L30: */
}
/* L40: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(& x[k + j * x_dim1]), f2c_abs(r__2));
i__3 = k - 1;
for (i__ = 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, f2c_abs(r__1)) + ( r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs( r__2))) * xk;
/* L50: */
}
rwork[k] += xk;
/* L60: */
}
}
}
else
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(& x[k + j * x_dim1]), f2c_abs(r__2));
i__3 = *n;
for (i__ = k;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, f2c_abs(r__1)) + ( r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs( r__2))) * xk;
/* L70: */
}
/* L80: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(& x[k + j * x_dim1]), f2c_abs(r__2));
i__3 = *n;
for (i__ = k + 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, f2c_abs(r__1)) + ( r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs( r__2))) * xk;
/* L90: */
}
rwork[k] += xk;
/* L100: */
}
}
}
}
else
{
/* Compute f2c_abs(A**H)*f2c_abs(X) + f2c_abs(B). */
if (upper)
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.f;
i__3 = k;
for (i__ = 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs(r__2))) * ((r__3 = x[i__5].r, f2c_abs(r__3)) + (r__4 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__4))) ;
/* L110: */
}
rwork[k] += s;
/* L120: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
s = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&x[ k + j * x_dim1]), f2c_abs(r__2));
i__3 = k - 1;
for (i__ = 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs(r__2))) * ((r__3 = x[i__5].r, f2c_abs(r__3)) + (r__4 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__4))) ;
/* L130: */
}
rwork[k] += s;
/* L140: */
}
}
}
else
{
if (nounit)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.f;
i__3 = *n;
for (i__ = k;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs(r__2))) * ((r__3 = x[i__5].r, f2c_abs(r__3)) + (r__4 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__4))) ;
/* L150: */
}
rwork[k] += s;
/* L160: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
i__3 = k + j * x_dim1;
s = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&x[ k + j * x_dim1]), f2c_abs(r__2));
i__3 = *n;
for (i__ = k + 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs(r__2))) * ((r__3 = x[i__5].r, f2c_abs(r__3)) + (r__4 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__4))) ;
/* L170: */
}
rwork[k] += s;
/* L180: */
}
}
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
/* Computing MAX */
i__3 = i__;
r__3 = s;
r__4 = ((r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2))) / rwork[i__]; // , expr subst
s = max(r__3,r__4);
}
else
{
/* Computing MAX */
i__3 = i__;
r__3 = s;
r__4 = ((r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2)) + safe1) / (rwork[i__] + safe1); // , expr subst
s = max(r__3,r__4);
}
/* L190: */
}
berr[j] = s;
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( f2c_abs(inv(op(A)))* */
/* ( f2c_abs(R) + NZ*EPS*( f2c_abs(op(A))*f2c_abs(X)+f2c_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) */
/* 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(op(A))*f2c_abs(X)+f2c_abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* f2c_abs(op(A))*f2c_abs(X) + f2c_abs(B) is less than SAFE2. */
/* Use CLACN2 to estimate the infinity-norm of the matrix */
/* inv(op(A)) * diag(W), */
/* where W = f2c_abs(R) + NZ*EPS*( f2c_abs(op(A))*f2c_abs(X)+f2c_abs(B) ))) */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2)) + nz * eps * rwork[i__] ;
}
else
{
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2)) + nz * eps * rwork[i__] + safe1;
}
/* L200: */
}
kase = 0;
L210:
clacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Multiply by diag(W)*inv(op(A)**H). */
ctrsv_(uplo, transt, diag, n, &a[a_offset], lda, &work[1], & c__1);
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r;
q__1.i = rwork[i__4] * work[i__5].i; // , expr subst
work[i__3].r = q__1.r;
work[i__3].i = q__1.i; // , expr subst
/* L220: */
}
}
else
{
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r;
q__1.i = rwork[i__4] * work[i__5].i; // , expr subst
work[i__3].r = q__1.r;
work[i__3].i = q__1.i; // , expr subst
/* L230: */
}
ctrsv_(uplo, transn, diag, n, &a[a_offset], lda, &work[1], & c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
/* Computing MAX */
i__3 = i__ + j * x_dim1;
r__3 = lstres;
r__4 = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__2)); // , expr subst
lstres = max(r__3,r__4);
/* L240: */
}
if (lstres != 0.f)
{
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of CTRRFS */
}
/* Subroutine */ int chpcon_(char *uplo, integer *n, complex *ap, integer *
ipiv, real *anorm, real *rcond, complex *work, integer *info)
{
/* System generated locals */
integer i__1, i__2;
/* Local variables */
integer i__, ip, kase;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *), xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */ int chptrs_(char *, integer *, integer *, complex
*, integer *, complex *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CHPCON estimates the reciprocal of the condition number of a complex */
/* Hermitian packed matrix A using the factorization A = U*D*U**H or */
/* A = L*D*L**H computed by CHPTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* Specifies whether the details of the factorization are stored */
/* as an upper or lower triangular matrix. */
/* = 'U': Upper triangular, form is A = U*D*U**H; */
/* = 'L': Lower triangular, form is A = L*D*L**H. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* AP (input) COMPLEX 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 CHPTRF, 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 CHPTRF. */
/* ANORM (input) REAL */
/* The 1-norm of the original matrix A. */
/* RCOND (output) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--work;
--ipiv;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*anorm < 0.f) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CHPCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm <= 0.f) {
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper) {
/* Upper triangular storage: examine D from bottom to top */
ip = *n * (*n + 1) / 2;
for (i__ = *n; i__ >= 1; --i__) {
i__1 = ip;
if (ipiv[i__] > 0 && (ap[i__1].r == 0.f && ap[i__1].i == 0.f)) {
