/* Subroutine */ int sposvx_(char *fact, char *uplo, integer *n, integer *
nrhs, real *a, integer *lda, real *af, integer *ldaf, char *equed,
real *s, real *b, integer *ldb, real *x, integer *ldx, real *rcond,
real *ferr, real *berr, real *work, integer *iwork, integer *info)
{
/* -- LAPACK driver routine (version 2.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
compute the solution to a real system of linear equations
A * X = B,
where A is an N-by-N symmetric positive definite matrix and X and B
are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also
provided.
Description
===========
The following steps are performed:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is a lower triangular
matrix.
3. The factored form of A is used to estimate the condition number
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.
Arguments
=========
FACT (input) CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF contains the factored form of A.
If EQUED = 'Y', the matrix A has been equilibrated
with scaling factors given by S. A and AF will not
be modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
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.
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/output) REAL array, dimension (LDA,N)
On entry, the symmetric matrix A, except if FACT = 'F' and
EQUED = 'Y', then A must contain the equilibrated matrix
diag(S)*A*diag(S). 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. A is not modified if
FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input or output) REAL array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
contains the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T, in the same storage
format as A. If EQUED .ne. 'N', then AF is the factored form
of the equilibrated matrix diag(S)*A*diag(S).
If FACT = 'N', then AF is an output argument and on exit
returns the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T of the original
matrix A.
If FACT = 'E', then AF is an output argument and on exit
returns the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T of the equilibrated
matrix A (see the description of A for the form of the
equilibrated matrix).
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
S (input or output) REAL array, dimension (N)
The scale factors for A; not accessed if EQUED = 'N'. S is
an input argument if FACT = 'F'; otherwise, S is an output
argument. If FACT = 'F' and EQUED = 'Y', each element of S
must be positive.
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
B is overwritten by diag(S) * B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) REAL array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that if EQUED = 'Y', A and B are
modified on exit, and the solution to the equilibrated system
is inv(diag(S))*X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) REAL
The estimate of the reciprocal condition number of the matrix
A after equilibration (if done). If RCOND is less than the
machine precision (in particular, if RCOND = 0), the matrix
is singular to working precision. This condition is
indicated by a return code of INFO > 0, and the solution and
error bounds are not computed.
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) REAL array, dimension (3*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: the leading minor of order i of A
is not positive definite, so the factorization
could not be completed, and the solution and error
bounds could not be computed.
= N+1: RCOND is less than machine precision. The
factorization has been completed, but the matrix
is singular to working precision, and the solution
and error bounds have not been computed.
=====================================================================
Parameter adjustments
Function Body */
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2;
real r__1, r__2;
/* Local variables */
static real amax, smin, smax;
static integer i, j;
extern logical lsame_(char *, char *);
static real scond, anorm;
static logical equil, rcequ;
extern doublereal slamch_(char *);
static logical nofact;
extern /* Subroutine */ int xerbla_(char *, integer *);
static real bignum;
static integer infequ;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *), spocon_(char *, integer *,
real *, integer *, real *, real *, real *, integer *, integer *);
extern doublereal slansy_(char *, char *, integer *, real *, integer *,
real *);
static real smlnum;
extern /* Subroutine */ int slaqsy_(char *, integer *, real *, integer *,
real *, real *, real *, char *), spoequ_(integer *
, real *, integer *, real *, real *, real *, integer *), sporfs_(
char *, integer *, integer *, real *, integer *, real *, integer *
, real *, integer *, real *, integer *, real *, real *, real *,
integer *, integer *), spotrf_(char *, integer *, real *,
integer *, integer *), spotrs_(char *, integer *, integer
*, real *, integer *, real *, integer *, integer *);
#define S(I) s[(I)-1]
#define FERR(I) ferr[(I)-1]
#define BERR(I) berr[(I)-1]
#define WORK(I) work[(I)-1]
#define IWORK(I) iwork[(I)-1]
#define A(I,J) a[(I)-1 + ((J)-1)* ( *lda)]
#define AF(I,J) af[(I)-1 + ((J)-1)* ( *ldaf)]
#define B(I,J) b[(I)-1 + ((J)-1)* ( *ldb)]
#define X(I,J) x[(I)-1 + ((J)-1)* ( *ldx)]
*info = 0;
nofact = lsame_(fact, "N");
equil = lsame_(fact, "E");
if (nofact || equil) {
*(unsigned char *)equed = 'N';
rcequ = FALSE_;
} else {
rcequ = lsame_(equed, "Y");
smlnum = slamch_("Safe minimum");
bignum = 1.f / smlnum;
}
/* Test the input parameters. */
if (! nofact && ! equil && ! lsame_(fact, "F")) {
*info = -1;
} else if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*nrhs < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else if (*ldaf < max(1,*n)) {
*info = -8;
} else if (lsame_(fact, "F") && ! (rcequ || lsame_(equed, "N"))) {
*info = -9;
} else {
if (rcequ) {
smin = bignum;
smax = 0.f;
i__1 = *n;
for (j = 1; j <= *n; ++j) {
/* Computing MIN */
r__1 = smin, r__2 = S(j);
smin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = smax, r__2 = S(j);
smax = dmax(r__1,r__2);
/* L10: */
}
if (smin <= 0.f) {
*info = -10;
} else if (*n > 0) {
scond = dmax(smin,smlnum) / dmin(smax,bignum);
} else {
scond = 1.f;
}
}
if (*info == 0) {
if (*ldb < max(1,*n)) {
*info = -12;
} else if (*ldx < max(1,*n)) {
*info = -14;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SPOSVX", &i__1);
return 0;
}
if (equil) {
/* Compute row and column scalings to equilibrate the matrix A.
