/* Subroutine */ int sgetrs_(char *trans, integer *n, integer *nrhs, real *a,
integer *lda, integer *ipiv, real *b, integer *ldb, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1;
/* Local variables */
extern logical lsame_(char *, char *);
extern /* Subroutine */ int strsm_(char *, char *, char *, char *,
integer *, integer *, real *, real *, integer *, real *, integer *
), xerbla_(char *, integer *);
logical notran;
extern /* Subroutine */ int slaswp_(integer *, real *, integer *, integer
*, integer *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGETRS solves a system of linear equations */
/* A * X = B or A' * X = B */
/* with a general N-by-N matrix A using the LU factorization computed */
/* by SGETRF. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A * X = B (No transpose) */
/* = 'T': A'* X = B (Transpose) */
/* = 'C': A'* X = B (Conjugate transpose = Transpose) */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrix B. NRHS >= 0. */
/* A (input) REAL array, dimension (LDA,N) */
/* The factors L and U from the factorization A = P*L*U */
/* as computed by SGETRF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from SGETRF; for 1<=i<=N, row i of the */
/* matrix was interchanged with row IPIV(i). */
/* B (input/output) REAL array, dimension (LDB,NRHS) */
/* On entry, the right hand side matrix B. */
/* On exit, the solution matrix X. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
/* Function Body */
*info = 0;
notran = lsame_(trans, "N");
if (! notran && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGETRS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
return 0;
}
if (notran) {
/* Solve A * X = B. */
/* Apply row interchanges to the right hand sides. */
slaswp_(nrhs, &b[b_offset], ldb, &c__1, n, &ipiv[1], &c__1);
/* Solve L*X = B, overwriting B with X. */
strsm_("Left", "Lower", "No transpose", "Unit", n, nrhs, &c_b12, &a[
a_offset], lda, &b[b_offset], ldb);
/* Solve U*X = B, overwriting B with X. */
strsm_("Left", "Upper", "No transpose", "Non-unit", n, nrhs, &c_b12, &
a[a_offset], lda, &b[b_offset], ldb);
} else {
/* Solve A' * X = B. */
/* Solve U'*X = B, overwriting B with X. */
strsm_("Left", "Upper", "Transpose", "Non-unit", n, nrhs, &c_b12, &a[
a_offset], lda, &b[b_offset], ldb);
/* Solve L'*X = B, overwriting B with X. */
strsm_("Left", "Lower", "Transpose", "Unit", n, nrhs, &c_b12, &a[
a_offset], lda, &b[b_offset], ldb);
/* Apply row interchanges to the solution vectors. */
slaswp_(nrhs, &b[b_offset], ldb, &c__1, n, &ipiv[1], &c_n1);
}
return 0;
/* End of SGETRS */
} /* sgetrs_ */
/* Subroutine */ int slatdf_(integer *ijob, integer *n, real *z__, integer *
ldz, real *rhs, real *rdsum, real *rdscal, integer *ipiv, integer *
jpiv)
{
/* -- LAPACK auxiliary 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
=======
SLATDF uses the LU factorization of the n-by-n matrix Z computed by
SGETC2 and computes a contribution to the reciprocal Dif-estimate
by solving Z * x = b for x, and choosing the r.h.s. b such that
the norm of x is as large as possible. On entry RHS = b holds the
contribution from earlier solved sub-systems, and on return RHS = x.
The factorization of Z returned by SGETC2 has the form Z = P*L*U*Q,
where P and Q are permutation matrices. L is lower triangular with
unit diagonal elements and U is upper triangular.
Arguments
=========
IJOB (input) INTEGER
IJOB = 2: First compute an approximative null-vector e
of Z using SGECON, e is normalized and solve for
Zx = +-e - f with the sign giving the greater value
of 2-norm(x). About 5 times as expensive as Default.
IJOB .ne. 2: Local look ahead strategy where all entries of
the r.h.s. b is choosen as either +1 or -1 (Default).
N (input) INTEGER
The number of columns of the matrix Z.
Z (input) REAL array, dimension (LDZ, N)
On entry, the LU part of the factorization of the n-by-n
matrix Z computed by SGETC2: Z = P * L * U * Q
LDZ (input) INTEGER
The leading dimension of the array Z. LDA >= max(1, N).
RHS (input/output) REAL array, dimension N.
On entry, RHS contains contributions from other subsystems.
On exit, RHS contains the solution of the subsystem with
entries acoording to the value of IJOB (see above).
RDSUM (input/output) REAL
On entry, the sum of squares of computed contributions to
the Dif-estimate under computation by STGSYL, where the
scaling factor RDSCAL (see below) has been factored out.
On exit, the corresponding sum of squares updated with the
contributions from the current sub-system.
If TRANS = 'T' RDSUM is not touched.
NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.
RDSCAL (input/output) REAL
On entry, scaling factor used to prevent overflow in RDSUM.
On exit, RDSCAL is updated w.r.t. the current contributions
in RDSUM.
If TRANS = 'T', RDSCAL is not touched.
NOTE: RDSCAL only makes sense when STGSY2 is called by
STGSYL.
IPIV (input) INTEGER array, dimension (N).
The pivot indices; for 1 <= i <= N, row i of the
matrix has been interchanged with row IPIV(i).
JPIV (input) INTEGER array, dimension (N).
The pivot indices; for 1 <= j <= N, column j of the
matrix has been interchanged with column JPIV(j).
Further Details
===============
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
This routine is a further developed implementation of algorithm
BSOLVE in [1] using complete pivoting in the LU factorization.