return 0;
}
ip -= i__;
/* L10: */
}
} else {
/* Lower triangular storage: examine D from top to bottom. */
ip = 1;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = ip;
if (ipiv[i__] > 0 && (ap[i__2].r == 0.f && ap[i__2].i == 0.f)) {
return 0;
}
ip = ip + *n - i__ + 1;
/* L20: */
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
/* Multiply by inv(L*D*L') or inv(U*D*U'). */
chptrs_(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.f) {
*rcond = 1.f / ainvnm / *anorm;
}
return 0;
/* End of CHPCON */
} /* chpcon_ */
doublereal cla_hercond_x__(char *uplo, integer *n, complex *a, integer *lda,
complex *af, integer *ldaf, integer *ipiv, complex *x, integer *info,
complex *work, real *rwork, ftnlen uplo_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2, i__3, i__4;
real ret_val, r__1, r__2;
complex q__1, q__2;
/* Builtin functions */
double r_imag(complex *);
void c_div(complex *, complex *, complex *);
/* Local variables */
integer i__, j;
logical up;
real tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
real anorm;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *), xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */ int chetrs_(char *, integer *, integer *, complex
*, integer *, integer *, complex *, integer *, integer *);
/* -- 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 */
/* ======= */
/* CLA_HERCOND_X computes the infinity norm condition number of */
/* op(A) * diag(X) where X is a COMPLEX vector. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* A (input) COMPLEX array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) COMPLEX array, dimension (LDAF,N) */
/* The block diagonal matrix D and the multipliers used to */
/* obtain the factor U or L as computed by CHETRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by CHETRF. */
/* X (input) COMPLEX array, dimension (N) */
/* The vector X in the formula op(A) * diag(X). */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) COMPLEX array, dimension (2*N). */
/* Workspace. */
/* RWORK (input) REAL array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function Definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
--x;
--work;
--rwork;
/* Function Body */
ret_val = 0.f;
*info = 0;
if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CLA_HERCOND_X", &i__1);
return ret_val;
}
up = FALSE_;
if (lsame_(uplo, "U")) {
up = TRUE_;
}
/* Compute norm of op(A)*op2(C). */
anorm = 0.f;
if (up) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.f;
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
i__4 = j;
q__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
q__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
q__1.r = q__2.r, q__1.i = q__2.i;
tmp += (r__1 = q__1.r, dabs(r__1)) + (r__2 = r_imag(&q__1),
dabs(r__2));
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
i__4 = j;
q__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
q__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
q__1.r = q__2.r, q__1.i = q__2.i;
tmp += (r__1 = q__1.r, dabs(r__1)) + (r__2 = r_imag(&q__1),
dabs(r__2));
}
rwork[i__] = tmp;
anorm = dmax(anorm,tmp);
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.f;
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
i__4 = j;
q__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
q__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
q__1.r = q__2.r, q__1.i = q__2.i;
tmp += (r__1 = q__1.r, dabs(r__1)) + (r__2 = r_imag(&q__1),
dabs(r__2));
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
i__4 = j;
q__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[i__4].i,
q__2.i = a[i__3].r * x[i__4].i + a[i__3].i * x[i__4]
.r;
q__1.r = q__2.r, q__1.i = q__2.i;
tmp += (r__1 = q__1.r, dabs(r__1)) + (r__2 = r_imag(&q__1),
dabs(r__2));
}
rwork[i__] = tmp;
anorm = dmax(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0) {
ret_val = 1.f;
return ret_val;
} else if (anorm == 0.f) {
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.f;
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__4] * work[i__3].r, q__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
if (up) {
chetrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
} else {
chetrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
}
/* Multiply by inv(X). */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
c_div(&q__1, &work[i__], &x[i__]);
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
} else {
/* Multiply by inv(X'). */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
c_div(&q__1, &work[i__], &x[i__]);
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
if (up) {
chetrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
} else {
chetrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__4] * work[i__3].r, q__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
ret_val = 1.f / ainvnm;
}
return ret_val;
} /* cla_hercond_x__ */
/* Subroutine */ int cpocon_(char *uplo, integer *n, complex *a, integer *lda,
real *anorm, real *rcond, complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
real r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer ix, kase;
real scale;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
extern integer icamax_(integer *, complex *, integer *);
real scalel;
extern doublereal slamch_(char *);
real scaleu;
extern /* Subroutine */ int xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */ int clatrs_(char *, char *, char *, char *,
integer *, complex *, integer *, complex *, real *, real *,
integer *), csrscl_(integer *,
real *, complex *, integer *);
char normin[1];
real smlnum;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CPOCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a complex Hermitian positive definite matrix using the */
/* Cholesky factorization A = U**H*U or A = L*L**H computed by CPOTRF. */
/* 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. */
/* A (input) COMPLEX array, dimension (LDA,N) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**H*U or A = L*L**H, as computed by CPOTRF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* ANORM (input) REAL */
/* The 1-norm (or infinity-norm) of the Hermitian matrix A. */
/* RCOND (output) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*anorm < 0.f) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPOCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm == 0.f) {