*/
spoequ_(n, &A(1,1), lda, &S(1), &scond, &amax, &infequ);
if (infequ == 0) {
/* Equilibrate the matrix. */
slaqsy_(uplo, n, &A(1,1), lda, &S(1), &scond, &amax, equed);
rcequ = lsame_(equed, "Y");
}
}
/* Scale the right hand side. */
if (rcequ) {
i__1 = *nrhs;
for (j = 1; j <= *nrhs; ++j) {
i__2 = *n;
for (i = 1; i <= *n; ++i) {
B(i,j) = S(i) * B(i,j);
/* L20: */
}
/* L30: */
}
}
if (nofact || equil) {
/* Compute the Cholesky factorization A = U'*U or A = L*L'. */
slacpy_(uplo, n, n, &A(1,1), lda, &AF(1,1), ldaf);
spotrf_(uplo, n, &AF(1,1), ldaf, info);
/* Return if INFO is non-zero. */
if (*info != 0) {
if (*info > 0) {
*rcond = 0.f;
}
return 0;
}
}
/* Compute the norm of the matrix A. */
anorm = slansy_("1", uplo, n, &A(1,1), lda, &WORK(1));
/* Compute the reciprocal of the condition number of A. */
spocon_(uplo, n, &AF(1,1), ldaf, &anorm, rcond, &WORK(1), &IWORK(1),
info);
/* Return if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon")) {
*info = *n + 1;
return 0;
}
/* Compute the solution matrix X. */
slacpy_("Full", n, nrhs, &B(1,1), ldb, &X(1,1), ldx);
spotrs_(uplo, n, nrhs, &AF(1,1), ldaf, &X(1,1), ldx, info);
/* Use iterative refinement to improve the computed solution and
compute error bounds and backward error estimates for it. */
sporfs_(uplo, n, nrhs, &A(1,1), lda, &AF(1,1), ldaf, &B(1,1), ldb, &X(1,1), ldx, &FERR(1), &BERR(1), &WORK(1), &
IWORK(1), info);
/* Transform the solution matrix X to a solution of the original
system. */
if (rcequ) {
i__1 = *nrhs;
for (j = 1; j <= *nrhs; ++j) {
i__2 = *n;
for (i = 1; i <= *n; ++i) {
X(i,j) = S(i) * X(i,j);
/* L40: */
}
/* L50: */
}
i__1 = *nrhs;
for (j = 1; j <= *nrhs; ++j) {
FERR(j) /= scond;
/* L60: */
}
}
return 0;
/* End of SPOSVX */
} /* sposvx_ */
/* Subroutine */
int sposvxx_(char *fact, char *uplo, integer *n, integer * nrhs, real *a, integer *lda, real *af, integer *ldaf, char *equed, real *s, real *b, integer *ldb, real *x, integer *ldx, real *rcond, real *rpvgrw, real *berr, integer *n_err_bnds__, real * err_bnds_norm__, real *err_bnds_comp__, integer *nparams, real * params, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, x_offset, err_bnds_norm_dim1, err_bnds_norm_offset, err_bnds_comp_dim1, err_bnds_comp_offset, i__1;
real r__1, r__2;
/* Local variables */
integer j;
real amax, smin, smax;
extern real sla_porpvgrw_(char *, integer *, real *, integer *, real *, integer *, real *);
extern logical lsame_(char *, char *);
real scond;
logical equil, rcequ;
extern real slamch_(char *);
logical nofact;
extern /* Subroutine */
int xerbla_(char *, integer *);
real bignum;
integer infequ;
extern /* Subroutine */
int slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *);
real smlnum;
extern /* Subroutine */
int slaqsy_(char *, integer *, real *, integer *, real *, real *, real *, char *), spotrf_(char *, integer *, real *, integer *, integer *), spotrs_(char *, integer *, integer *, real *, integer *, real *, integer *, integer *), slascl2_(integer *, integer *, real *, real *, integer *), spoequb_(integer *, real *, integer *, real *, real * , real *, integer *), sporfsx_(char *, char *, integer *, integer *, real *, integer *, real *, integer *, real *, real *, integer * , real *, integer *, real *, real *, integer *, real *, real *, integer *, real *, real *, integer *, integer *);
/* -- LAPACK driver routine (version 3.4.1) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* April 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
err_bnds_comp_dim1 = *nrhs;
err_bnds_comp_offset = 1 + err_bnds_comp_dim1;
err_bnds_comp__ -= err_bnds_comp_offset;
err_bnds_norm_dim1 = *nrhs;
err_bnds_norm_offset = 1 + err_bnds_norm_dim1;
err_bnds_norm__ -= err_bnds_norm_offset;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--s;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--berr;
--params;
--work;
--iwork;
/* Function Body */
*info = 0;
nofact = lsame_(fact, "N");
equil = lsame_(fact, "E");
smlnum = slamch_("Safe minimum");
bignum = 1.f / smlnum;
if (nofact || equil)
{
*(unsigned char *)equed = 'N';
rcequ = FALSE_;
}
else
{
rcequ = lsame_(equed, "Y");
}
/* Default is failure. If an input parameter is wrong or */
/* factorization fails, make everything look horrible. Only the */
/* pivot growth is set here, the rest is initialized in SPORFSX. */
*rpvgrw = 0.f;
/* Test the input parameters. PARAMS is not tested until SPORFSX. */
if (! nofact && ! equil && ! lsame_(fact, "F"))
{
*info = -1;
}
else if (! lsame_(uplo, "U") && ! lsame_(uplo, "L"))
{
*info = -2;
}
else if (*n < 0)
{
*info = -3;
}
else if (*nrhs < 0)
{
*info = -4;
}
else if (*lda < max(1,*n))
{
*info = -6;
}
else if (*ldaf < max(1,*n))
{
*info = -8;
}
else if (lsame_(fact, "F") && ! (rcequ || lsame_( equed, "N")))
{
*info = -9;
}
else
{
if (rcequ)
{
smin = bignum;
smax = 0.f;
i__1 = *n;
for (j = 1;
j <= i__1;
++j)
{
/* Computing MIN */
r__1 = smin;
r__2 = s[j]; // , expr subst
smin = min(r__1,r__2);
/* Computing MAX */
r__1 = smax;
r__2 = s[j]; // , expr subst
smax = max(r__1,r__2);
/* L10: */
}
if (smin <= 0.f)
{
*info = -10;
}
else if (*n > 0)
{
scond = max(smin,smlnum) / min(smax,bignum);
}
else
{
scond = 1.f;
}
}
if (*info == 0)
{
if (*ldb < max(1,*n))
{
*info = -12;
}
else if (*ldx < max(1,*n))
{
*info = -14;
}
}
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("SPOSVXX", &i__1);