[1] Bo Kagstrom and Lars Westin,
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation, IEEE Transactions
on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
[2] Peter Poromaa,
On Efficient and Robust Estimators for the Separation
between two Regular Matrix Pairs with Applications in
Condition Estimation. Report IMINF-95.05, Departement of
Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
static real c_b23 = 1.f;
static real c_b37 = -1.f;
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer info;
static real temp;
extern doublereal sdot_(integer *, real *, integer *, real *, integer *);
static real work[32];
static integer i__, j, k;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static real pmone;
extern doublereal sasum_(integer *, real *, integer *);
static real sminu;
static integer iwork[8];
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), saxpy_(integer *, real *, real *, integer *, real *,
integer *);
static real splus;
extern /* Subroutine */ int sgesc2_(integer *, real *, integer *, real *,
integer *, integer *, real *);
static real bm, bp, xm[8], xp[8];
extern /* Subroutine */ int sgecon_(char *, integer *, real *, integer *,
real *, real *, real *, integer *, integer *), slassq_(
integer *, real *, integer *, real *, real *), slaswp_(integer *,
real *, integer *, integer *, integer *, integer *, integer *);
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--rhs;
--ipiv;
--jpiv;
/* Function Body */
if (*ijob != 2) {
/* Apply permutations IPIV to RHS */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], ldz, &c__1, &i__1, &ipiv[1], &c__1);
/* Solve for L-part choosing RHS either to +1 or -1. */
pmone = -1.f;
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
bp = rhs[j] + 1.f;
bm = rhs[j] - 1.f;
splus = 1.f;
/* Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
SMIN computed more efficiently than in BSOLVE [1]. */
i__2 = *n - j;
splus += sdot_(&i__2, &z___ref(j + 1, j), &c__1, &z___ref(j + 1,
j), &c__1);
i__2 = *n - j;
sminu = sdot_(&i__2, &z___ref(j + 1, j), &c__1, &rhs[j + 1], &
c__1);
splus *= rhs[j];
if (splus > sminu) {
rhs[j] = bp;
} else if (sminu > splus) {
rhs[j] = bm;
} else {
/* In this case the updating sums are equal and we can
choose RHS(J) +1 or -1. The first time this happens
we choose -1, thereafter +1. This is a simple way to
get good estimates of matrices like Byers well-known
example (see [1]). (Not done in BSOLVE.) */
rhs[j] += pmone;
pmone = 1.f;
}
/* Compute the remaining r.h.s. */
temp = -rhs[j];
i__2 = *n - j;
saxpy_(&i__2, &temp, &z___ref(j + 1, j), &c__1, &rhs[j + 1], &
c__1);
/* L10: */
}
/* Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
in BSOLVE and will hopefully give us a better estimate because
any ill-conditioning of the original matrix is transfered to U
and not to L. U(N, N) is an approximation to sigma_min(LU). */
i__1 = *n - 1;
scopy_(&i__1, &rhs[1], &c__1, xp, &c__1);
xp[*n - 1] = rhs[*n] + 1.f;
rhs[*n] += -1.f;
splus = 0.f;
sminu = 0.f;
for (i__ = *n; i__ >= 1; --i__) {
temp = 1.f / z___ref(i__, i__);
xp[i__ - 1] *= temp;
rhs[i__] *= temp;
i__1 = *n;
for (k = i__ + 1; k <= i__1; ++k) {
xp[i__ - 1] -= xp[k - 1] * (z___ref(i__, k) * temp);
rhs[i__] -= rhs[k] * (z___ref(i__, k) * temp);
/* L20: */
}
splus += (r__1 = xp[i__ - 1], dabs(r__1));
sminu += (r__1 = rhs[i__], dabs(r__1));
/* L30: */
}
if (splus > sminu) {
scopy_(n, xp, &c__1, &rhs[1], &c__1);
}
/* Apply the permutations JPIV to the computed solution (RHS) */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], ldz, &c__1, &i__1, &jpiv[1], &c_n1);
/* Compute the sum of squares */
slassq_(n, &rhs[1], &c__1, rdscal, rdsum);
} else {
/* IJOB = 2, Compute approximate nullvector XM of Z */
sgecon_("I", n, &z__[z_offset], ldz, &c_b23, &temp, work, iwork, &
info);
scopy_(n, &work[*n], &c__1, xm, &c__1);
/* Compute RHS */
i__1 = *n - 1;
slaswp_(&c__1, xm, ldz, &c__1, &i__1, &ipiv[1], &c_n1);
temp = 1.f / sqrt(sdot_(n, xm, &c__1, xm, &c__1));
sscal_(n, &temp, xm, &c__1);
scopy_(n, xm, &c__1, xp, &c__1);
saxpy_(n, &c_b23, &rhs[1], &c__1, xp, &c__1);
saxpy_(n, &c_b37, xm, &c__1, &rhs[1], &c__1);
sgesc2_(n, &z__[z_offset], ldz, &rhs[1], &ipiv[1], &jpiv[1], &temp);
sgesc2_(n, &z__[z_offset], ldz, xp, &ipiv[1], &jpiv[1], &temp);
if (sasum_(n, xp, &c__1) > sasum_(n, &rhs[1], &c__1)) {
scopy_(n, xp, &c__1, &rhs[1], &c__1);
}
/* Compute the sum of squares */
slassq_(n, &rhs[1], &c__1, rdscal, rdsum);
}
return 0;
/* End of SLATDF */
} /* slatdf_ */
/* Subroutine */ int sgetrf_(integer *m, integer *n, real *a, integer *lda,
integer *ipiv, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
real r__1;
/* Local variables */
integer i__, j, ipivstart, jpivstart, jp;