return 0;
}
smlnum = slamch_("Safe minimum");
/* Estimate the 1-norm of inv(A). */
kase = 0;
*(unsigned char *)normin = 'N';
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (upper) {
/* Multiply by inv(U'). */
clatrs_("Upper", "Conjugate transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(U). */
clatrs_("Upper", "No transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &scaleu, &rwork[1], info);
} else {
/* Multiply by inv(L). */
clatrs_("Lower", "No transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &scalel, &rwork[1], info);
*(unsigned char *)normin = 'Y';
/* Multiply by inv(L'). */
clatrs_("Lower", "Conjugate transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &scaleu, &rwork[1], info);
}
/* Multiply by 1/SCALE if doing so will not cause overflow. */
scale = scalel * scaleu;
if (scale != 1.f) {
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((r__1 = work[i__1].r, dabs(r__1)) + (r__2 = r_imag(&
work[ix]), dabs(r__2))) * smlnum || scale == 0.f) {
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
L20:
return 0;
/* End of CPOCON */
} /* cpocon_ */
/* Subroutine */ int ctbcon_(char *norm, char *uplo, char *diag, integer *n,
integer *kd, complex *ab, integer *ldab, real *rcond, complex *work,
real *rwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1;
real r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer ix, kase, kase1;
real scale;
extern logical lsame_(char *, char *);
integer isave[3];
real anorm;
logical upper;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
real xnorm;
extern integer icamax_(integer *, complex *, integer *);
extern doublereal clantb_(char *, char *, char *, integer *, integer *,
complex *, integer *, real *), slamch_(
char *);
extern /* Subroutine */ int clatbs_(char *, char *, char *, char *,
integer *, integer *, complex *, integer *, complex *, real *,
real *, integer *), xerbla_(char *
, integer *);
real ainvnm;
extern /* Subroutine */ int csrscl_(integer *, real *, complex *, integer
*);
logical onenrm;
char normin[1];
real smlnum;
logical nounit;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTBCON estimates the reciprocal of the condition number of a */
/* triangular band matrix A, in either the 1-norm or the infinity-norm. */
/* The norm of A is computed and an estimate is obtained for */
/* norm(inv(A)), then the reciprocal of the condition number is */
/* computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': A is upper triangular; */
/* = 'L': A is lower triangular. */
/* DIAG (input) CHARACTER*1 */
/* = 'N': A is non-unit triangular; */
/* = 'U': A is unit triangular. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals or subdiagonals of the */
/* triangular band matrix A. KD >= 0. */
/* AB (input) COMPLEX 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) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
nounit = lsame_(diag, "N");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (! upper && ! lsame_(uplo, "L")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*kd < 0) {
*info = -5;
} else if (*ldab < *kd + 1) {
*info = -7;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTBCON", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*rcond = 1.f;
return 0;
}
*rcond = 0.f;
smlnum = slamch_("Safe minimum") * (real) max(*n,1);
/* Compute the 1-norm of the triangular matrix A or A'. */
anorm = clantb_(norm, uplo, diag, n, kd, &ab[ab_offset], ldab, &rwork[1]);
/* Continue only if ANORM > 0. */
if (anorm > 0.f) {
/* Estimate the 1-norm of the inverse of A. */
ainvnm = 0.f;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(A). */
clatbs_(uplo, "No transpose", diag, normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scale, &rwork[1], info);
} else {
/* Multiply by inv(A'). */
clatbs_(uplo, "Conjugate transpose", diag, normin, n, kd, &ab[
ab_offset], ldab, &work[1], &scale, &rwork[1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.f) {
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
xnorm = (r__1 = work[i__1].r, dabs(r__1)) + (r__2 = r_imag(&
work[ix]), dabs(r__2));
if (scale < xnorm * smlnum || scale == 0.f) {
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / anorm / ainvnm;
}
}
L20:
return 0;
/* End of CTBCON */
} /* ctbcon_ */
doublereal cla_gercond_c__(char *trans, integer *n, complex *a, integer *lda,
complex *af, integer *ldaf, integer *ipiv, real *c__, logical *capply,
integer *info, complex *work, real *rwork, ftnlen trans_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2, i__3, i__4;
real ret_val, r__1, r__2;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j;
real tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
real anorm;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *), xerbla_(char *, integer *),
cgetrs_(char *, integer *, integer *, complex *, integer *,
integer *, complex *, integer *, integer *);
real 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. -- */
/* .. */
/* .. Scalar Aguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CLA_GERCOND_C computes the infinity norm condition number of */
/* op(A) * inv(diag(C)) where C is a REAL vector. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A**T * X = B (Transpose) */
/* = 'C': A**H * X = B (Conjugate Transpose = Transpose) */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* A (input) COMPLEX array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) COMPLEX array, dimension (LDAF,N) */
/* The factors L and U from the factorization */
/* A = P*L*U as computed by CGETRF. */
/* 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 CGETRF; row i of the matrix was interchanged */
/* with row IPIV(i). */
/* C (input) REAL array, dimension (N) */
/* The vector C in the formula op(A) * inv(diag(C)). */
/* CAPPLY (input) LOGICAL */
/* If .TRUE. then access the vector C in the formula above. */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* i > 0: The ith argument is invalid. */
/* WORK (input) COMPLEX array, dimension (2*N). */
/* Workspace. */
/* RWORK (input) REAL array, dimension (N). */
/* Workspace. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function Definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