return 0;
}
if (equil)
{
/* Compute row and column scalings to equilibrate the matrix A. */
spoequb_(n, &a[a_offset], lda, &s[1], &scond, &amax, &infequ);
if (infequ == 0)
{
/* Equilibrate the matrix. */
slaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed);
rcequ = lsame_(equed, "Y");
}
}
/* Scale the right-hand side. */
if (rcequ)
{
slascl2_(n, nrhs, &s[1], &b[b_offset], ldb);
}
if (nofact || equil)
{
/* Compute the Cholesky factorization of A. */
slacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf);
spotrf_(uplo, n, &af[af_offset], ldaf, info);
/* Return if INFO is non-zero. */
if (*info != 0)
{
/* Pivot in column INFO is exactly 0 */
/* Compute the reciprocal pivot growth factor of the */
/* leading rank-deficient INFO columns of A. */
*rpvgrw = sla_porpvgrw_(uplo, info, &a[a_offset], lda, &af[ af_offset], ldaf, &work[1]);
return 0;
}
}
/* Compute the reciprocal growth factor RPVGRW. */
*rpvgrw = sla_porpvgrw_(uplo, n, &a[a_offset], lda, &af[af_offset], ldaf, &work[1]);
/* Compute the solution matrix X. */
slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
spotrs_(uplo, n, nrhs, &af[af_offset], ldaf, &x[x_offset], ldx, info);
/* Use iterative refinement to improve the computed solution and */
/* compute error bounds and backward error estimates for it. */
sporfsx_(uplo, equed, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, & s[1], &b[b_offset], ldb, &x[x_offset], ldx, rcond, &berr[1], n_err_bnds__, &err_bnds_norm__[err_bnds_norm_offset], & err_bnds_comp__[err_bnds_comp_offset], nparams, ¶ms[1], &work[ 1], &iwork[1], info);
/* Scale solutions. */
if (rcequ)
{
slascl2_(n, nrhs, &s[1], &x[x_offset], ldx);
}
return 0;
/* End of SPOSVXX */
}
/* Subroutine */ int sposvxx_(char *fact, char *uplo, integer *n, integer *
nrhs, real *a, integer *lda, real *af, integer *ldaf, char *equed,
real *s, real *b, integer *ldb, real *x, integer *ldx, real *rcond,
real *rpvgrw, real *berr, integer *n_err_bnds__, real *
err_bnds_norm__, real *err_bnds_comp__, integer *nparams, real *
params, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, err_bnds_norm_dim1, err_bnds_norm_offset,
err_bnds_comp_dim1, err_bnds_comp_offset, i__1;
real r__1, r__2;
/* Local variables */
integer j;
real amax, smin, smax;
extern doublereal sla_porpvgrw__(char *, integer *, real *, integer *,
real *, integer *, real *, ftnlen);
extern logical lsame_(char *, char *);
real scond;
logical equil, rcequ;
extern doublereal slamch_(char *);
logical nofact;
extern /* Subroutine */ int xerbla_(char *, integer *);
real bignum;
integer infequ;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *);
real smlnum;
extern /* Subroutine */ int slaqsy_(char *, integer *, real *, integer *,
real *, real *, real *, char *), spotrf_(char *,
integer *, real *, integer *, integer *), spotrs_(char *,
integer *, integer *, real *, integer *, real *, integer *,
integer *), slascl2_(integer *, integer *, real *, real *,
integer *), spoequb_(integer *, real *, integer *, real *, real *
, real *, integer *), sporfsx_(char *, char *, integer *, integer
*, real *, integer *, real *, integer *, real *, real *, integer *
, real *, integer *, real *, real *, integer *, real *, real *,
integer *, real *, real *, integer *, integer *);
/* -- LAPACK driver routine (version 3.2) -- */
/* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/* -- Jason Riedy of Univ. of California Berkeley. -- */
/* -- November 2008 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley and NAG Ltd. -- */
/* .. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T */
/* to compute the solution to a real system of linear equations */
/* A * X = B, where A is an N-by-N symmetric positive definite matrix */
/* and X and B are N-by-NRHS matrices. */
/* If requested, both normwise and maximum componentwise error bounds */
/* are returned. SPOSVXX will return a solution with a tiny */
/* guaranteed error (O(eps) where eps is the working machine */
/* precision) unless the matrix is very ill-conditioned, in which */
/* case a warning is returned. Relevant condition numbers also are */
/* calculated and returned. */
/* SPOSVXX accepts user-provided factorizations and equilibration */
/* factors; see the definitions of the FACT and EQUED options. */
/* Solving with refinement and using a factorization from a previous */
/* SPOSVXX call will also produce a solution with either O(eps) */
/* errors or warnings, but we cannot make that claim for general */
/* user-provided factorizations and equilibration factors if they */
/* differ from what SPOSVXX would itself produce. */
/* Description */
/* =========== */
/* The following steps are performed: */
/* 1. If FACT = 'E', real scaling factors are computed to equilibrate */
/* the system: */
/* diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B */
/* Whether or not the system will be equilibrated depends on the */
/* scaling of the matrix A, but if equilibration is used, A is */
/* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */
/* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */
/* factor the matrix A (after equilibration if FACT = 'E') as */
/* A = U**T* U, if UPLO = 'U', or */
/* A = L * L**T, if UPLO = 'L', */
/* where U is an upper triangular matrix and L is a lower triangular */
/* matrix. */