real tmp;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *),
sgemm_(char *, char *, integer *, integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *);
integer kcols;
real sfmin;
integer nstep;
extern /* Subroutine */ int strsm_(char *, char *, char *, char *,
integer *, integer *, real *, real *, integer *, real *, integer *
);
integer kahead;
extern doublereal slamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer isamax_(integer *, real *, integer *);
integer npived;
extern logical sisnan_(real *);
integer kstart;
extern /* Subroutine */ int slaswp_(integer *, real *, integer *, integer
*, integer *, integer *, integer *);
integer ntopiv;
/* -- LAPACK routine (version 3.X) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* May 2008 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGETRF computes an LU factorization of a general M-by-N matrix A */
/* using partial pivoting with row interchanges. */
/* The factorization has the form */
/* A = P * L * U */
/* where P is a permutation matrix, L is lower triangular with unit */
/* diagonal elements (lower trapezoidal if m > n), and U is upper */
/* triangular (upper trapezoidal if m < n). */
/* This code implements an iterative version of Sivan Toledo's recursive */
/* LU algorithm[1]. For square matrices, this iterative versions should */
/* be within a factor of two of the optimum number of memory transfers. */
/* The pattern is as follows, with the large blocks of U being updated */
/* in one call to STRSM, and the dotted lines denoting sections that */
/* have had all pending permutations applied: */
/* 1 2 3 4 5 6 7 8 */
/* +-+-+---+-------+------ */
/* | |1| | | */
/* |.+-+ 2 | | */
/* | | | | | */
/* |.|.+-+-+ 4 | */
/* | | | |1| | */
/* | | |.+-+ | */
/* | | | | | | */
/* |.|.|.|.+-+-+---+ 8 */
/* | | | | | |1| | */
/* | | | | |.+-+ 2 | */
/* | | | | | | | | */
/* | | | | |.|.+-+-+ */
/* | | | | | | | |1| */
/* | | | | | | |.+-+ */
/* | | | | | | | | | */
/* |.|.|.|.|.|.|.|.+----- */
/* | | | | | | | | | */
/* The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in */
/* the binary expansion of the current column. Each Schur update is */
/* applied as soon as the necessary portion of U is available. */
/* [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with */
/* Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997), */
/* 1065-1081. http://dx.doi.org/10.1137/S0895479896297744 */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix A. N >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix to be factored. */
/* On exit, the factors L and U from the factorization */
/* A = P*L*U; the unit diagonal elements of L are not stored. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* IPIV (output) INTEGER array, dimension (min(M,N)) */
/* The pivot indices; for 1 <= i <= min(M,N), row i of the */
/* matrix was interchanged with row IPIV(i). */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, U(i,i) is exactly zero. The factorization */
/* has been completed, but the factor U is exactly */
/* singular, and division by zero will occur if it is used */
/* to solve a system of equations. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGETRF", &i__1);
return 0;
}
/* Quick return if possible */
if (*m == 0 || *n == 0) {
return 0;
}
/* Compute machine safe minimum */
sfmin = slamch_("S");
nstep = min(*m,*n);
i__1 = nstep;
for (j = 1; j <= i__1; ++j) {
kahead = j & -j;
kstart = j + 1 - kahead;
/* Computing MIN */
i__2 = kahead, i__3 = *m - j;
kcols = min(i__2,i__3);
/* Find pivot. */
i__2 = *m - j + 1;
jp = j - 1 + isamax_(&i__2, &a[j + j * a_dim1], &c__1);
ipiv[j] = jp;
/* Permute just this column. */
if (jp != j) {
tmp = a[j + j * a_dim1];
a[j + j * a_dim1] = a[jp + j * a_dim1];
a[jp + j * a_dim1] = tmp;
}
/* Apply pending permutations to L */
ntopiv = 1;
ipivstart = j;
jpivstart = j - ntopiv;
while(ntopiv < kahead) {
slaswp_(&ntopiv, &a[jpivstart * a_dim1 + 1], lda, &ipivstart, &j,
&ipiv[1], &c__1);
ipivstart -= ntopiv;
ntopiv <<= 1;
jpivstart -= ntopiv;
}
/* Permute U block to match L */
slaswp_(&kcols, &a[(j + 1) * a_dim1 + 1], lda, &kstart, &j, &ipiv[1],
&c__1);
/* Factor the current column */
if (a[j + j * a_dim1] != 0.f && ! sisnan_(&a[j + j * a_dim1])) {
if ((r__1 = a[j + j * a_dim1], dabs(r__1)) >= sfmin) {
i__2 = *m - j;
r__1 = 1.f / a[j + j * a_dim1];
sscal_(&i__2, &r__1, &a[j + 1 + j * a_dim1], &c__1);
} else {
i__2 = *m - j;
for (i__ = 1; i__ <= i__2; ++i__) {
a[j + i__ + j * a_dim1] /= a[j + j * a_dim1];
}
}
} else if (a[j + j * a_dim1] == 0.f && *info == 0) {
*info = j;
}
/* Solve for U block. */
strsm_("Left", "Lower", "No transpose", "Unit", &kahead, &kcols, &