--c__;
--work;
--rwork;
/* Function Body */
ret_val = 0.f;
*info = 0;
notrans = lsame_(trans, "N");
if (! notrans && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
} else if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CLA_GERCOND_C", &i__1);
return ret_val;
}
/* Compute norm of op(A)*op2(C). */
anorm = 0.f;
if (notrans) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.f;
if (*capply) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += ((r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
a[i__ + j * a_dim1]), dabs(r__2))) / c__[j];
}
} else {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
tmp += (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[
i__ + j * a_dim1]), dabs(r__2));
}
}
rwork[i__] = tmp;
anorm = dmax(anorm,tmp);
}
} else {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tmp = 0.f;
if (*capply) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += ((r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
a[j + i__ * a_dim1]), dabs(r__2))) / c__[j];
}
} else {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
tmp += (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[
j + i__ * a_dim1]), dabs(r__2));
}
}
rwork[i__] = tmp;
anorm = dmax(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0) {
ret_val = 1.f;
return ret_val;
} else if (anorm == 0.f) {
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.f;
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == 2) {
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__4] * work[i__3].r, q__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
if (notrans) {
cgetrs_("No transpose", n, &c__1, &af[af_offset], ldaf, &ipiv[
1], &work[1], n, info);
} else {
cgetrs_("Conjugate transpose", n, &c__1, &af[af_offset], ldaf,
&ipiv[1], &work[1], n, info);
}
/* Multiply by inv(C). */
if (*capply) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = c__[i__4] * work[i__3].r, q__1.i = c__[i__4] *
work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
}
} else {
/* Multiply by inv(C'). */
if (*capply) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = c__[i__4] * work[i__3].r, q__1.i = c__[i__4] *
work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
}
if (notrans) {
cgetrs_("Conjugate transpose", n, &c__1, &af[af_offset], ldaf,
&ipiv[1], &work[1], n, info);
} else {
cgetrs_("No transpose", n, &c__1, &af[af_offset], ldaf, &ipiv[
1], &work[1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__4] * work[i__3].r, q__1.i = rwork[i__4] *
work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
ret_val = 1.f / ainvnm;
}
return ret_val;
} /* cla_gercond_c__ */
/* ===================================================================== */
real cla_syrcond_c_(char *uplo, integer *n, complex *a, integer *lda, complex *af, integer *ldaf, integer *ipiv, real *c__, logical *capply, integer *info, complex *work, real *rwork)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2, i__3, i__4;
real ret_val, r__1, r__2;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j;
logical up;
real tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
real anorm;
logical upper;
extern /* Subroutine */
int clacn2_(integer *, complex *, complex *, real *, integer *, integer *), xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */
int csytrs_(char *, integer *, integer *, complex *, integer *, integer *, complex *, 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 .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function Definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
--c__;
--work;
--rwork;
/* Function Body */
ret_val = 0.f;
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*n))
{
*info = -4;
}
else if (*ldaf < max(1,*n))
{
*info = -6;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("CLA_SYRCOND_C", &i__1);
return ret_val;
}
up = FALSE_;
if (lsame_(uplo, "U"))
{
up = TRUE_;
}
/* Compute norm of op(A)*op2(C). */
anorm = 0.f;
if (up)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.f;
if (*capply)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
i__3 = j + i__ * a_dim1;
tmp += ((r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ j + i__ * a_dim1]), abs(r__2))) / c__[j];
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
i__3 = i__ + j * a_dim1;
tmp += ((r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ i__ + j * a_dim1]), abs(r__2))) / c__[j];
}
}
else
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
i__3 = j + i__ * a_dim1;
tmp += (r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ j + i__ * a_dim1]), abs(r__2));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
i__3 = i__ + j * a_dim1;
tmp += (r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ i__ + j * a_dim1]), abs(r__2));
}
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
else
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.f;
if (*capply)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
i__3 = i__ + j * a_dim1;
tmp += ((r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ i__ + j * a_dim1]), abs(r__2))) / c__[j];
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
i__3 = j + i__ * a_dim1;
tmp += ((r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ j + i__ * a_dim1]), abs(r__2))) / c__[j];
}
}
else
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
i__3 = i__ + j * a_dim1;
tmp += (r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ i__ + j * a_dim1]), abs(r__2));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
i__3 = j + i__ * a_dim1;
tmp += (r__1 = a[i__3].r, abs(r__1)) + (r__2 = r_imag(&a[ j + i__ * a_dim1]), abs(r__2));
}
}
rwork[i__] = tmp;
anorm = max(anorm,tmp);
}
}
/* Quick return if possible. */
if (*n == 0)
{
ret_val = 1.f;
return ret_val;
}
else if (anorm == 0.f)
{
return ret_val;
}
/* Estimate the norm of inv(op(A)). */
ainvnm = 0.f;
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (kase == 2)
{
/* Multiply by R. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__4] * work[i__3].r;
q__1.i = rwork[i__4] * work[i__3].i; // , expr subst
work[i__2].r = q__1.r;
work[i__2].i = q__1.i; // , expr subst
}
if (up)
{
csytrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
else
{
csytrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
/* Multiply by inv(C). */
