/* Subroutine */ int sdrvge_(logical *dotype, integer *nn, integer *nval,
integer *nrhs, real *thresh, logical *tsterr, integer *nmax, real *a,
real *afac, real *asav, real *b, real *bsav, real *x, real *xact,
real *s, real *work, real *rwork, integer *iwork, integer *nout)
{
/* Initialized data */
static integer iseedy[4] = { 1988,1989,1990,1991 };
static char transs[1*3] = "N" "T" "C";
static char facts[1*3] = "F" "N" "E";
static char equeds[1*4] = "N" "R" "C" "B";
/* Format strings */
static char fmt_9999[] = "(1x,a6,\002, N =\002,i5,\002, type \002,i2,"
"\002, test(\002,i2,\002) =\002,g12.5)";
static char fmt_9997[] = "(1x,a6,\002, FACT='\002,a1,\002', TRANS='\002,"
"a1,\002', N=\002,i5,\002, EQUED='\002,a1,\002', type \002,i2,"
"\002, test(\002,i1,\002)=\002,g12.5)";
static char fmt_9998[] = "(1x,a6,\002, FACT='\002,a1,\002', TRANS='\002,"
"a1,\002', N=\002,i5,\002, type \002,i2,\002, test(\002,i1,\002)"
"=\002,g12.5)";
/* System generated locals */
address a__1[2];
integer i__1, i__2, i__3, i__4, i__5[2];
real r__1;
char ch__1[2];
/* Builtin functions */
/* Subroutine */ int s_copy(char *, char *, ftnlen, ftnlen);
integer s_wsfe(cilist *), do_fio(integer *, char *, ftnlen), e_wsfe(void);
/* Subroutine */ int s_cat(char *, char **, integer *, integer *, ftnlen);
/* Local variables */
integer i__, k, n, k1, nb, in, kl, ku, nt, lda;
char fact[1];
integer ioff, mode;
real amax;
char path[3];
integer imat, info;
char dist[1], type__[1];
integer nrun, ifact, nfail, iseed[4], nfact;
extern logical lsame_(char *, char *);
char equed[1];
integer nbmin;
real rcond, roldc;
extern /* Subroutine */ int sget01_(integer *, integer *, real *, integer
*, real *, integer *, integer *, real *, real *);
integer nimat;
real roldi;
extern doublereal sget06_(real *, real *);
extern /* Subroutine */ int sget02_(char *, integer *, integer *, integer
*, real *, integer *, real *, integer *, real *, integer *, real *
, real *);
real anorm;
integer itran;
extern /* Subroutine */ int sget04_(integer *, integer *, real *, integer
*, real *, integer *, real *, real *);
logical equil;
real roldo;
extern /* Subroutine */ int sget07_(char *, integer *, integer *, real *,
integer *, real *, integer *, real *, integer *, real *, integer *
, real *, real *, real *);
char trans[1];
integer izero, nerrs;
extern /* Subroutine */ int sgesv_(integer *, integer *, real *, integer *
, integer *, real *, integer *, integer *);
integer lwork;
logical zerot;
char xtype[1];
extern /* Subroutine */ int slatb4_(char *, integer *, integer *, integer
*, char *, integer *, integer *, real *, integer *, real *, char *
), aladhd_(integer *, char *),
alaerh_(char *, char *, integer *, integer *, char *, integer *,
integer *, integer *, integer *, integer *, integer *, integer *,
integer *, integer *);
logical prefac;
real colcnd;
extern doublereal slamch_(char *);
real rcondc;
extern doublereal slange_(char *, integer *, integer *, real *, integer *,
real *);
logical nofact;
integer iequed;
extern /* Subroutine */ int slaqge_(integer *, integer *, real *, integer
*, real *, real *, real *, real *, real *, char *);
real rcondi;
extern /* Subroutine */ int alasvm_(char *, integer *, integer *, integer
*, integer *);
real cndnum, anormi, rcondo, ainvnm;
extern /* Subroutine */ int sgeequ_(integer *, integer *, real *, integer
*, real *, real *, real *, real *, real *, integer *);
logical trfcon;
real anormo, rowcnd;
extern /* Subroutine */ int sgetrf_(integer *, integer *, real *, integer
*, integer *, integer *), sgetri_(integer *, real *, integer *,
integer *, real *, integer *, integer *), slacpy_(char *, integer
*, integer *, real *, integer *, real *, integer *),
slarhs_(char *, char *, char *, char *, integer *, integer *,
integer *, integer *, integer *, real *, integer *, real *,
integer *, real *, integer *, integer *, integer *);
extern doublereal slantr_(char *, char *, char *, integer *, integer *,
real *, integer *, real *);
extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *,
real *, real *, integer *), slatms_(integer *, integer *,
char *, integer *, char *, real *, integer *, real *, real *,
integer *, integer *, char *, real *, integer *, real *, integer *
), xlaenv_(integer *, integer *);
real result[7];
extern /* Subroutine */ int sgesvx_(char *, char *, integer *, integer *,
real *, integer *, real *, integer *, integer *, char *, real *,
real *, real *, integer *, real *, integer *, real *, real *,
real *, real *, integer *, integer *);
real rpvgrw;
extern /* Subroutine */ int serrvx_(char *, integer *);
/* Fortran I/O blocks */
static cilist io___55 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___61 = { 0, 0, 0, fmt_9997, 0 };
static cilist io___62 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___63 = { 0, 0, 0, fmt_9997, 0 };
static cilist io___64 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___65 = { 0, 0, 0, fmt_9997, 0 };