c_b12, &a[kstart + kstart * a_dim1], lda, &a[kstart + (j + 1)
* a_dim1], lda);
/* Schur complement. */
i__2 = *m - j;
sgemm_("No transpose", "No transpose", &i__2, &kcols, &kahead, &c_b15,
&a[j + 1 + kstart * a_dim1], lda, &a[kstart + (j + 1) *
a_dim1], lda, &c_b12, &a[j + 1 + (j + 1) * a_dim1], lda);
}
/* Handle pivot permutations on the way out of the recursion */
npived = nstep & -nstep;
j = nstep - npived;
while(j > 0) {
ntopiv = j & -j;
i__1 = j + 1;
slaswp_(&ntopiv, &a[(j - ntopiv + 1) * a_dim1 + 1], lda, &i__1, &
nstep, &ipiv[1], &c__1);
j -= ntopiv;
}
/* If short and wide, handle the rest of the columns. */
if (*m < *n) {
i__1 = *n - *m;
slaswp_(&i__1, &a[(*m + kcols + 1) * a_dim1 + 1], lda, &c__1, m, &
ipiv[1], &c__1);
i__1 = *n - *m;
strsm_("Left", "Lower", "No transpose", "Unit", m, &i__1, &c_b12, &a[
a_offset], lda, &a[(*m + kcols + 1) * a_dim1 + 1], lda);
}
return 0;
/* End of SGETRF */
} /* sgetrf_ */
int sgesc2_(int *n, float *a, int *lda, float *rhs,
int *ipiv, int *jpiv, float *scale)
{
/* System generated locals */
int a_dim1, a_offset, i__1, i__2;
float r__1, r__2;
/* Local variables */
int i__, j;
float eps, temp;
extern int sscal_(int *, float *, float *, int *),
slabad_(float *, float *);
extern double slamch_(char *);
float bignum;
extern int isamax_(int *, float *, int *);
extern int slaswp_(int *, float *, int *, int
*, int *, int *, int *);
float smlnum;
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGESC2 solves a system of linear equations */
/* A * X = scale* RHS */
/* with a general N-by-N matrix A using the LU factorization with */
/* complete pivoting computed by SGETC2. */
/* Arguments */
/* ========= */
/* N (input) INTEGER */
/* The order of the matrix A. */
/* A (input) REAL array, dimension (LDA,N) */
/* On entry, the LU part of the factorization of the n-by-n */
/* matrix A computed by SGETC2: A = P * L * U * Q */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1, N). */
/* RHS (input/output) REAL array, dimension (N). */
/* On entry, the right hand side vector b. */
/* On exit, the solution vector X. */
/* IPIV (input) INTEGER array, dimension (N). */
/* The pivot indices; for 1 <= i <= N, row i of the */
/* matrix has been interchanged with row IPIV(i). */
/* JPIV (input) INTEGER array, dimension (N). */
/* The pivot indices; for 1 <= j <= N, column j of the */
/* matrix has been interchanged with column JPIV(j). */
/* SCALE (output) REAL */
/* On exit, SCALE contains the scale factor. SCALE is chosen */
/* 0 <= SCALE <= 1 to prevent owerflow in the solution. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */
/* Umea University, S-901 87 Umea, Sweden. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Set constant to control owerflow */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--rhs;
--ipiv;
--jpiv;
/* Function Body */
eps = slamch_("P");
smlnum = slamch_("S") / eps;
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
/* Apply permutations IPIV to RHS */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], lda, &c__1, &i__1, &ipiv[1], &c__1);
/* Solve for L part */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
rhs[j] -= a[j + i__ * a_dim1] * rhs[i__];
/* L10: */
}
/* L20: */
}
/* Solve for U part */
*scale = 1.f;
/* Check for scaling */
i__ = isamax_(n, &rhs[1], &c__1);
if (smlnum * 2.f * (r__1 = rhs[i__], ABS(r__1)) > (r__2 = a[*n + *n *
a_dim1], ABS(r__2))) {
temp = .5f / (r__1 = rhs[i__], ABS(r__1));
sscal_(n, &temp, &rhs[1], &c__1);
*scale *= temp;
}
for (i__ = *n; i__ >= 1; --i__) {
temp = 1.f / a[i__ + i__ * a_dim1];
rhs[i__] *= temp;
i__1 = *n;
for (j = i__ + 1; j <= i__1; ++j) {
rhs[i__] -= rhs[j] * (a[i__ + j * a_dim1] * temp);
/* L30: */
}
/* L40: */
}
/* Apply permutations JPIV to the solution (RHS) */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], lda, &c__1, &i__1, &jpiv[1], &c_n1);
return 0;
/* End of SGESC2 */
} /* sgesc2_ */
/* Subroutine */ int sgbtrf_(integer *m, integer *n, integer *kl, integer *ku,
real *ab, integer *ldab, integer *ipiv, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1, i__2, i__3, i__4, i__5, i__6;
real r__1;
/* Local variables */
integer i__, j, i2, i3, j2, j3, k2, jb, nb, ii, jj, jm, ip, jp, km, ju,
kv, nw;
extern /* Subroutine */ int sger_(integer *, integer *, real *, real *,
integer *, real *, integer *, real *, integer *);
real temp;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *),
sgemm_(char *, char *, integer *, integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *);
real work13[4160] /* was [65][64] */, work31[4160] /* was [65][
64] */;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
), strsm_(char *, char *, char *, char *, integer *, integer *,