if (*capply)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = c__[i__4] * work[i__3].r;
q__1.i = c__[i__4] * work[i__3].i; // , expr subst
work[i__2].r = q__1.r;
work[i__2].i = q__1.i; // , expr subst
}
}
}
else
{
/* Multiply by inv(C**T). */
if (*capply)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = c__[i__4] * work[i__3].r;
q__1.i = c__[i__4] * work[i__3].i; // , expr subst
work[i__2].r = q__1.r;
work[i__2].i = q__1.i; // , expr subst
}
}
if (up)
{
csytrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
else
{
csytrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__4] * work[i__3].r;
q__1.i = rwork[i__4] * work[i__3].i; // , expr subst
work[i__2].r = q__1.r;
work[i__2].i = q__1.i; // , expr subst
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f)
{
ret_val = 1.f / ainvnm;
}
return ret_val;
}
/* Subroutine */ int cgecon_(char *norm, integer *n, complex *a, integer *lda,
real *anorm, real *rcond, complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
real r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
real sl;
integer ix;
real su;
integer kase, kase1;
real scale;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
extern integer icamax_(integer *, complex *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */ int clatrs_(char *, char *, char *, char *,
integer *, complex *, integer *, complex *, real *, real *,
integer *), csrscl_(integer *,
real *, complex *, integer *);
logical onenrm;
char normin[1];
real smlnum;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CGECON estimates the reciprocal of the condition number of a general */
/* complex matrix A, in either the 1-norm or the infinity-norm, using */
/* the LU factorization computed by CGETRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input) COMPLEX array, dimension (LDA,N) */
/* The factors L and U from the factorization A = P*L*U */
/* as computed by CGETRF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* ANORM (input) REAL */
/* 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) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--work;
--rwork;
/* Function Body */
*info = 0;
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*anorm < 0.f) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGECON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm == 0.f) {
return 0;
}
smlnum = slamch_("Safe minimum");
/* Estimate the norm of inv(A). */
ainvnm = 0.f;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(L). */
clatrs_("Lower", "No transpose", "Unit", normin, n, &a[a_offset],
lda, &work[1], &sl, &rwork[1], info);
/* Multiply by inv(U). */
clatrs_("Upper", "No transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &su, &rwork[*n + 1], info);
} else {
/* Multiply by inv(U'). */
clatrs_("Upper", "Conjugate transpose", "Non-unit", normin, n, &a[
a_offset], lda, &work[1], &su, &rwork[*n + 1], info);
/* Multiply by inv(L'). */
clatrs_("Lower", "Conjugate transpose", "Unit", normin, n, &a[
a_offset], lda, &work[1], &sl, &rwork[1], info);
}
/* Divide X by 1/(SL*SU) if doing so will not cause overflow. */
scale = sl * su;
*(unsigned char *)normin = 'Y';
if (scale != 1.f) {
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
if (scale < ((r__1 = work[i__1].r, dabs(r__1)) + (r__2 = r_imag(&
work[ix]), dabs(r__2))) * smlnum || scale == 0.f) {
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
L20:
return 0;
/* End of CGECON */
} /* cgecon_ */
/* Subroutine */
int checon_(char *uplo, integer *n, complex *a, integer *lda, integer *ipiv, real *anorm, real *rcond, complex *work, integer * info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
/* Local variables */
integer i__, kase;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */
int clacn2_(integer *, complex *, complex *, real *, integer *, integer *), xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */
int chetrs_(char *, integer *, integer *, complex *, integer *, integer *, complex *, integer *, integer *);
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
--work;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*n))
{
*info = -4;
}
else if (*anorm < 0.f)
{
*info = -6;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("CHECON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0)
{
*rcond = 1.f;
return 0;
}
else if (*anorm <= 0.f)
{
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper)
{
/* Upper triangular storage: examine D from bottom to top */
for (i__ = *n;
i__ >= 1;
--i__)
{
i__1 = i__ + i__ * a_dim1;
if (ipiv[i__] > 0 && (a[i__1].r == 0.f && a[i__1].i == 0.f))
{
return 0;
}
/* L10: */
}
}
else
{
/* Lower triangular storage: examine D from top to bottom. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
i__2 = i__ + i__ * a_dim1;
if (ipiv[i__] > 0 && (a[i__2].r == 0.f && a[i__2].i == 0.f))
{
return 0;
}
/* L20: */
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0)
{
/* Multiply by inv(L*D*L**H) or inv(U*D*U**H). */
chetrs_(uplo, n, &c__1, &a[a_offset], lda, &ipiv[1], &work[1], n, info);
goto L30;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f)
{
*rcond = 1.f / ainvnm / *anorm;
}
return 0;
/* End of CHECON */
}
/* Subroutine */
int cherfs_(char *uplo, integer *n, integer *nrhs, complex * a, integer *lda, complex *af, integer *ldaf, integer *ipiv, complex * b, integer *ldb, complex *x, integer *ldx, real *ferr, real *berr, complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j, k;
real s, xk;
integer nz;
real eps;
integer kase;
real safe1, safe2;
extern logical lsame_(char *, char *);
extern /* Subroutine */
int chemv_(char *, integer *, complex *, complex * , integer *, complex *, integer *, complex *, complex *, integer * );
integer isave[3];
extern /* Subroutine */
int ccopy_(integer *, complex *, integer *, complex *, integer *), caxpy_(integer *, complex *, complex *, integer *, complex *, integer *);
integer count;
logical upper;
extern /* Subroutine */
int clacn2_(integer *, complex *, complex *, real *, integer *, integer *);
extern real slamch_(char *);
real safmin;
extern /* Subroutine */
int xerbla_(char *, integer *), chetrs_( char *, integer *, integer *, complex *, integer *, integer *, complex *, integer *, integer *);
real lstres;
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