/* 3. If the leading i-by-i principal minor is not positive definite, */
/* then the routine returns with INFO = i. Otherwise, the factored */
/* form of A is used to estimate the condition number of the matrix */
/* A (see argument RCOND). If the reciprocal of the condition number */
/* is less than machine precision, the routine still goes on to solve */
/* for X and compute error bounds as described below. */
/* 4. The system of equations is solved for X using the factored form */
/* of A. */
/* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), */
/* the routine will use iterative refinement to try to get a small */
/* error and error bounds. Refinement calculates the residual to at */
/* least twice the working precision. */
/* 6. If equilibration was used, the matrix X is premultiplied by */
/* diag(S) so that it solves the original system before */
/* equilibration. */
/* Arguments */
/* ========= */
/* Some optional parameters are bundled in the PARAMS array. These */
/* settings determine how refinement is performed, but often the */
/* defaults are acceptable. If the defaults are acceptable, users */
/* can pass NPARAMS = 0 which prevents the source code from accessing */
/* the PARAMS argument. */
/* FACT (input) CHARACTER*1 */
/* Specifies whether or not the factored form of the matrix A is */
/* supplied on entry, and if not, whether the matrix A should be */
/* equilibrated before it is factored. */
/* = 'F': On entry, AF contains the factored form of A. */
/* If EQUED is not 'N', the matrix A has been */
/* equilibrated with scaling factors given by S. */
/* A and AF are not modified. */
/* = 'N': The matrix A will be copied to AF and factored. */
/* = 'E': The matrix A will be equilibrated if necessary, then */
/* copied to AF and factored. */
/* 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. */
/* 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/output) REAL array, dimension (LDA,N) */
/* On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = */
/* 'Y', then A must contain the equilibrated matrix */
/* diag(S)*A*diag(S). 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. A is */
/* not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = */
/* 'N' on exit. */
/* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */
/* diag(S)*A*diag(S). */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input or output) REAL array, dimension (LDAF,N) */
/* If FACT = 'F', then AF is an input argument and on entry */
/* contains the triangular factor U or L from the Cholesky */
/* factorization A = U**T*U or A = L*L**T, in the same storage */
/* format as A. If EQUED .ne. 'N', then AF is the factored */
/* form of the equilibrated matrix diag(S)*A*diag(S). */
/* If FACT = 'N', then AF is an output argument and on exit */
/* returns the triangular factor U or L from the Cholesky */
/* factorization A = U**T*U or A = L*L**T of the original */
/* matrix A. */
/* If FACT = 'E', then AF is an output argument and on exit */
/* returns the triangular factor U or L from the Cholesky */
/* factorization A = U**T*U or A = L*L**T of the equilibrated */
/* matrix A (see the description of A for the form of the */
/* equilibrated matrix). */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* EQUED (input or output) CHARACTER*1 */
/* Specifies the form of equilibration that was done. */
/* = 'N': No equilibration (always true if FACT = 'N'). */
/* = 'Y': Both row and column equilibration, i.e., A has been */
/* replaced by diag(S) * A * diag(S). */
/* EQUED is an input argument if FACT = 'F'; otherwise, it is an */
/* output argument. */
/* S (input or output) REAL array, dimension (N) */
/* The row scale factors for A. If EQUED = 'Y', A is multiplied on */
/* the left and right by diag(S). S is an input argument if FACT = */
/* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED */
/* = 'Y', each element of S must be positive. If S is output, each */
/* element of S is a power of the radix. If S is input, each element */
/* of S should be a power of the radix to ensure a reliable solution */
/* and error estimates. Scaling by powers of the radix does not cause */
/* rounding errors unless the result underflows or overflows. */
/* Rounding errors during scaling lead to refining with a matrix that */
/* is not equivalent to the input matrix, producing error estimates */
/* that may not be reliable. */
/* B (input/output) REAL array, dimension (LDB,NRHS) */
/* On entry, the N-by-NRHS right hand side matrix B. */
/* On exit, */
/* if EQUED = 'N', B is not modified; */
/* if EQUED = 'Y', B is overwritten by diag(S)*B; */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (output) REAL array, dimension (LDX,NRHS) */
/* If INFO = 0, the N-by-NRHS solution matrix X to the original */
/* system of equations. Note that A and B are modified on exit if */
/* EQUED .ne. 'N', and the solution to the equilibrated system is */
/* inv(diag(S))*X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* RCOND (output) REAL */
/* Reciprocal scaled condition number. This is an estimate of the */
/* reciprocal Skeel condition number of the matrix A after */
/* equilibration (if done). If this is less than the machine */
/* precision (in particular, if it is zero), the matrix is singular */