real *, real *, integer *, real *, integer *), sgbtf2_(integer *, integer *, integer *, integer
*, real *, integer *, integer *, integer *), xerbla_(char *,
integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *), isamax_(integer *, real *,
integer *);
extern /* Subroutine */ int slaswp_(integer *, real *, integer *, integer
*, integer *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGBTRF computes an LU factorization of a real m-by-n band matrix A */
/* using partial pivoting with row interchanges. */
/* This is the blocked version of the algorithm, calling Level 3 BLAS. */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix A. N >= 0. */
/* KL (input) INTEGER */
/* The number of subdiagonals within the band of A. KL >= 0. */
/* KU (input) INTEGER */
/* The number of superdiagonals within the band of A. KU >= 0. */
/* AB (input/output) REAL array, dimension (LDAB,N) */
/* On entry, the matrix A in band storage, in rows KL+1 to */
/* 2*KL+KU+1; rows 1 to KL of the array need not be set. */
/* The j-th column of A is stored in the j-th column of the */
/* array AB as follows: */
/* AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) */
/* On exit, details of the factorization: U is stored as an */
/* upper triangular band matrix with KL+KU superdiagonals in */
/* rows 1 to KL+KU+1, and the multipliers used during the */
/* factorization are stored in rows KL+KU+2 to 2*KL+KU+1. */
/* See below for further details. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= 2*KL+KU+1. */
/* IPIV (output) INTEGER array, dimension (min(M,N)) */
/* The pivot indices; for 1 <= i <= min(M,N), row i of the */
/* matrix was interchanged with row IPIV(i). */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = +i, U(i,i) is exactly zero. The factorization */
/* has been completed, but the factor U is exactly */
/* singular, and division by zero will occur if it is used */
/* to solve a system of equations. */
/* Further Details */
/* =============== */
/* The band storage scheme is illustrated by the following example, when */
/* M = N = 6, KL = 2, KU = 1: */
/* On entry: On exit: */
/* * * * + + + * * * u14 u25 u36 */
/* * * + + + + * * u13 u24 u35 u46 */
/* * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 */
/* a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 */
/* a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * */
/* a31 a42 a53 a64 * * m31 m42 m53 m64 * * */
/* Array elements marked * are not used by the routine; elements marked */
/* + need not be set on entry, but are required by the routine to store */
/* elements of U because of fill-in resulting from the row interchanges. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* KV is the number of superdiagonals in the factor U, allowing for */
/* fill-in */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--ipiv;
/* Function Body */
kv = *ku + *kl;
/* Test the input parameters. */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kl < 0) {
*info = -3;
} else if (*ku < 0) {
*info = -4;
} else if (*ldab < *kl + kv + 1) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGBTRF", &i__1);
return 0;
}
/* Quick return if possible */
if (*m == 0 || *n == 0) {
return 0;
}
/* Determine the block size for this environment */
nb = ilaenv_(&c__1, "SGBTRF", " ", m, n, kl, ku);
/* The block size must not exceed the limit set by the size of the */
/* local arrays WORK13 and WORK31. */
nb = min(nb,64);
if (nb <= 1 || nb > *kl) {
/* Use unblocked code */
sgbtf2_(m, n, kl, ku, &ab[ab_offset], ldab, &ipiv[1], info);
} else {
/* Use blocked code */
/* Zero the superdiagonal elements of the work array WORK13 */
i__1 = nb;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work13[i__ + j * 65 - 66] = 0.f;
/* L10: */
}
/* L20: */
}
/* Zero the subdiagonal elements of the work array WORK31 */
i__1 = nb;
for (j = 1; j <= i__1; ++j) {
i__2 = nb;
for (i__ = j + 1; i__ <= i__2; ++i__) {
work31[i__ + j * 65 - 66] = 0.f;
/* L30: */
}
/* L40: */
}
/* Gaussian elimination with partial pivoting */
/* Set fill-in elements in columns KU+2 to KV to zero */
i__1 = min(kv,*n);
for (j = *ku + 2; j <= i__1; ++j) {
i__2 = *kl;
for (i__ = kv - j + 2; i__ <= i__2; ++i__) {
ab[i__ + j * ab_dim1] = 0.f;
/* L50: */
}
/* L60: */
}
/* JU is the index of the last column affected by the current */
/* stage of the factorization */
ju = 1;
i__1 = min(*m,*n);
i__2 = nb;
for (j = 1; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) {
/* Computing MIN */
i__3 = nb, i__4 = min(*m,*n) - j + 1;
jb = min(i__3,i__4);
/* The active part of the matrix is partitioned */
/* A11 A12 A13 */
/* A21 A22 A23 */
/* A31 A32 A33 */
/* Here A11, A21 and A31 denote the current block of JB columns */