static cilist io___66 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___67 = { 0, 0, 0, fmt_9997, 0 };
static cilist io___68 = { 0, 0, 0, fmt_9998, 0 };
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SDRVGE tests the driver routines SGESV and -SVX. */
/* Arguments */
/* ========= */
/* DOTYPE (input) LOGICAL array, dimension (NTYPES) */
/* The matrix types to be used for testing. Matrices of type j */
/* (for 1 <= j <= NTYPES) are used for testing if DOTYPE(j) = */
/* .TRUE.; if DOTYPE(j) = .FALSE., then type j is not used. */
/* NN (input) INTEGER */
/* The number of values of N contained in the vector NVAL. */
/* NVAL (input) INTEGER array, dimension (NN) */
/* The values of the matrix column dimension N. */
/* NRHS (input) INTEGER */
/* The number of right hand side vectors to be generated for */
/* each linear system. */
/* THRESH (input) REAL */
/* The threshold value for the test ratios. A result is */
/* included in the output file if RESULT >= THRESH. To have */
/* every test ratio printed, use THRESH = 0. */
/* TSTERR (input) LOGICAL */
/* Flag that indicates whether error exits are to be tested. */
/* NMAX (input) INTEGER */
/* The maximum value permitted for N, used in dimensioning the */
/* work arrays. */
/* A (workspace) REAL array, dimension (NMAX*NMAX) */
/* AFAC (workspace) REAL array, dimension (NMAX*NMAX) */
/* ASAV (workspace) REAL array, dimension (NMAX*NMAX) */
/* B (workspace) REAL array, dimension (NMAX*NRHS) */
/* BSAV (workspace) REAL array, dimension (NMAX*NRHS) */
/* X (workspace) REAL array, dimension (NMAX*NRHS) */
/* XACT (workspace) REAL array, dimension (NMAX*NRHS) */
/* S (workspace) REAL array, dimension (2*NMAX) */
/* WORK (workspace) REAL array, dimension */
/* (NMAX*max(3,NRHS)) */
/* RWORK (workspace) REAL array, dimension (2*NRHS+NMAX) */
/* IWORK (workspace) INTEGER array, dimension (2*NMAX) */
/* NOUT (input) INTEGER */
/* The unit number for output. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Scalars in Common .. */
/* .. */
/* .. Common blocks .. */
/* .. */
/* .. Data statements .. */
/* Parameter adjustments */
--iwork;
--rwork;
--work;
--s;
--xact;
--x;
--bsav;
--b;
--asav;
--afac;
--a;
--nval;
--dotype;
/* Function Body */
/* .. */
/* .. Executable Statements .. */
/* Initialize constants and the random number seed. */
s_copy(path, "Single precision", (ftnlen)1, (ftnlen)16);
s_copy(path + 1, "GE", (ftnlen)2, (ftnlen)2);
nrun = 0;
nfail = 0;
nerrs = 0;
for (i__ = 1; i__ <= 4; ++i__) {
iseed[i__ - 1] = iseedy[i__ - 1];
/* L10: */
}
/* Test the error exits */
if (*tsterr) {
serrvx_(path, nout);
}
infoc_1.infot = 0;
/* Set the block size and minimum block size for testing. */
nb = 1;
nbmin = 2;
xlaenv_(&c__1, &nb);
xlaenv_(&c__2, &nbmin);
/* Do for each value of N in NVAL */
i__1 = *nn;
for (in = 1; in <= i__1; ++in) {
n = nval[in];
lda = max(n,1);
*(unsigned char *)xtype = 'N';
nimat = 11;
if (n <= 0) {
nimat = 1;
}
i__2 = nimat;
for (imat = 1; imat <= i__2; ++imat) {
/* Do the tests only if DOTYPE( IMAT ) is true. */
if (! dotype[imat]) {
goto L80;
}
/* Skip types 5, 6, or 7 if the matrix size is too small. */
zerot = imat >= 5 && imat <= 7;
if (zerot && n < imat - 4) {
goto L80;
}
/* Set up parameters with SLATB4 and generate a test matrix */
/* with SLATMS. */
slatb4_(path, &imat, &n, &n, type__, &kl, &ku, &anorm, &mode, &
cndnum, dist);
rcondc = 1.f / cndnum;
s_copy(srnamc_1.srnamt, "SLATMS", (ftnlen)6, (ftnlen)6);
slatms_(&n, &n, dist, iseed, type__, &rwork[1], &mode, &cndnum, &
anorm, &kl, &ku, "No packing", &a[1], &lda, &work[1], &
info);
/* Check error code from SLATMS. */
if (info != 0) {
alaerh_(path, "SLATMS", &info, &c__0, " ", &n, &n, &c_n1, &
c_n1, &c_n1, &imat, &nfail, &nerrs, nout);
goto L80;
}
/* For types 5-7, zero one or more columns of the matrix to */
/* test that INFO is returned correctly. */
if (zerot) {
if (imat == 5) {
izero = 1;
} else if (imat == 6) {
izero = n;
} else {
izero = n / 2 + 1;
}
ioff = (izero - 1) * lda;
if (imat < 7) {
i__3 = n;
for (i__ = 1; i__ <= i__3; ++i__) {
a[ioff + i__] = 0.f;
/* L20: */
}
} else {
i__3 = n - izero + 1;
slaset_("Full", &n, &i__3, &c_b20, &c_b20, &a[ioff + 1], &
lda);
}
} else {
izero = 0;
}
/* Save a copy of the matrix A in ASAV. */
slacpy_("Full", &n, &n, &a[1], &lda, &asav[1], &lda);
for (iequed = 1; iequed <= 4; ++iequed) {
*(unsigned char *)equed = *(unsigned char *)&equeds[iequed -
1];
if (iequed == 1) {
nfact = 3;
} else {
nfact = 1;
}
i__3 = nfact;
for (ifact = 1; ifact <= i__3; ++ifact) {
*(unsigned char *)fact = *(unsigned char *)&facts[ifact -
1];
prefac = lsame_(fact, "F");
nofact = lsame_(fact, "N");
equil = lsame_(fact, "E");
if (zerot) {
if (prefac) {
goto L60;
}
rcondo = 0.f;
rcondi = 0.f;
} else if (! nofact) {
/* Compute the condition number for comparison with */
/* the value returned by SGESVX (FACT = 'N' reuses */
/* the condition number from the previous iteration */
/* with FACT = 'F'). */
slacpy_("Full", &n, &n, &asav[1], &lda, &afac[1], &
lda);