/* to working precision. Note that the error may still be small even */
/* if this number is very small and the matrix appears ill- */
/* conditioned. */
/* RPVGRW (output) REAL */
/* Reciprocal pivot growth. On exit, this contains the reciprocal */
/* pivot growth factor norm(A)/norm(U). The "max absolute element" */
/* norm is used. If this is much less than 1, then the stability of */
/* the LU factorization of the (equilibrated) matrix A could be poor. */
/* This also means that the solution X, estimated condition numbers, */
/* and error bounds could be unreliable. If factorization fails with */
/* 0<INFO<=N, then this contains the reciprocal pivot growth factor */
/* for the leading INFO columns of A. */
/* BERR (output) REAL array, dimension (NRHS) */
/* Componentwise relative backward error. This is 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). */
/* N_ERR_BNDS (input) INTEGER */
/* Number of error bounds to return for each right hand side */
/* and each type (normwise or componentwise). See ERR_BNDS_NORM and */
/* ERR_BNDS_COMP below. */
/* ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* normwise relative error, which is defined as follows: */
/* Normwise relative error in the ith solution vector: */
/* max_j (abs(XTRUE(j,i) - X(j,i))) */
/* ------------------------------ */
/* max_j abs(X(j,i)) */
/* The array is indexed by the type of error information as described */
/* below. There currently are up to three pieces of information */
/* returned. */
/* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_NORM(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * slamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * slamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated normwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * slamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*A, where S scales each row by a power of the */
/* radix so all absolute row sums of Z are approximately 1. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* componentwise relative error, which is defined as follows: */
/* Componentwise relative error in the ith solution vector: */
/* abs(XTRUE(j,i) - X(j,i)) */
/* max_j ---------------------- */
/* abs(X(j,i)) */
/* The array is indexed by the right-hand side i (on which the */
/* componentwise relative error depends), and the type of error */
/* information as described below. There currently are up to three */
/* pieces of information returned for each right-hand side. If */
/* componentwise accuracy is not requested (PARAMS(3) = 0.0), then */
/* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most */
/* the first (:,N_ERR_BNDS) entries are returned. */
/* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_COMP(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * slamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * slamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated componentwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * slamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*(A*diag(x)), where x is the solution for the */
/* current right-hand side and S scales each row of */
/* A*diag(x) by a power of the radix so all absolute row */
/* sums of Z are approximately 1. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* NPARAMS (input) INTEGER */
/* Specifies the number of parameters set in PARAMS. If .LE. 0, the */
/* PARAMS array is never referenced and default values are used. */
/* PARAMS (input / output) REAL array, dimension NPARAMS */
/* Specifies algorithm parameters. If an entry is .LT. 0.0, then */
/* that entry will be filled with default value used for that */
/* parameter. Only positions up to NPARAMS are accessed; defaults */
/* are used for higher-numbered parameters. */
/* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative */
/* refinement or not. */
/* Default: 1.0 */
/* = 0.0 : No refinement is performed, and no error bounds are */
/* computed. */
/* = 1.0 : Use the double-precision refinement algorithm, */
/* possibly with doubled-single computations if the */
/* compilation environment does not support DOUBLE */
/* PRECISION. */
/* (other values are reserved for future use) */
/* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual */
/* computations allowed for refinement. */
/* Default: 10 */
/* Aggressive: Set to 100 to permit convergence using approximate */
/* factorizations or factorizations other than LU. If */
/* the factorization uses a technique other than */
/* Gaussian elimination, the guarantees in */
/* err_bnds_norm and err_bnds_comp may no longer be */
/* trustworthy. */
/* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code */
/* will attempt to find a solution with small componentwise */
/* relative error in the double-precision algorithm. Positive */
/* is true, 0.0 is false. */
/* Default: 1.0 (attempt componentwise convergence) */
/* WORK (workspace) REAL array, dimension (4*N) */
/* IWORK (workspace) INTEGER array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: Successful exit. The solution to every right-hand side is */
/* guaranteed. */
/* < 0: If INFO = -i, the i-th argument had an illegal value */
/* > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization */
/* has been completed, but the factor U is exactly singular, so */
/* the solution and error bounds could not be computed. RCOND = 0 */