/* which is about to be factorized. The number of rows in the */
/* partitioning are JB, I2, I3 respectively, and the numbers */
/* of columns are JB, J2, J3. The superdiagonal elements of A13 */
/* and the subdiagonal elements of A31 lie outside the band. */
/* Computing MIN */
i__3 = *kl - jb, i__4 = *m - j - jb + 1;
i2 = min(i__3,i__4);
/* Computing MIN */
i__3 = jb, i__4 = *m - j - *kl + 1;
i3 = min(i__3,i__4);
/* J2 and J3 are computed after JU has been updated. */
/* Factorize the current block of JB columns */
i__3 = j + jb - 1;
for (jj = j; jj <= i__3; ++jj) {
/* Set fill-in elements in column JJ+KV to zero */
if (jj + kv <= *n) {
i__4 = *kl;
for (i__ = 1; i__ <= i__4; ++i__) {
ab[i__ + (jj + kv) * ab_dim1] = 0.f;
/* L70: */
}
}
/* Find pivot and test for singularity. KM is the number of */
/* subdiagonal elements in the current column. */
/* Computing MIN */
i__4 = *kl, i__5 = *m - jj;
km = min(i__4,i__5);
i__4 = km + 1;
jp = isamax_(&i__4, &ab[kv + 1 + jj * ab_dim1], &c__1);
ipiv[jj] = jp + jj - j;
if (ab[kv + jp + jj * ab_dim1] != 0.f) {
/* Computing MAX */
/* Computing MIN */
i__6 = jj + *ku + jp - 1;
i__4 = ju, i__5 = min(i__6,*n);
ju = max(i__4,i__5);
if (jp != 1) {
/* Apply interchange to columns J to J+JB-1 */
if (jp + jj - 1 < j + *kl) {
i__4 = *ldab - 1;
i__5 = *ldab - 1;
sswap_(&jb, &ab[kv + 1 + jj - j + j * ab_dim1], &
i__4, &ab[kv + jp + jj - j + j * ab_dim1],
&i__5);
} else {
/* The interchange affects columns J to JJ-1 of A31 */
/* which are stored in the work array WORK31 */
i__4 = jj - j;
i__5 = *ldab - 1;
sswap_(&i__4, &ab[kv + 1 + jj - j + j * ab_dim1],
&i__5, &work31[jp + jj - j - *kl - 1], &
c__65);
i__4 = j + jb - jj;
i__5 = *ldab - 1;
i__6 = *ldab - 1;
sswap_(&i__4, &ab[kv + 1 + jj * ab_dim1], &i__5, &
ab[kv + jp + jj * ab_dim1], &i__6);
}
}
/* Compute multipliers */
r__1 = 1.f / ab[kv + 1 + jj * ab_dim1];
sscal_(&km, &r__1, &ab[kv + 2 + jj * ab_dim1], &c__1);
/* Update trailing submatrix within the band and within */
/* the current block. JM is the index of the last column */
/* which needs to be updated. */
/* Computing MIN */
i__4 = ju, i__5 = j + jb - 1;
jm = min(i__4,i__5);
if (jm > jj) {
i__4 = jm - jj;
i__5 = *ldab - 1;
i__6 = *ldab - 1;
sger_(&km, &i__4, &c_b18, &ab[kv + 2 + jj * ab_dim1],
&c__1, &ab[kv + (jj + 1) * ab_dim1], &i__5, &
ab[kv + 1 + (jj + 1) * ab_dim1], &i__6);
}
} else {
/* If pivot is zero, set INFO to the index of the pivot */
/* unless a zero pivot has already been found. */
if (*info == 0) {
*info = jj;
}
}
/* Copy current column of A31 into the work array WORK31 */
/* Computing MIN */
i__4 = jj - j + 1;
nw = min(i__4,i3);
if (nw > 0) {
scopy_(&nw, &ab[kv + *kl + 1 - jj + j + jj * ab_dim1], &
c__1, &work31[(jj - j + 1) * 65 - 65], &c__1);
}
/* L80: */
}
if (j + jb <= *n) {
/* Apply the row interchanges to the other blocks. */
/* Computing MIN */
i__3 = ju - j + 1;
j2 = min(i__3,kv) - jb;
/* Computing MAX */
i__3 = 0, i__4 = ju - j - kv + 1;
j3 = max(i__3,i__4);
/* Use SLASWP to apply the row interchanges to A12, A22, and */
/* A32. */
i__3 = *ldab - 1;
slaswp_(&j2, &ab[kv + 1 - jb + (j + jb) * ab_dim1], &i__3, &
c__1, &jb, &ipiv[j], &c__1);
/* Adjust the pivot indices. */
i__3 = j + jb - 1;
for (i__ = j; i__ <= i__3; ++i__) {
ipiv[i__] = ipiv[i__] + j - 1;
/* L90: */
}
/* Apply the row interchanges to A13, A23, and A33 */
/* columnwise. */
k2 = j - 1 + jb + j2;
i__3 = j3;
for (i__ = 1; i__ <= i__3; ++i__) {
jj = k2 + i__;
i__4 = j + jb - 1;
for (ii = j + i__ - 1; ii <= i__4; ++ii) {
ip = ipiv[ii];
if (ip != ii) {
temp = ab[kv + 1 + ii - jj + jj * ab_dim1];
ab[kv + 1 + ii - jj + jj * ab_dim1] = ab[kv + 1 +
ip - jj + jj * ab_dim1];
ab[kv + 1 + ip - jj + jj * ab_dim1] = temp;
}
/* L100: */
}
/* L110: */
}
/* Update the relevant part of the trailing submatrix */
if (j2 > 0) {
/* Update A12 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
strsm_("Left", "Lower", "No transpose", "Unit", &jb, &j2,
&c_b31, &ab[kv + 1 + j * ab_dim1], &i__3, &ab[kv
+ 1 - jb + (j + jb) * ab_dim1], &i__4);
if (i2 > 0) {
/* Update A22 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
i__5 = *ldab - 1;
sgemm_("No transpose", "No transpose", &i2, &j2, &jb,
&c_b18, &ab[kv + 1 + jb + j * ab_dim1], &i__3,
&ab[kv + 1 - jb + (j + jb) * ab_dim1], &i__4,
&c_b31, &ab[kv + 1 + (j + jb) * ab_dim1], &
i__5);
}
if (i3 > 0) {
/* Update A32 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
sgemm_("No transpose", "No transpose", &i3, &j2, &jb,
&c_b18, work31, &c__65, &ab[kv + 1 - jb + (j
+ jb) * ab_dim1], &i__3, &c_b31, &ab[kv + *kl
+ 1 - jb + (j + jb) * ab_dim1], &i__4);
}
}
if (j3 > 0) {
/* Copy the lower triangle of A13 into the work array */