if (equil || iequed > 1) {
/* Compute row and column scale factors to */
/* equilibrate the matrix A. */
sgeequ_(&n, &n, &afac[1], &lda, &s[1], &s[n + 1],
&rowcnd, &colcnd, &amax, &info);
if (info == 0 && n > 0) {
if (lsame_(equed, "R"))
{
rowcnd = 0.f;
colcnd = 1.f;
} else if (lsame_(equed, "C")) {
rowcnd = 1.f;
colcnd = 0.f;
} else if (lsame_(equed, "B")) {
rowcnd = 0.f;
colcnd = 0.f;
}
/* Equilibrate the matrix. */
slaqge_(&n, &n, &afac[1], &lda, &s[1], &s[n +
1], &rowcnd, &colcnd, &amax, equed);
}
}
/* Save the condition number of the non-equilibrated */
/* system for use in SGET04. */
if (equil) {
roldo = rcondo;
roldi = rcondi;
}
/* Compute the 1-norm and infinity-norm of A. */
anormo = slange_("1", &n, &n, &afac[1], &lda, &rwork[
1]);
anormi = slange_("I", &n, &n, &afac[1], &lda, &rwork[
1]);
/* Factor the matrix A. */
sgetrf_(&n, &n, &afac[1], &lda, &iwork[1], &info);
/* Form the inverse of A. */
slacpy_("Full", &n, &n, &afac[1], &lda, &a[1], &lda);
lwork = *nmax * max(3,*nrhs);
sgetri_(&n, &a[1], &lda, &iwork[1], &work[1], &lwork,
&info);
/* Compute the 1-norm condition number of A. */
ainvnm = slange_("1", &n, &n, &a[1], &lda, &rwork[1]);
if (anormo <= 0.f || ainvnm <= 0.f) {
rcondo = 1.f;
} else {
rcondo = 1.f / anormo / ainvnm;
}
/* Compute the infinity-norm condition number of A. */
ainvnm = slange_("I", &n, &n, &a[1], &lda, &rwork[1]);
if (anormi <= 0.f || ainvnm <= 0.f) {
rcondi = 1.f;
} else {
rcondi = 1.f / anormi / ainvnm;
}
}
for (itran = 1; itran <= 3; ++itran) {
/* Do for each value of TRANS. */
*(unsigned char *)trans = *(unsigned char *)&transs[
itran - 1];
if (itran == 1) {
rcondc = rcondo;
} else {
rcondc = rcondi;
}
/* Restore the matrix A. */
slacpy_("Full", &n, &n, &asav[1], &lda, &a[1], &lda);
/* Form an exact solution and set the right hand side. */
s_copy(srnamc_1.srnamt, "SLARHS", (ftnlen)6, (ftnlen)
6);
slarhs_(path, xtype, "Full", trans, &n, &n, &kl, &ku,
nrhs, &a[1], &lda, &xact[1], &lda, &b[1], &
lda, iseed, &info);
*(unsigned char *)xtype = 'C';
slacpy_("Full", &n, nrhs, &b[1], &lda, &bsav[1], &lda);
if (nofact && itran == 1) {
/* --- Test SGESV --- */
/* Compute the LU factorization of the matrix and */
/* solve the system. */
slacpy_("Full", &n, &n, &a[1], &lda, &afac[1], &
lda);
slacpy_("Full", &n, nrhs, &b[1], &lda, &x[1], &
lda);
s_copy(srnamc_1.srnamt, "SGESV ", (ftnlen)6, (
ftnlen)6);
sgesv_(&n, nrhs, &afac[1], &lda, &iwork[1], &x[1],
&lda, &info);
/* Check error code from SGESV . */
if (info != izero) {
alaerh_(path, "SGESV ", &info, &izero, " ", &
n, &n, &c_n1, &c_n1, nrhs, &imat, &
nfail, &nerrs, nout);
}
/* Reconstruct matrix from factors and compute */
/* residual. */
sget01_(&n, &n, &a[1], &lda, &afac[1], &lda, &
iwork[1], &rwork[1], result);
nt = 1;
if (izero == 0) {
/* Compute residual of the computed solution. */
slacpy_("Full", &n, nrhs, &b[1], &lda, &work[
1], &lda);
sget02_("No transpose", &n, &n, nrhs, &a[1], &
lda, &x[1], &lda, &work[1], &lda, &
rwork[1], &result[1]);
/* Check solution from generated exact solution. */
sget04_(&n, nrhs, &x[1], &lda, &xact[1], &lda,
&rcondc, &result[2]);
nt = 3;
}
/* Print information about the tests that did not */
/* pass the threshold. */
i__4 = nt;
for (k = 1; k <= i__4; ++k) {
if (result[k - 1] >= *thresh) {
if (nfail == 0 && nerrs == 0) {
aladhd_(nout, path);
}
io___55.ciunit = *nout;
s_wsfe(&io___55);
do_fio(&c__1, "SGESV ", (ftnlen)6);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&k, (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&result[k - 1], (
ftnlen)sizeof(real));
e_wsfe();
++nfail;
}
/* L30: */
}
nrun += nt;
}
/* --- Test SGESVX --- */
if (! prefac) {
slaset_("Full", &n, &n, &c_b20, &c_b20, &afac[1],
&lda);
}
slaset_("Full", &n, nrhs, &c_b20, &c_b20, &x[1], &lda);
if (iequed > 1 && n > 0) {
/* Equilibrate the matrix if FACT = 'F' and */
/* EQUED = 'R', 'C', or 'B'. */
slaqge_(&n, &n, &a[1], &lda, &s[1], &s[n + 1], &
rowcnd, &colcnd, &amax, equed);
}
/* Solve the system and compute the condition number */
/* and error bounds using SGESVX. */
s_copy(srnamc_1.srnamt, "SGESVX", (ftnlen)6, (ftnlen)
6);
sgesvx_(fact, trans, &n, nrhs, &a[1], &lda, &afac[1],
&lda, &iwork[1], equed, &s[1], &s[n + 1], &b[
1], &lda, &x[1], &lda, &rcond, &rwork[1], &
rwork[*nrhs + 1], &work[1], &iwork[n + 1], &
info);
/* Check the error code from SGESVX. */
if (info != izero) {
/* Writing concatenation */
i__5[0] = 1, a__1[0] = fact;
i__5[1] = 1, a__1[1] = trans;
s_cat(ch__1, a__1, i__5, &c__2, (ftnlen)2);
alaerh_(path, "SGESVX", &info, &izero, ch__1, &n,
&n, &c_n1, &c_n1, nrhs, &imat, &nfail, &
nerrs, nout);
}
/* Compare WORK(1) from SGESVX with the computed */
/* reciprocal pivot growth factor RPVGRW */
if (info != 0) {
rpvgrw = slantr_("M", "U", "N", &info, &info, &
afac[1], &lda, &work[1]);
if (rpvgrw == 0.f) {
rpvgrw = 1.f;
} else {
rpvgrw = slange_("M", &n, &info, &a[1], &lda,
&work[1]) / rpvgrw;
}
} else {
rpvgrw = slantr_("M", "U", "N", &n, &n, &afac[1],