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;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*nrhs < 0)
{
*info = -3;
}
else if (*lda < max(1,*n))
{
*info = -5;
}
else if (*ldaf < max(1,*n))
{
*info = -7;
}
else if (*ldb < max(1,*n))
{
*info = -10;
}
else if (*ldx < max(1,*n))
{
*info = -12;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("CHERFS", &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.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("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.f;
L20: /* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - A * X */
ccopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
q__1.r = -1.f;
q__1.i = -0.f; // , expr subst
chemv_(uplo, n, &q__1, &a[a_offset], lda, &x[j * x_dim1 + 1], &c__1, & c_b1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( f2c_abs(R(i)) / ( f2c_abs(A)*f2c_abs(X) + f2c_abs(B) )(i) ) */
/* where f2c_abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__ + j * b_dim1;
rwork[i__] = (r__1 = b[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&b[ i__ + j * b_dim1]), f2c_abs(r__2));
/* L30: */
}
/* Compute f2c_abs(A)*f2c_abs(X) + f2c_abs(B). */
if (upper)
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.f;
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&x[k + j * x_dim1]), f2c_abs(r__2));
i__3 = k - 1;
for (i__ = 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs(r__2))) * xk;
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[ i__ + k * a_dim1]), f2c_abs(r__2))) * ((r__3 = x[i__5] .r, f2c_abs(r__3)) + (r__4 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__4)));
/* L40: */
}
i__3 = k + k * a_dim1;
rwork[k] = rwork[k] + (r__1 = a[i__3].r, f2c_abs(r__1)) * xk + s;
/* L50: */
}
}
else
{
i__2 = *n;
for (k = 1;
k <= i__2;
++k)
{
s = 0.f;
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&x[k + j * x_dim1]), f2c_abs(r__2));
i__3 = k + k * a_dim1;
rwork[k] += (r__1 = a[i__3].r, f2c_abs(r__1)) * xk;
i__3 = *n;
for (i__ = k + 1;
i__ <= i__3;
++i__)
{
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[i__ + k * a_dim1]), f2c_abs(r__2))) * xk;
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, f2c_abs(r__1)) + (r__2 = r_imag(&a[ i__ + k * a_dim1]), f2c_abs(r__2))) * ((r__3 = x[i__5] .r, f2c_abs(r__3)) + (r__4 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__4)));
/* L60: */
}
rwork[k] += s;
/* L70: */
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
/* Computing MAX */
i__3 = i__;
r__3 = s;
r__4 = ((r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2))) / rwork[i__]; // , expr subst
s = max(r__3,r__4);
}
else
{
/* Computing MAX */
i__3 = i__;
r__3 = s;
r__4 = ((r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2)) + safe1) / (rwork[i__] + safe1); // , expr subst
s = max(r__3,r__4);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2.f <= lstres && count <= 5)
{
/* Update solution and try again. */
chetrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[1], n, info);
caxpy_(n, &c_b1, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( f2c_abs(inv(A))* */
/* ( f2c_abs(R) + NZ*EPS*( f2c_abs(A)*f2c_abs(X)+f2c_abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(A) is the inverse of A */
/* f2c_abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of f2c_abs(R)+NZ*EPS*(f2c_abs(A)*f2c_abs(X)+f2c_abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* f2c_abs(A)*f2c_abs(X) + f2c_abs(B) is less than SAFE2. */
/* Use CLACN2 to estimate the infinity-norm of the matrix */
/* inv(A) * diag(W), */
/* where W = f2c_abs(R) + NZ*EPS*( f2c_abs(A)*f2c_abs(X)+f2c_abs(B) ))) */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (rwork[i__] > safe2)
{
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2)) + nz * eps * rwork[i__] ;
}
else
{
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&work[i__]), f2c_abs(r__2)) + nz * eps * rwork[i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
clacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0)
{
if (kase == 1)
{
/* Multiply by diag(W)*inv(A**H). */
chetrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r;
q__1.i = rwork[i__4] * work[i__5].i; // , expr subst
work[i__3].r = q__1.r;
work[i__3].i = q__1.i; // , expr subst
/* L110: */
}
}
else if (kase == 2)
{
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r;
q__1.i = rwork[i__4] * work[i__5].i; // , expr subst
work[i__3].r = q__1.r;
work[i__3].i = q__1.i; // , expr subst
/* L120: */
}
chetrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1;
i__ <= i__2;
++i__)
{
/* Computing MAX */
i__3 = i__ + j * x_dim1;
r__3 = lstres;
r__4 = (r__1 = x[i__3].r, f2c_abs(r__1)) + (r__2 = r_imag(&x[i__ + j * x_dim1]), f2c_abs(r__2)); // , expr subst
lstres = max(r__3,r__4);
/* L130: */
}
if (lstres != 0.f)
{
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of CHERFS */
}
/* Subroutine */
int ctpcon_(char *norm, char *uplo, char *diag, integer *n, complex *ap, real *rcond, complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer i__1;
real r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer ix, kase, kase1;
real scale;
extern logical lsame_(char *, char *);
integer isave[3];
real anorm;
logical upper;
extern /* Subroutine */
int clacn2_(integer *, complex *, complex *, real *, integer *, integer *);
real xnorm;
extern integer icamax_(integer *, complex *, integer *);
extern real slamch_(char *);
extern /* Subroutine */
int xerbla_(char *, integer *);
extern real clantp_(char *, char *, char *, integer *, complex *, real *);
extern /* Subroutine */
int clatps_(char *, char *, char *, char *, integer *, complex *, complex *, real *, real *, integer *);
real ainvnm;
extern /* Subroutine */
int csrscl_(integer *, real *, complex *, integer *);
logical onenrm;
char normin[1];
real 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 .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--rwork;
--work;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
nounit = lsame_(diag, "N");
if (! onenrm && ! lsame_(norm, "I"))
{
*info = -1;
}
else if (! upper && ! lsame_(uplo, "L"))
{
*info = -2;
}
else if (! nounit && ! lsame_(diag, "U"))
{
*info = -3;
}
else if (*n < 0)
{
*info = -4;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("CTPCON", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0)