/* is returned. */
/* = N+J: The solution corresponding to the Jth right-hand side is */
/* not guaranteed. The solutions corresponding to other right- */
/* hand sides K with K > J may not be guaranteed as well, but */
/* only the first such right-hand side is reported. If a small */
/* componentwise error is not requested (PARAMS(3) = 0.0) then */
/* the Jth right-hand side is the first with a normwise error */
/* bound that is not guaranteed (the smallest J such */
/* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) */
/* the Jth right-hand side is the first with either a normwise or */
/* componentwise error bound that is not guaranteed (the smallest */
/* J such that either ERR_BNDS_NORM(J,1) = 0.0 or */
/* ERR_BNDS_COMP(J,1) = 0.0). See the definition of */
/* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information */
/* about all of the right-hand sides check ERR_BNDS_NORM or */
/* ERR_BNDS_COMP. */
/* ================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
err_bnds_comp_dim1 = *nrhs;
err_bnds_comp_offset = 1 + err_bnds_comp_dim1;
err_bnds_comp__ -= err_bnds_comp_offset;
err_bnds_norm_dim1 = *nrhs;
err_bnds_norm_offset = 1 + err_bnds_norm_dim1;
err_bnds_norm__ -= err_bnds_norm_offset;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--s;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--berr;
--params;
--work;
--iwork;
/* Function Body */
*info = 0;
nofact = lsame_(fact, "N");
equil = lsame_(fact, "E");
smlnum = slamch_("Safe minimum");
bignum = 1.f / smlnum;
if (nofact || equil) {
*(unsigned char *)equed = 'N';
rcequ = FALSE_;
} else {
rcequ = lsame_(equed, "Y");
}
/* Default is failure. If an input parameter is wrong or */
/* factorization fails, make everything look horrible. Only the */
/* pivot growth is set here, the rest is initialized in SPORFSX. */
*rpvgrw = 0.f;
/* Test the input parameters. PARAMS is not tested until SPORFSX. */
if (! nofact && ! equil && ! lsame_(fact, "F")) {
*info = -1;
} else if (! lsame_(uplo, "U") && ! lsame_(uplo,
"L")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*nrhs < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else if (*ldaf < max(1,*n)) {
*info = -8;
} else if (lsame_(fact, "F") && ! (rcequ || lsame_(
equed, "N"))) {
*info = -9;
} else {
if (rcequ) {
smin = bignum;
smax = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
r__1 = smin, r__2 = s[j];
smin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = smax, r__2 = s[j];
smax = dmax(r__1,r__2);
/* L10: */
}
if (smin <= 0.f) {
*info = -10;
} else if (*n > 0) {
scond = dmax(smin,smlnum) / dmin(smax,bignum);
} else {
scond = 1.f;
}
}
if (*info == 0) {
if (*ldb < max(1,*n)) {
*info = -12;
} else if (*ldx < max(1,*n)) {
*info = -14;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SPOSVXX", &i__1);
return 0;
}
if (equil) {
/* Compute row and column scalings to equilibrate the matrix A. */
spoequb_(n, &a[a_offset], lda, &s[1], &scond, &amax, &infequ);
if (infequ == 0) {
/* Equilibrate the matrix. */
slaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed);
rcequ = lsame_(equed, "Y");
}
}
/* Scale the right-hand side. */
if (rcequ) {
slascl2_(n, nrhs, &s[1], &b[b_offset], ldb);
}
if (nofact || equil) {
/* Compute the LU factorization of A. */
slacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf);
spotrf_(uplo, n, &af[af_offset], ldaf, info);
/* Return if INFO is non-zero. */
if (*info != 0) {
/* Pivot in column INFO is exactly 0 */
/* Compute the reciprocal pivot growth factor of the */
/* leading rank-deficient INFO columns of A. */
*rpvgrw = sla_porpvgrw__(uplo, info, &a[a_offset], lda, &af[
af_offset], ldaf, &work[1], (ftnlen)1);
return 0;
}
}
/* Compute the reciprocal growth factor RPVGRW. */
*rpvgrw = sla_porpvgrw__(uplo, n, &a[a_offset], lda, &af[af_offset], ldaf,
&work[1], (ftnlen)1);
/* Compute the solution matrix X. */
slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
spotrs_(uplo, n, nrhs, &af[af_offset], ldaf, &x[x_offset], ldx, info);
/* Use iterative refinement to improve the computed solution and */
/* compute error bounds and backward error estimates for it. */
sporfsx_(uplo, equed, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &
s[1], &b[b_offset], ldb, &x[x_offset], ldx, rcond, &berr[1],
n_err_bnds__, &err_bnds_norm__[err_bnds_norm_offset], &
err_bnds_comp__[err_bnds_comp_offset], nparams, ¶ms[1], &work[
1], &iwork[1], info);
/* Scale solutions. */
if (rcequ) {
slascl2_(n, nrhs, &s[1], &x[x_offset], ldx);
}
return 0;
/* End of SPOSVXX */
} /* sposvxx_ */
/* Subroutine */ int sposvx_(char *fact, char *uplo, integer *n, integer *
nrhs, real *a, integer *lda, real *af, integer *ldaf, char *equed,
real *s, real *b, integer *ldb, real *x, integer *ldx, real *rcond,
real *ferr, real *berr, real *work, integer *iwork, integer *info,
ftnlen fact_len, ftnlen uplo_len, ftnlen equed_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2;
real r__1, r__2;
/* Local variables */
static integer i__, j;
static real amax, smin, smax;
extern logical lsame_(char *, char *, ftnlen, ftnlen);
static real scond, anorm;
static logical equil, rcequ;
extern doublereal slamch_(char *, ftnlen);
static logical nofact;
extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
static real bignum;