&lda, &work[1]);
if (rpvgrw == 0.f) {
rpvgrw = 1.f;
} else {
rpvgrw = slange_("M", &n, &n, &a[1], &lda, &
work[1]) / rpvgrw;
}
}
result[6] = (r__1 = rpvgrw - work[1], dabs(r__1)) /
dmax(work[1],rpvgrw) / slamch_("E")
;
if (! prefac) {
/* Reconstruct matrix from factors and compute */
/* residual. */
sget01_(&n, &n, &a[1], &lda, &afac[1], &lda, &
iwork[1], &rwork[(*nrhs << 1) + 1],
result);
k1 = 1;
} else {
k1 = 2;
}
if (info == 0) {
trfcon = FALSE_;
/* Compute residual of the computed solution. */
slacpy_("Full", &n, nrhs, &bsav[1], &lda, &work[1]
, &lda);
sget02_(trans, &n, &n, nrhs, &asav[1], &lda, &x[1]
, &lda, &work[1], &lda, &rwork[(*nrhs <<
1) + 1], &result[1]);
/* Check solution from generated exact solution. */
if (nofact || prefac && lsame_(equed, "N")) {
sget04_(&n, nrhs, &x[1], &lda, &xact[1], &lda,
&rcondc, &result[2]);
} else {
if (itran == 1) {
roldc = roldo;
} else {
roldc = roldi;
}
sget04_(&n, nrhs, &x[1], &lda, &xact[1], &lda,
&roldc, &result[2]);
}
/* Check the error bounds from iterative */
/* refinement. */
sget07_(trans, &n, nrhs, &asav[1], &lda, &b[1], &
lda, &x[1], &lda, &xact[1], &lda, &rwork[
1], &rwork[*nrhs + 1], &result[3]);
} else {
trfcon = TRUE_;
}
/* Compare RCOND from SGESVX with the computed value */
/* in RCONDC. */
result[5] = sget06_(&rcond, &rcondc);
/* Print information about the tests that did not pass */
/* the threshold. */
if (! trfcon) {
for (k = k1; k <= 7; ++k) {
if (result[k - 1] >= *thresh) {
if (nfail == 0 && nerrs == 0) {
aladhd_(nout, path);
}
if (prefac) {
io___61.ciunit = *nout;
s_wsfe(&io___61);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)
sizeof(integer));
do_fio(&c__1, equed, (ftnlen)1);
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&k, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[k - 1],
(ftnlen)sizeof(real));
e_wsfe();
} else {
io___62.ciunit = *nout;
s_wsfe(&io___62);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&k, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[k - 1],
(ftnlen)sizeof(real));
e_wsfe();
}
++nfail;
}
/* L40: */
}
nrun = nrun + 7 - k1;
} else {
if (result[0] >= *thresh && ! prefac) {
if (nfail == 0 && nerrs == 0) {
aladhd_(nout, path);
}
if (prefac) {
io___63.ciunit = *nout;
s_wsfe(&io___63);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, equed, (ftnlen)1);
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&c__1, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[0], (ftnlen)
sizeof(real));
e_wsfe();
} else {
io___64.ciunit = *nout;
s_wsfe(&io___64);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&c__1, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[0], (ftnlen)
sizeof(real));
e_wsfe();
}
++nfail;
++nrun;
}
if (result[5] >= *thresh) {
if (nfail == 0 && nerrs == 0) {
aladhd_(nout, path);
}
if (prefac) {
io___65.ciunit = *nout;
s_wsfe(&io___65);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, equed, (ftnlen)1);
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&c__6, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[5], (ftnlen)
sizeof(real));
e_wsfe();
} else {
io___66.ciunit = *nout;
s_wsfe(&io___66);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&c__6, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[5], (ftnlen)
sizeof(real));
e_wsfe();
}
++nfail;
++nrun;
}
if (result[6] >= *thresh) {
if (nfail == 0 && nerrs == 0) {
aladhd_(nout, path);
}
if (prefac) {
io___67.ciunit = *nout;
s_wsfe(&io___67);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, equed, (ftnlen)1);
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&c__7, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[6], (ftnlen)
sizeof(real));
e_wsfe();
} else {
io___68.ciunit = *nout;
s_wsfe(&io___68);
do_fio(&c__1, "SGESVX", (ftnlen)6);
do_fio(&c__1, fact, (ftnlen)1);
do_fio(&c__1, trans, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&imat, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&c__7, (ftnlen)
sizeof(integer));
do_fio(&c__1, (char *)&result[6], (ftnlen)
sizeof(real));
e_wsfe();
}
++nfail;
++nrun;
}
}
/* L50: */
}
L60:
;
}
/* L70: */
}
L80:
;
}
/* L90: */
}
/* Print a summary of the results. */
alasvm_(path, nout, &nfail, &nrun, &nerrs);
return 0;
/* End of SDRVGE */
} /* sdrvge_ */
int sgesvx_(char *fact, char *trans, int *n, int *
nrhs, float *a, int *lda, float *af, int *ldaf, int *ipiv,
char *equed, float *r__, float *c__, float *b, int *ldb, float *x,
int *ldx, float *rcond, float *ferr, float *berr, float *work,
int *iwork, int *info)
{
/* System generated locals */
int a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2;
float r__1, r__2;
/* Local variables */
int i__, j;
float amax;
char norm[1];
extern int lsame_(char *, char *);
float rcmin, rcmax, anorm;
int equil;
float colcnd;
extern double slamch_(char *), slange_(char *, int *,
int *, float *, int *, float *);
int nofact;
extern int slaqge_(int *, int *, float *, int
*, float *, float *, float *, float *, float *, char *),
xerbla_(char *, int *), sgecon_(char *, int *,