{
*rcond = 1.f;
return 0;
}
*rcond = 0.f;
smlnum = slamch_("Safe minimum") * (real) max(1,*n);
/* Compute the norm of the triangular matrix A. */
anorm = clantp_(norm, uplo, diag, n, &ap[1], &rwork[1]);
/* Continue only if ANORM > 0. */
if (anorm > 0.f)
{
/* Estimate the norm of the inverse of A. */
ainvnm = 0.f;
*(unsigned char *)normin = 'N';
if (onenrm)
{
kase1 = 1;
}
else
{
kase1 = 2;
}
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (kase == kase1)
{
/* Multiply by inv(A). */
clatps_(uplo, "No transpose", diag, normin, n, &ap[1], &work[ 1], &scale, &rwork[1], info);
}
else
{
/* Multiply by inv(A**H). */
clatps_(uplo, "Conjugate transpose", diag, normin, n, &ap[1], &work[1], &scale, &rwork[1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.f)
{
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
xnorm = (r__1 = work[i__1].r, abs(r__1)) + (r__2 = r_imag(& work[ix]), abs(r__2));
if (scale < xnorm * smlnum || scale == 0.f)
{
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f)
{
*rcond = 1.f / anorm / ainvnm;
}
}
L20:
return 0;
/* End of CTPCON */
}
int csycon_(char *uplo, int *n, complex *a, int *lda,
int *ipiv, float *anorm, float *rcond, complex *work, int *
info)
{
/* System generated locals */
int a_dim1, a_offset, i__1, i__2;
/* Local variables */
int i__, kase;
extern int lsame_(char *, char *);
int isave[3];
int upper;
extern int clacn2_(int *, complex *, complex *, float
*, int *, int *), xerbla_(char *, int *);
float ainvnm;
extern int csytrs_(char *, int *, int *, complex
*, int *, int *, complex *, int *, int *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CSYCON estimates the reciprocal of the condition number (in the */
/* 1-norm) of a complex symmetric matrix A using the factorization */
/* A = U*D*U**T or A = L*D*L**T computed by CSYTRF. */
/* An estimate is obtained for norm(inv(A)), and the reciprocal of the */
/* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* Specifies whether the details of the factorization are stored */
/* as an upper or lower triangular matrix. */
/* = 'U': Upper triangular, form is A = U*D*U**T; */
/* = 'L': Lower triangular, form is A = L*D*L**T. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input) COMPLEX array, dimension (LDA,N) */
/* The block diagonal matrix D and the multipliers used to */
/* obtain the factor U or L as computed by CSYTRF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by CSYTRF. */
/* ANORM (input) REAL */
/* The 1-norm of the original matrix A. */
/* RCOND (output) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an */
/* estimate of the 1-norm of inv(A) computed in this routine. */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
--work;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < MAX(1,*n)) {
*info = -4;
} else if (*anorm < 0.f) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CSYCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm <= 0.f) {
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper) {
/* Upper triangular storage: examine D from bottom to top */
for (i__ = *n; i__ >= 1; --i__) {
i__1 = i__ + i__ * a_dim1;
if (ipiv[i__] > 0 && (a[i__1].r == 0.f && a[i__1].i == 0.f)) {
return 0;
}
/* L10: */
}
} else {
/* Lower triangular storage: examine D from top to bottom. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__ + i__ * a_dim1;
if (ipiv[i__] > 0 && (a[i__2].r == 0.f && a[i__2].i == 0.f)) {
return 0;
}
/* L20: */
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
/* Multiply by inv(L*D*L') or inv(U*D*U'). */
csytrs_(uplo, n, &c__1, &a[a_offset], lda, &ipiv[1], &work[1], n,
info);
goto L30;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
return 0;
/* End of CSYCON */
} /* csycon_ */
/* Subroutine */ int csyrfs_(char *uplo, integer *n, integer *nrhs, complex *
a, integer *lda, complex *af, integer *ldaf, integer *ipiv, complex *
b, integer *ldb, complex *x, integer *ldx, real *ferr, real *berr,
complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j, k;
real s, xk;
integer nz;
real eps;
integer kase;
real safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *), caxpy_(integer *, complex *, complex *,
integer *, complex *, integer *);
integer count;
logical upper;
extern /* Subroutine */ int csymv_(char *, integer *, complex *, complex *
, integer *, complex *, integer *, complex *, complex *, integer *
), clacn2_(integer *, complex *, complex *, real *,
integer *, integer *);
extern doublereal slamch_(char *);
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
real lstres;
extern /* Subroutine */ int csytrs_(char *, integer *, integer *, complex
*, integer *, integer *, complex *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CSYRFS 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) COMPLEX 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) COMPLEX 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 CSYTRF. */
/* 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 CSYTRF. */
/* B (input) COMPLEX array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) COMPLEX array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by CSYTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) REAL 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) REAL array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Internal Parameters */
/* =================== */
/* ITMAX is the maximum number of steps of iterative refinement. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
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;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldaf < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -10;
} else if (*ldx < max(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CSYRFS", &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.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("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.f;
L20:
/* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - A * X */
ccopy_(n, &b[j * b_dim1 + 1], &c__1, &work[1], &c__1);
q__1.r = -1.f, q__1.i = -0.f;
csymv_(uplo, n, &q__1, &a[a_offset], lda, &x[j * x_dim1 + 1], &c__1, &
c_b1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&b[
i__ + j * b_dim1]), dabs(r__2));
/* L30: */
}