/* WORK13 */
i__3 = j3;
for (jj = 1; jj <= i__3; ++jj) {
i__4 = jb;
for (ii = jj; ii <= i__4; ++ii) {
work13[ii + jj * 65 - 66] = ab[ii - jj + 1 + (jj
+ j + kv - 1) * ab_dim1];
/* L120: */
}
/* L130: */
}
/* Update A13 in the work array */
i__3 = *ldab - 1;
strsm_("Left", "Lower", "No transpose", "Unit", &jb, &j3,
&c_b31, &ab[kv + 1 + j * ab_dim1], &i__3, work13,
&c__65);
if (i2 > 0) {
/* Update A23 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
sgemm_("No transpose", "No transpose", &i2, &j3, &jb,
&c_b18, &ab[kv + 1 + jb + j * ab_dim1], &i__3,
work13, &c__65, &c_b31, &ab[jb + 1 + (j + kv)
* ab_dim1], &i__4);
}
if (i3 > 0) {
/* Update A33 */
i__3 = *ldab - 1;
sgemm_("No transpose", "No transpose", &i3, &j3, &jb,
&c_b18, work31, &c__65, work13, &c__65, &
c_b31, &ab[*kl + 1 + (j + kv) * ab_dim1], &
i__3);
}
/* Copy the lower triangle of A13 back into place */
i__3 = j3;
for (jj = 1; jj <= i__3; ++jj) {
i__4 = jb;
for (ii = jj; ii <= i__4; ++ii) {
ab[ii - jj + 1 + (jj + j + kv - 1) * ab_dim1] =
work13[ii + jj * 65 - 66];
/* L140: */
}
/* L150: */
}
}
} else {
/* Adjust the pivot indices. */
i__3 = j + jb - 1;
for (i__ = j; i__ <= i__3; ++i__) {
ipiv[i__] = ipiv[i__] + j - 1;
/* L160: */
}
}
/* Partially undo the interchanges in the current block to */
/* restore the upper triangular form of A31 and copy the upper */
/* triangle of A31 back into place */
i__3 = j;
for (jj = j + jb - 1; jj >= i__3; --jj) {
jp = ipiv[jj] - jj + 1;
if (jp != 1) {
/* Apply interchange to columns J to JJ-1 */
if (jp + jj - 1 < j + *kl) {
/* The interchange does not affect A31 */
i__4 = jj - j;
i__5 = *ldab - 1;
i__6 = *ldab - 1;
sswap_(&i__4, &ab[kv + 1 + jj - j + j * ab_dim1], &
i__5, &ab[kv + jp + jj - j + j * ab_dim1], &
i__6);
} else {
/* The interchange does affect A31 */
i__4 = jj - j;
i__5 = *ldab - 1;
sswap_(&i__4, &ab[kv + 1 + jj - j + j * ab_dim1], &
i__5, &work31[jp + jj - j - *kl - 1], &c__65);
}
}
/* Copy the current column of A31 back into place */
/* Computing MIN */
i__4 = i3, i__5 = jj - j + 1;
nw = min(i__4,i__5);
if (nw > 0) {
scopy_(&nw, &work31[(jj - j + 1) * 65 - 65], &c__1, &ab[
kv + *kl + 1 - jj + j + jj * ab_dim1], &c__1);
}
/* L170: */
}
/* L180: */
}
}
return 0;
/* End of SGBTRF */
} /* sgbtrf_ */
/* Subroutine */ int sget01_(integer *m, integer *n, real *a, integer *lda,
real *afac, integer *ldafac, integer *ipiv, real *rwork, real *resid)
{
/* System generated locals */
integer a_dim1, a_offset, afac_dim1, afac_offset, i__1, i__2;
/* Local variables */
integer i__, j, k;
real t, eps;
extern doublereal sdot_(integer *, real *, integer *, real *, integer *);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
real anorm;
extern /* Subroutine */ int sgemv_(char *, integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *), strmv_(char *, char *, char *, integer *, real *,
integer *, real *, integer *);
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
extern /* Subroutine */ int slaswp_(integer *, real *, integer *, integer
*, integer *, integer *, integer *);
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGET01 reconstructs a matrix A from its L*U factorization and */
/* computes the residual */
/* norm(L*U - A) / ( N * norm(A) * EPS ), */
/* where EPS is the machine epsilon. */
/* Arguments */
/* ========== */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix A. N >= 0. */
/* A (input) REAL array, dimension (LDA,N) */
/* The original M x N matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* AFAC (input/output) REAL array, dimension (LDAFAC,N) */
/* The factored form of the matrix A. AFAC contains the factors */
/* L and U from the L*U factorization as computed by SGETRF. */
/* Overwritten with the reconstructed matrix, and then with the */
/* difference L*U - A. */
/* LDAFAC (input) INTEGER */
/* The leading dimension of the array AFAC. LDAFAC >= max(1,M). */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from SGETRF. */
/* RWORK (workspace) REAL array, dimension (M) */
/* RESID (output) REAL */
/* norm(L*U - A) / ( N * norm(A) * EPS ) */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Quick exit if M = 0 or N = 0. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
afac_dim1 = *ldafac;
afac_offset = 1 + afac_dim1;
afac -= afac_offset;
--ipiv;
--rwork;
/* Function Body */
if (*m <= 0 || *n <= 0) {
*resid = 0.f;
return 0;
}
/* Determine EPS and the norm of A. */
eps = slamch_("Epsilon");
anorm = slange_("1", m, n, &a[a_offset], lda, &rwork[1]);
/* Compute the product L*U and overwrite AFAC with the result. */
/* A column at a time of the product is obtained, starting with */
/* column N. */
for (k = *n; k >= 1; --k) {
if (k > *m) {