float *, int *, float *, float *, float *, int *, int *);
float bignum;
int infequ;
int colequ;
extern int sgeequ_(int *, int *, float *, int
*, float *, float *, float *, float *, float *, int *), sgerfs_(
char *, int *, int *, float *, int *, float *, int *
, int *, float *, int *, float *, int *, float *, float *,
float *, int *, int *), sgetrf_(int *,
int *, float *, int *, int *, int *);
float rowcnd;
extern int slacpy_(char *, int *, int *, float *,
int *, float *, int *);
int notran;
extern double slantr_(char *, char *, char *, int *, int *,
float *, int *, float *);
extern int sgetrs_(char *, int *, int *, float *,
int *, int *, float *, int *, int *);
float smlnum;
int rowequ;
float rpvgrw;
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGESVX uses the LU factorization to compute the solution to a float */
/* system of linear equations */
/* A * X = B, */
/* where A is an N-by-N 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', float scaling factors are computed to equilibrate */
/* the system: */
/* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B */
/* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B */
/* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') */
/* or diag(C)*B (if TRANS = 'T' or 'C'). */
/* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the */
/* matrix A (after equilibration if FACT = 'E') as */
/* A = P * L * U, */
/* where P is a permutation matrix, L is a unit lower triangular */
/* matrix, and U is upper triangular. */
/* 3. If some U(i,i)=0, so that U is exactly singular, 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A. */
/* If EQUED is not 'N', the matrix A has been */
/* equilibrated with scaling factors given by R and C. */
/* A, AF, and IPIV 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. */
/* 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 (Transpose) */
/* 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 N-by-N matrix A. If FACT = 'F' and EQUED is */
/* not 'N', then A must have been equilibrated by the scaling */
/* factors in R and/or C. A is not modified if FACT = 'F' or */
/* 'N', or if FACT = 'E' and EQUED = 'N' on exit. */
/* On exit, if EQUED .ne. 'N', A is scaled as follows: */
/* EQUED = 'R': A := diag(R) * A */
/* EQUED = 'C': A := A * diag(C) */
/* EQUED = 'B': A := diag(R) * A * diag(C). */
/* 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 factors L and U from the factorization */
/* A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then */
/* AF is the factored form of the equilibrated matrix A. */
/* If FACT = 'N', then AF is an output argument and on exit */
/* returns the factors L and U from the factorization A = P*L*U */
/* of the original matrix A. */
/* If FACT = 'E', then AF is an output argument and on exit */
/* returns the factors L and U from the factorization A = P*L*U */
/* 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). */
/* IPIV (input or output) INTEGER array, dimension (N) */
/* If FACT = 'F', then IPIV is an input argument and on entry */
/* contains the pivot indices from the factorization A = P*L*U */
/* as computed by SGETRF; row i of the matrix was interchanged */
/* with row IPIV(i). */
/* If FACT = 'N', then IPIV is an output argument and on exit */
/* contains the pivot indices from the factorization A = P*L*U */
/* of the original matrix A. */
/* If FACT = 'E', then IPIV is an output argument and on exit */
/* contains the pivot indices from the factorization A = P*L*U */
/* of the equilibrated matrix A. */
/* EQUED (input or output) CHARACTER*1 */
/* Specifies the form of equilibration that was done. */
/* = 'N': No equilibration (always true if FACT = 'N'). */
/* = 'R': Row equilibration, i.e., A has been premultiplied by */
/* diag(R). */
/* = 'C': Column equilibration, i.e., A has been postmultiplied */
/* by diag(C). */
/* = 'B': Both row and column equilibration, i.e., A has been */
/* replaced by diag(R) * A * diag(C). */
/* EQUED is an input argument if FACT = 'F'; otherwise, it is an */
/* output argument. */
/* R (input or output) REAL array, dimension (N) */
/* The row scale factors for A. If EQUED = 'R' or 'B', A is */
/* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R */
/* is not accessed. R is an input argument if FACT = 'F'; */
/* otherwise, R is an output argument. If FACT = 'F' and */
/* EQUED = 'R' or 'B', each element of R must be positive. */
/* C (input or output) REAL array, dimension (N) */
/* The column scale factors for A. If EQUED = 'C' or 'B', A is */
/* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C */
/* is not accessed. C is an input argument if FACT = 'F'; */
/* otherwise, C is an output argument. If FACT = 'F' and */
/* EQUED = 'C' or 'B', each element of C 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by */
/* diag(R)*B; */
/* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is */
/* overwritten by diag(C)*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 A and B are */