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x[k + j
* x_dim1]), dabs(r__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + k * a_dim1]), dabs(r__2))) * xk;
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&a[
i__ + k * a_dim1]), dabs(r__2))) * ((r__3 = x[
i__5].r, dabs(r__3)) + (r__4 = r_imag(&x[i__ + j *
x_dim1]), dabs(r__4)));
/* L40: */
}
i__3 = k + k * a_dim1;
rwork[k] = rwork[k] + ((r__1 = a[i__3].r, dabs(r__1)) + (r__2
= r_imag(&a[k + k * a_dim1]), dabs(r__2))) * xk + s;
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x[k + j
* x_dim1]), dabs(r__2));
i__3 = k + k * a_dim1;
rwork[k] += ((r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
a[k + k * a_dim1]), dabs(r__2))) * xk;
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = i__ + k * a_dim1;
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + k * a_dim1]), dabs(r__2))) * xk;
i__4 = i__ + k * a_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&a[
i__ + k * a_dim1]), dabs(r__2))) * ((r__3 = x[
i__5].r, dabs(r__3)) + (r__4 = r_imag(&x[i__ + j *
x_dim1]), dabs(r__4)));
/* L60: */
}
rwork[k] += s;
/* L70: */
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2))) / rwork[i__];
s = dmax(r__3,r__4);
} else {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__]
+ safe1);
s = dmax(r__3,r__4);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2.f <= lstres && count <= 5) {
/* Update solution and try again. */
csytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[1],
n, info);
caxpy_(n, &c_b1, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(A))* */
/* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(A) is the inverse of A */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(A)*abs(X) + abs(B) is less than SAFE2. */
/* Use CLACN2 to estimate the infinity-norm of the matrix */
/* inv(A) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__];
} else {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
clacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
csytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L110: */
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L120: */
}
csytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = i__ + j * x_dim1;
r__3 = lstres, r__4 = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__2));
lstres = dmax(r__3,r__4);
/* L130: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of CSYRFS */
} /* csyrfs_ */
int ctpcon_(char *norm, char *uplo, char *diag, int *n,
complex *ap, float *rcond, complex *work, float *rwork, int *info)
{
/* System generated locals */
int i__1;
float r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
int ix, kase, kase1;
float scale;
extern int lsame_(char *, char *);
int isave[3];
float anorm;
int upper;
extern int clacn2_(int *, complex *, complex *, float
*, int *, int *);
float xnorm;
extern int icamax_(int *, complex *, int *);
extern double slamch_(char *);
extern int xerbla_(char *, int *);
extern double clantp_(char *, char *, char *, int *, complex *,
float *);
extern int clatps_(char *, char *, char *, char *,
int *, complex *, complex *, float *, float *, int *);
float ainvnm;
extern int csrscl_(int *, float *, complex *, int
*);
int onenrm;
char normin[1];
float smlnum;
int nounit;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTPCON estimates the reciprocal of the condition number of a packed */
/* triangular matrix A, in either the 1-norm or the infinity-norm. */
/* The norm of A is computed and an estimate is obtained for */
/* norm(inv(A)), then the reciprocal of the condition number is */
/* computed as */
/* RCOND = 1 / ( norm(A) * norm(inv(A)) ). */
/* Arguments */
/* ========= */
/* NORM (input) CHARACTER*1 */
/* Specifies whether the 1-norm condition number or the */
/* infinity-norm condition number is required: */
/* = '1' or 'O': 1-norm; */
/* = 'I': Infinity-norm. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': A is upper triangular; */
/* = 'L': A is lower triangular. */
/* DIAG (input) CHARACTER*1 */
/* = 'N': A is non-unit triangular; */
/* = 'U': A is unit triangular. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* AP (input) COMPLEX array, dimension (N*(N+1)/2) */
/* The upper or lower triangular matrix A, packed columnwise in */
/* a linear array. The j-th column of A is stored in the array */
/* AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */
/* If DIAG = 'U', the diagonal elements of A are not referenced */
/* and are assumed to be 1. */
/* RCOND (output) REAL */
/* The reciprocal of the condition number of the matrix A, */
/* computed as RCOND = 1/(norm(A) * norm(inv(A))). */
/* WORK (workspace) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--rwork;
--work;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
onenrm = *(unsigned char *)norm == '1' || lsame_(norm, "O");
nounit = lsame_(diag, "N");
if (! onenrm && ! lsame_(norm, "I")) {
*info = -1;
} else if (! upper && ! lsame_(uplo, "L")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTPCON", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*rcond = 1.f;
return 0;
}
*rcond = 0.f;
smlnum = slamch_("Safe minimum") * (float) MAX(1,*n);
/* Compute the norm of the triangular matrix A. */
anorm = clantp_(norm, uplo, diag, n, &ap[1], &rwork[1]);
/* Continue only if ANORM > 0. */
if (anorm > 0.f) {
/* Estimate the norm of the inverse of A. */
ainvnm = 0.f;
*(unsigned char *)normin = 'N';
if (onenrm) {
kase1 = 1;
} else {
kase1 = 2;
}
kase = 0;
L10:
clacn2_(n, &work[*n + 1], &work[1], &ainvnm, &kase, isave);
if (kase != 0) {
if (kase == kase1) {
/* Multiply by inv(A). */
clatps_(uplo, "No transpose", diag, normin, n, &ap[1], &work[
1], &scale, &rwork[1], info);
} else {
/* Multiply by inv(A'). */
clatps_(uplo, "Conjugate transpose", diag, normin, n, &ap[1],
&work[1], &scale, &rwork[1], info);
}
*(unsigned char *)normin = 'Y';
/* Multiply by 1/SCALE if doing so will not cause overflow. */
if (scale != 1.f) {
ix = icamax_(n, &work[1], &c__1);
i__1 = ix;
xnorm = (r__1 = work[i__1].r, ABS(r__1)) + (r__2 = r_imag(&
work[ix]), ABS(r__2));
if (scale < xnorm * smlnum || scale == 0.f) {
goto L20;
}
csrscl_(n, &scale, &work[1], &c__1);
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / anorm / ainvnm;
}
}
L20:
return 0;
/* End of CTPCON */
} /* ctpcon_ */