static integer infequ;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *, ftnlen), spocon_(char *, integer *,
real *, integer *, real *, real *, real *, integer *, integer *,
ftnlen);
extern doublereal slansy_(char *, char *, integer *, real *, integer *,
real *, ftnlen, ftnlen);
static real smlnum;
extern /* Subroutine */ int slaqsy_(char *, integer *, real *, integer *,
real *, real *, real *, char *, ftnlen, ftnlen), spoequ_(integer *
, real *, integer *, real *, real *, real *, integer *), sporfs_(
char *, integer *, integer *, real *, integer *, real *, integer *
, real *, integer *, real *, integer *, real *, real *, real *,
integer *, integer *, ftnlen), spotrf_(char *, integer *, real *,
integer *, integer *, ftnlen), spotrs_(char *, integer *, integer
*, real *, integer *, real *, integer *, integer *, ftnlen);
/* -- LAPACK driver routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* June 30, 1999 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to */
/* compute the solution to a real system of linear equations */
/* A * X = B, */
/* where A is an N-by-N symmetric positive definite matrix and X and B */
/* are N-by-NRHS matrices. */
/* Error bounds on the solution and a condition estimate are also */
/* provided. */
/* Description */
/* =========== */
/* The following steps are performed: */
/* 1. If FACT = 'E', real scaling factors are computed to equilibrate */
/* the system: */
/* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B */
/* Whether or not the system will be equilibrated depends on the */
/* scaling of the matrix A, but if equilibration is used, A is */
/* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */
/* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */
/* factor the matrix A (after equilibration if FACT = 'E') as */
/* A = U**T* U, if UPLO = 'U', or */
/* A = L * L**T, if UPLO = 'L', */
/* where U is an upper triangular matrix and L is a lower triangular */
/* matrix. */
/* 3. If the leading i-by-i principal minor is not positive definite, */
/* then the routine returns with INFO = i. Otherwise, the factored */
/* form of A is used to estimate the condition number of the matrix */
/* A. If the reciprocal of the condition number is less than machine */
/* precision, INFO = N+1 is returned as a warning, but the routine */
/* still goes on to solve for X and compute error bounds as */
/* described below. */
/* 4. The system of equations is solved for X using the factored form */
/* of A. */
/* 5. Iterative refinement is applied to improve the computed solution */
/* matrix and calculate error bounds and backward error estimates */
/* for it. */
/* 6. If equilibration was used, the matrix X is premultiplied by */
/* diag(S) so that it solves the original system before */
/* equilibration. */
/* Arguments */
/* ========= */
/* FACT (input) CHARACTER*1 */
/* Specifies whether or not the factored form of the matrix A is */
/* supplied on entry, and if not, whether the matrix A should be */
/* equilibrated before it is factored. */
/* = 'F': On entry, AF contains the factored form of A. */
/* If EQUED = 'Y', the matrix A has been equilibrated */
/* with scaling factors given by S. A and AF will not */
/* be modified. */
/* = 'N': The matrix A will be copied to AF and factored. */
/* = 'E': The matrix A will be equilibrated if necessary, then */
/* copied to AF and factored. */
/* 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. */
/* 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/output) REAL array, dimension (LDA,N) */
/* On entry, the symmetric matrix A, except if FACT = 'F' and */
/* EQUED = 'Y', then A must contain the equilibrated matrix */
/* diag(S)*A*diag(S). 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. A is not modified if */
/* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. */
/* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */
/* diag(S)*A*diag(S). */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input or output) REAL array, dimension (LDAF,N) */
/* If FACT = 'F', then AF is an input argument and on entry */
/* contains the triangular factor U or L from the Cholesky */
/* factorization A = U**T*U or A = L*L**T, in the same storage */
/* format as A. If EQUED .ne. 'N', then AF is the factored form */
/* of the equilibrated matrix diag(S)*A*diag(S). */
/* If FACT = 'N', then AF is an output argument and on exit */
/* returns the triangular factor U or L from the Cholesky */
/* factorization A = U**T*U or A = L*L**T of the original */
/* matrix A. */
/* If FACT = 'E', then AF is an output argument and on exit */
/* returns the triangular factor U or L from the Cholesky */
/* factorization A = U**T*U or A = L*L**T of the equilibrated */
/* matrix A (see the description of A for the form of the */
/* equilibrated matrix). */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* EQUED (input or output) CHARACTER*1 */
/* Specifies the form of equilibration that was done. */
/* = 'N': No equilibration (always true if FACT = 'N'). */
/* = 'Y': Equilibration was done, i.e., A has been replaced by */
/* diag(S) * A * diag(S). */
/* EQUED is an input argument if FACT = 'F'; otherwise, it is an */
/* output argument. */
/* S (input or output) REAL array, dimension (N) */
/* The scale factors for A; not accessed if EQUED = 'N'. S is */