/* modified on exit if EQUED .ne. 'N', and the solution to the */
/* equilibrated system is inv(diag(C))*X if TRANS = 'N' and */
/* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' */
/* and EQUED = 'R' or 'B'. */
/* 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/output) REAL array, dimension (4*N) */
/* On exit, WORK(1) contains the reciprocal pivot growth */
/* factor norm(A)/norm(U). The "max absolute element" norm is */
/* used. If WORK(1) 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, condition */
/* estimator RCOND, and forward error bound FERR could be */
/* unreliable. If factorization fails with 0<INFO<=N, then */
/* WORK(1) contains the reciprocal pivot growth factor for the */
/* leading INFO columns of A. */
/* 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: U(i,i) 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+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;
--ipiv;
--r__;
--c__;
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");
equil = lsame_(fact, "E");
notran = lsame_(trans, "N");
if (nofact || equil) {
*(unsigned char *)equed = 'N';
rowequ = FALSE;
colequ = FALSE;
} else {
rowequ = lsame_(equed, "R") || lsame_(equed,
"B");
colequ = lsame_(equed, "C") || lsame_(equed,
"B");
smlnum = slamch_("Safe minimum");
bignum = 1.f / smlnum;
}
/* Test the input parameters. */
if (! nofact && ! equil && ! lsame_(fact, "F")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "T") && !
lsame_(trans, "C")) {
*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") && ! (rowequ || colequ
|| lsame_(equed, "N"))) {
*info = -10;
} else {
if (rowequ) {
rcmin = bignum;
rcmax = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
r__1 = rcmin, r__2 = r__[j];
rcmin = MIN(r__1,r__2);
/* Computing MAX */
r__1 = rcmax, r__2 = r__[j];
rcmax = MAX(r__1,r__2);
/* L10: */
}
if (rcmin <= 0.f) {
*info = -11;
} else if (*n > 0) {
rowcnd = MAX(rcmin,smlnum) / MIN(rcmax,bignum);
} else {
rowcnd = 1.f;
}
}
if (colequ && *info == 0) {
rcmin = bignum;
rcmax = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
r__1 = rcmin, r__2 = c__[j];
rcmin = MIN(r__1,r__2);
/* Computing MAX */
r__1 = rcmax, r__2 = c__[j];
rcmax = MAX(r__1,r__2);
/* L20: */
}
if (rcmin <= 0.f) {
*info = -12;
} else if (*n > 0) {
colcnd = MAX(rcmin,smlnum) / MIN(rcmax,bignum);
} else {
colcnd = 1.f;
}
}
if (*info == 0) {
if (*ldb < MAX(1,*n)) {
*info = -14;
} else if (*ldx < MAX(1,*n)) {
*info = -16;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGESVX", &i__1);
return 0;
}
if (equil) {
/* Compute row and column scalings to equilibrate the matrix A. */
sgeequ_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &colcnd, &
amax, &infequ);
if (infequ == 0) {
/* Equilibrate the matrix. */
slaqge_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &
colcnd, &amax, equed);
rowequ = lsame_(equed, "R") || lsame_(equed,
"B");
colequ = lsame_(equed, "C") || lsame_(equed,
"B");
}
}
/* Scale the right hand side. */
if (notran) {
if (rowequ) {
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] = r__[i__] * b[i__ + j * b_dim1];
/* L30: */
}
/* L40: */
}
}
} else if (colequ) {
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] = c__[i__] * b[i__ + j * b_dim1];
/* L50: */
}
/* L60: */
}
}
if (nofact || equil) {
/* Compute the LU factorization of A. */
slacpy_("Full", n, n, &a[a_offset], lda, &af[af_offset], ldaf);
sgetrf_(n, n, &af[af_offset], ldaf, &ipiv[1], info);
/* Return if INFO is non-zero. */
if (*info > 0) {
/* Compute the reciprocal pivot growth factor of the */
/* leading rank-deficient INFO columns of A. */
rpvgrw = slantr_("M", "U", "N", info, info, &af[af_offset], ldaf,
&work[1]);
if (rpvgrw == 0.f) {
rpvgrw = 1.f;
} else {
rpvgrw = slange_("M", n, info, &a[a_offset], lda, &work[1]) / rpvgrw;
}
work[1] = rpvgrw;
*rcond = 0.f;
return 0;
}
}
/* Compute the norm of the matrix A and the */
/* reciprocal pivot growth factor RPVGRW. */
if (notran) {
*(unsigned char *)norm = '1';
} else {
*(unsigned char *)norm = 'I';
}
anorm = slange_(norm, n, n, &a[a_offset], lda, &work[1]);
rpvgrw = slantr_("M", "U", "N", n, n, &af[af_offset], ldaf, &work[1]);
if (rpvgrw == 0.f) {
rpvgrw = 1.f;
} else {
rpvgrw = slange_("M", n, n, &a[a_offset], lda, &work[1]) /
rpvgrw;
}
/* Compute the reciprocal of the condition number of A. */
sgecon_(norm, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &iwork[1],
info);
/* Compute the solution matrix X. */
slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
sgetrs_(trans, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx,
info);
/* Use iterative refinement to improve the computed solution and */
/* compute error bounds and backward error estimates for it. */
sgerfs_(trans, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &ipiv[1],
&b[b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[
1], &iwork[1], info);
/* Transform the solution matrix X to a solution of the original */
/* system. */
if (notran) {
if (colequ) {
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] = c__[i__] * x[i__ + j * x_dim1];
/* L70: */
}
/* L80: */
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] /= colcnd;
/* L90: */
}
}
} else if (rowequ) {
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] = r__[i__] * x[i__ + j * x_dim1];
/* L100: */
}
/* L110: */
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] /= rowcnd;
/* L120: */
}
}
/* Set INFO = N+1 if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon")) {
*info = *n + 1;
}
work[1] = rpvgrw;
return 0;
/* End of SGESVX */
} /* sgesvx_ */
/* Subroutine */ int sgesvx_(char *fact, char *trans, integer *n, integer *
nrhs, real *a, integer *lda, real *af, integer *ldaf, integer *ipiv,
char *equed, real *r__, real *c__, real *b, integer *ldb, real *x,
integer *ldx, real *rcond, real *ferr, real *berr, real *work,
integer *iwork, integer *info)
{
/* -- 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
Purpose
=======
SGESVX uses the LU factorization to compute the solution to a real
system of linear equations
A * X = B,
where A is an N-by-N 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:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
matrix A (after equilibration if FACT = 'E') as
A = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
3. If some U(i,i)=0, so that U is exactly singular, 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A.
If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by R and C.
A, AF, and IPIV 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.
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 (Transpose)
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 N-by-N matrix A. If FACT = 'F' and EQUED is
not 'N', then A must have been equilibrated by the scaling
factors in R and/or C. A is not modified if FACT = 'F' or
'N', or if FACT = 'E' and EQUED = 'N' on exit.
On exit, if EQUED .ne. 'N', A is scaled as follows:
EQUED = 'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).
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 factors L and U from the factorization
A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then
AF is the factored form of the equilibrated matrix A.
If FACT = 'N', then AF is an output argument and on exit
returns the factors L and U from the factorization A = P*L*U
of the original matrix A.
If FACT = 'E', then AF is an output argument and on exit
returns the factors L and U from the factorization A = P*L*U
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).
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains the pivot indices from the factorization A = P*L*U
as computed by SGETRF; row i of the matrix was interchanged
with row IPIV(i).
If FACT = 'N', then IPIV is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U
of the original matrix A.
If FACT = 'E', then IPIV is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U
of the equilibrated matrix A.
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'R': Row equilibration, i.e., A has been premultiplied by
diag(R).
= 'C': Column equilibration, i.e., A has been postmultiplied
by diag(C).
= 'B': Both row and column equilibration, i.e., A has been
replaced by diag(R) * A * diag(C).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
R (input or output) REAL array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is
multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
is not accessed. R is an input argument if FACT = 'F';
otherwise, R is an output argument. If FACT = 'F' and
EQUED = 'R' or 'B', each element of R must be positive.
C (input or output) REAL array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B', A is
multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
is not accessed. C is an input argument if FACT = 'F';
otherwise, C is an output argument. If FACT = 'F' and
EQUED = 'C' or 'B', each element of C 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
diag(R)*B;
if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
overwritten by diag(C)*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 A and B are
modified on exit if EQUED .ne. 'N', and the solution to the
equilibrated system is inv(diag(C))*X if TRANS = 'N' and
EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
and EQUED = 'R' or 'B'.
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/output) REAL array, dimension (4*N)
On exit, WORK(1) contains the reciprocal pivot growth
factor norm(A)/norm(U). The "max absolute element" norm is
used. If WORK(1) 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, condition
estimator RCOND, and forward error bound FERR could be
unreliable. If factorization fails with 0<INFO<=N, then
WORK(1) contains the reciprocal pivot growth factor for the
leading INFO columns of A.
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: U(i,i) 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+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.