/* an input argument if FACT = 'F'; otherwise, S is an output */
/* argument. If FACT = 'F' and EQUED = 'Y', each element of S */
/* must be positive. */
/* B (input/output) REAL array, dimension (LDB,NRHS) */
/* On entry, the N-by-NRHS right hand side matrix B. */
/* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', */
/* B is overwritten by diag(S) * B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (output) REAL array, dimension (LDX,NRHS) */
/* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */
/* the original system of equations. Note that if EQUED = 'Y', */
/* A and B are modified on exit, and the solution to the */
/* equilibrated system is inv(diag(S))*X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* RCOND (output) REAL */
/* The estimate of the reciprocal condition number of the matrix */
/* A after equilibration (if done). If RCOND is less than the */
/* machine precision (in particular, if RCOND = 0), the matrix */
/* is singular to working precision. This condition is */
/* indicated by a return code of INFO > 0. */
/* 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) REAL array, dimension (3*N) */
/* IWORK (workspace) INTEGER array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, and i is */
/* <= N: the leading minor of order i of A is */
/* not positive definite, so the factorization */
/* could not be completed, and the solution has not */
/* been computed. RCOND = 0 is returned. */
/* = N+1: U is nonsingular, but RCOND is less than machine */
/* precision, meaning that the matrix is singular */
/* to working precision. Nevertheless, the */
/* solution and error bounds are computed because */
/* there are a number of situations where the */
/* computed solution can be more accurate than the */
/* value of RCOND would suggest. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--s;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--iwork;
/* Function Body */
*info = 0;
nofact = lsame_(fact, "N", (ftnlen)1, (ftnlen)1);
equil = lsame_(fact, "E", (ftnlen)1, (ftnlen)1);
if (nofact || equil) {
*(unsigned char *)equed = 'N';
rcequ = FALSE_;
} else {
rcequ = lsame_(equed, "Y", (ftnlen)1, (ftnlen)1);
smlnum = slamch_("Safe minimum", (ftnlen)12);
bignum = 1.f / smlnum;
}
/* Test the input parameters. */
if (! nofact && ! equil && ! lsame_(fact, "F", (ftnlen)1, (ftnlen)1)) {
*info = -1;
} else if (! lsame_(uplo, "U", (ftnlen)1, (ftnlen)1) && ! lsame_(uplo,
"L", (ftnlen)1, (ftnlen)1)) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*nrhs < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else if (*ldaf < max(1,*n)) {
*info = -8;
} else if (lsame_(fact, "F", (ftnlen)1, (ftnlen)1) && ! (rcequ || lsame_(
equed, "N", (ftnlen)1, (ftnlen)1))) {
*info = -9;
} else {
if (rcequ) {
smin = bignum;
smax = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
r__1 = smin, r__2 = s[j];
smin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = smax, r__2 = s[j];
smax = dmax(r__1,r__2);
/* L10: */
}
if (smin <= 0.f) {
*info = -10;
} else if (*n > 0) {
scond = dmax(smin,smlnum) / dmin(smax,bignum);
} else {
scond = 1.f;
}
}
if (*info == 0) {
if (*ldb < max(1,*n)) {
*info = -12;
} else if (*ldx < max(1,*n)) {
*info = -14;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SPOSVX", &i__1, (ftnlen)6);
return 0;
}
if (equil) {
/* Compute row and column scalings to equilibrate the matrix A. */
spoequ_(n, &a[a_offset], lda, &s[1], &scond, &amax, &infequ);
if (infequ == 0) {
/* Equilibrate the matrix. */
slaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed, (
ftnlen)1, (ftnlen)1);
rcequ = lsame_(equed, "Y", (ftnlen)1, (ftnlen)1);
}
}
/* Scale the right hand side. */
if (rcequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
b[i__ + j * b_dim1] = s[i__] * b[i__ + j * b_dim1];
/* L20: */
}
/* L30: */
}
}
if (nofact || equil) {
/* Compute the Cholesky factorization A = U'*U or A = L*L'. */
slacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf, (ftnlen)
1);
spotrf_(uplo, n, &af[af_offset], ldaf, info, (ftnlen)1);
/* Return if INFO is non-zero. */
if (*info != 0) {
if (*info > 0) {
*rcond = 0.f;
}
return 0;
}
}
/* Compute the norm of the matrix A. */
anorm = slansy_("1", uplo, n, &a[a_offset], lda, &work[1], (ftnlen)1, (
ftnlen)1);
/* Compute the reciprocal of the condition number of A. */
spocon_(uplo, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &iwork[1],
info, (ftnlen)1);
/* Set INFO = N+1 if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon", (ftnlen)7)) {
*info = *n + 1;
}
/* Compute the solution matrix X. */
slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx, (ftnlen)4);
spotrs_(uplo, n, nrhs, &af[af_offset], ldaf, &x[x_offset], ldx, info, (
ftnlen)1);
/* Use iterative refinement to improve the computed solution and */
/* compute error bounds and backward error estimates for it. */
sporfs_(uplo, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &b[
b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[1], &
iwork[1], info, (ftnlen)1);
/* Transform the solution matrix X to a solution of the original */
/* system. */
if (rcequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
x[i__ + j * x_dim1] = s[i__] * x[i__ + j * x_dim1];
/* L40: */
}
/* L50: */
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] /= scond;
/* L60: */
}
}
return 0;
/* End of SPOSVX */
} /* sposvx_ */