=====================================================================
Parameter adjustments */
/* 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;
static char norm[1];
static integer i__, j;
extern logical lsame_(char *, char *);
static real rcmin, rcmax, anorm;
static logical equil;
static real colcnd;
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
static logical nofact;
extern /* Subroutine */ int slaqge_(integer *, integer *, real *, integer
*, real *, real *, real *, real *, real *, char *),
xerbla_(char *, integer *), sgecon_(char *, integer *,
real *, integer *, real *, real *, real *, integer *, integer *);
static real bignum;
static integer infequ;
static logical colequ;
extern /* Subroutine */ int sgeequ_(integer *, integer *, real *, integer
*, real *, real *, real *, real *, real *, integer *), sgerfs_(
char *, integer *, integer *, real *, integer *, real *, integer *
, integer *, real *, integer *, real *, integer *, real *, real *,
real *, integer *, integer *), sgetrf_(integer *,
integer *, real *, integer *, integer *, integer *);
static real rowcnd;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *);
static logical notran;
extern doublereal slantr_(char *, char *, char *, integer *, integer *,
real *, integer *, real *);
extern /* Subroutine */ int sgetrs_(char *, integer *, integer *, real *,
integer *, integer *, real *, integer *, integer *);
static real smlnum;
static logical rowequ;
static real rpvgrw;
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define x_ref(a_1,a_2) x[(a_2)*x_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1 * 1;
af -= af_offset;
--ipiv;
--r__;
--c__;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
--ferr;
--berr;
--work;
--iwork;
/* Function Body */
*info = 0;
nofact = lsame_(fact, "N");
equil = lsame_(fact, "E");
notran = lsame_(trans, "N");
if (nofact || equil) {
*(unsigned char *)equed = 'N';
rowequ = FALSE_;
colequ = FALSE_;
} else {
rowequ = lsame_(equed, "R") || lsame_(equed,
"B");
colequ = lsame_(equed, "C") || lsame_(equed,
"B");
smlnum = slamch_("Safe minimum");
bignum = 1.f / smlnum;
}
/* Test the input parameters. */
if (! nofact && ! equil && ! lsame_(fact, "F")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "T") && !
lsame_(trans, "C")) {
*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") && ! (rowequ || colequ
|| lsame_(equed, "N"))) {
*info = -10;
} else {
if (rowequ) {
rcmin = bignum;
rcmax = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
r__1 = rcmin, r__2 = r__[j];
rcmin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = rcmax, r__2 = r__[j];
rcmax = dmax(r__1,r__2);
/* L10: */
}
if (rcmin <= 0.f) {
*info = -11;
} else if (*n > 0) {
rowcnd = dmax(rcmin,smlnum) / dmin(rcmax,bignum);
} else {
rowcnd = 1.f;
}
}
if (colequ && *info == 0) {
rcmin = bignum;
rcmax = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
r__1 = rcmin, r__2 = c__[j];
rcmin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = rcmax, r__2 = c__[j];
rcmax = dmax(r__1,r__2);
/* L20: */
}
if (rcmin <= 0.f) {
*info = -12;
} else if (*n > 0) {
colcnd = dmax(rcmin,smlnum) / dmin(rcmax,bignum);
} else {
colcnd = 1.f;
}
}
if (*info == 0) {
if (*ldb < max(1,*n)) {
*info = -14;
} else if (*ldx < max(1,*n)) {
*info = -16;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGESVX", &i__1);
return 0;
}
if (equil) {
/* Compute row and column scalings to equilibrate the matrix A. */
sgeequ_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &colcnd, &
amax, &infequ);
if (infequ == 0) {
/* Equilibrate the matrix. */
slaqge_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &
colcnd, &amax, equed);
rowequ = lsame_(equed, "R") || lsame_(equed,
"B");
colequ = lsame_(equed, "C") || lsame_(equed,
"B");
}
}
/* Scale the right hand side. */
if (notran) {
if (rowequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
b_ref(i__, j) = r__[i__] * b_ref(i__, j);
/* L30: */
}
/* L40: */
}
}
} else if (colequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
b_ref(i__, j) = c__[i__] * b_ref(i__, j);
/* L50: */
}
/* L60: */
}
}
if (nofact || equil) {
/* Compute the LU factorization of A. */
slacpy_("Full", n, n, &a[a_offset], lda, &af[af_offset], ldaf);
sgetrf_(n, n, &af[af_offset], ldaf, &ipiv[1], info);
/* Return if INFO is non-zero. */
if (*info != 0) {
if (*info > 0) {
/* Compute the reciprocal pivot growth factor of the
leading rank-deficient INFO columns of A. */
rpvgrw = slantr_("M", "U", "N", info, info, &af[af_offset],
ldaf, &work[1]);
if (rpvgrw == 0.f) {
rpvgrw = 1.f;
} else {
rpvgrw = slange_("M", n, info, &a[a_offset], lda, &work[1]
) / rpvgrw;
}
work[1] = rpvgrw;
*rcond = 0.f;
}
return 0;
}
}
/* Compute the norm of the matrix A and the
reciprocal pivot growth factor RPVGRW. */
if (notran) {
*(unsigned char *)norm = '1';
} else {
*(unsigned char *)norm = 'I';
}
anorm = slange_(norm, n, n, &a[a_offset], lda, &work[1]);
rpvgrw = slantr_("M", "U", "N", n, n, &af[af_offset], ldaf, &work[1]);
if (rpvgrw == 0.f) {
rpvgrw = 1.f;
} else {
rpvgrw = slange_("M", n, n, &a[a_offset], lda, &work[1]) /
rpvgrw;
}
/* Compute the reciprocal of the condition number of A. */
sgecon_(norm, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &iwork[1],
info);
/* Set INFO = N+1 if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon")) {
*info = *n + 1;
}
/* Compute the solution matrix X. */
slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
sgetrs_(trans, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx,
info);
/* Use iterative refinement to improve the computed solution and
compute error bounds and backward error estimates for it. */
sgerfs_(trans, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &ipiv[1],
&b[b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[
1], &iwork[1], info);
/* Transform the solution matrix X to a solution of the original
system. */
if (notran) {
if (colequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
x_ref(i__, j) = c__[i__] * x_ref(i__, j);
/* L70: */
}
/* L80: */
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] /= colcnd;
/* L90: */
}
}
} else if (rowequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
x_ref(i__, j) = r__[i__] * x_ref(i__, j);
/* L100: */
}
/* L110: */
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] /= rowcnd;
/* L120: */
}
}
work[1] = rpvgrw;
return 0;
/* End of SGESVX */
} /* sgesvx_ */