strmv_("Lower", "No transpose", "Unit", m, &afac[afac_offset],
ldafac, &afac[k * afac_dim1 + 1], &c__1);
} else {
/* Compute elements (K+1:M,K) */
t = afac[k + k * afac_dim1];
if (k + 1 <= *m) {
i__1 = *m - k;
sscal_(&i__1, &t, &afac[k + 1 + k * afac_dim1], &c__1);
i__1 = *m - k;
i__2 = k - 1;
sgemv_("No transpose", &i__1, &i__2, &c_b11, &afac[k + 1 +
afac_dim1], ldafac, &afac[k * afac_dim1 + 1], &c__1, &
c_b11, &afac[k + 1 + k * afac_dim1], &c__1);
}
/* Compute the (K,K) element */
i__1 = k - 1;
afac[k + k * afac_dim1] = t + sdot_(&i__1, &afac[k + afac_dim1],
ldafac, &afac[k * afac_dim1 + 1], &c__1);
/* Compute elements (1:K-1,K) */
i__1 = k - 1;
strmv_("Lower", "No transpose", "Unit", &i__1, &afac[afac_offset],
ldafac, &afac[k * afac_dim1 + 1], &c__1);
}
/* L10: */
}
i__1 = min(*m,*n);
slaswp_(n, &afac[afac_offset], ldafac, &c__1, &i__1, &ipiv[1], &c_n1);
/* Compute the difference L*U - A and store in AFAC. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
afac[i__ + j * afac_dim1] -= a[i__ + j * a_dim1];
/* L20: */
}
/* L30: */
}
/* Compute norm( L*U - A ) / ( N * norm(A) * EPS ) */
*resid = slange_("1", m, n, &afac[afac_offset], ldafac, &rwork[1]);
if (anorm <= 0.f) {
if (*resid != 0.f) {
*resid = 1.f / eps;
}
} else {
*resid = *resid / (real) (*n) / anorm / eps;
}
return 0;
/* End of SGET01 */
} /* sget01_ */
/* Subroutine */ int sgetrf_(integer *m, integer *n, real *a, integer *lda,
integer *ipiv, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4, i__5;
/* Local variables */
integer i__, j, jb, nb, iinfo;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *), strsm_(char *, char *, char *,
char *, integer *, integer *, real *, real *, integer *, real *,
integer *), sgetf2_(integer *,
integer *, real *, integer *, integer *, integer *), xerbla_(char
*, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
extern /* Subroutine */ int slaswp_(integer *, real *, integer *, integer
*, integer *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGETRF computes an LU factorization of a general M-by-N matrix A */
/* using partial pivoting with row interchanges. */
/* The factorization has the form */
/* A = P * L * U */
/* where P is a permutation matrix, L is lower triangular with unit */
/* diagonal elements (lower trapezoidal if m > n), and U is upper */
/* triangular (upper trapezoidal if m < n). */
/* This is the right-looking Level 3 BLAS version of the algorithm. */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix A. N >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix to be factored. */
/* On exit, the factors L and U from the factorization */
/* A = P*L*U; the unit diagonal elements of L are not stored. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* IPIV (output) INTEGER array, dimension (min(M,N)) */
/* The pivot indices; for 1 <= i <= min(M,N), row i of the */
/* matrix was interchanged with row IPIV(i). */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, U(i,i) is exactly zero. The factorization */
/* has been completed, but the factor U is exactly */
/* singular, and division by zero will occur if it is used */
/* to solve a system of equations. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGETRF", &i__1);
return 0;
}
/* Quick return if possible */
if (*m == 0 || *n == 0) {
return 0;
}
/* Determine the block size for this environment. */
nb = ilaenv_(&c__1, "SGETRF", " ", m, n, &c_n1, &c_n1);
if (nb <= 1 || nb >= min(*m,*n)) {
/* Use unblocked code. */
sgetf2_(m, n, &a[a_offset], lda, &ipiv[1], info);
} else {
/* Use blocked code. */
i__1 = min(*m,*n);
i__2 = nb;
for (j = 1; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) {
/* Computing MIN */
i__3 = min(*m,*n) - j + 1;
jb = min(i__3,nb);
/* Factor diagonal and subdiagonal blocks and test for exact */
/* singularity. */
i__3 = *m - j + 1;
sgetf2_(&i__3, &jb, &a[j + j * a_dim1], lda, &ipiv[j], &iinfo);
/* Adjust INFO and the pivot indices. */
if (*info == 0 && iinfo > 0) {
*info = iinfo + j - 1;
}
/* Computing MIN */
i__4 = *m, i__5 = j + jb - 1;
i__3 = min(i__4,i__5);
for (i__ = j; i__ <= i__3; ++i__) {
ipiv[i__] = j - 1 + ipiv[i__];
/* L10: */
}
/* Apply interchanges to columns 1:J-1. */
i__3 = j - 1;
i__4 = j + jb - 1;
slaswp_(&i__3, &a[a_offset], lda, &j, &i__4, &ipiv[1], &c__1);
if (j + jb <= *n) {
/* Apply interchanges to columns J+JB:N. */
i__3 = *n - j - jb + 1;
i__4 = j + jb - 1;
slaswp_(&i__3, &a[(j + jb) * a_dim1 + 1], lda, &j, &i__4, &
ipiv[1], &c__1);
/* Compute block row of U. */
i__3 = *n - j - jb + 1;
strsm_("Left", "Lower", "No transpose", "Unit", &jb, &i__3, &
c_b16, &a[j + j * a_dim1], lda, &a[j + (j + jb) *
a_dim1], lda);
if (j + jb <= *m) {
/* Update trailing submatrix. */
i__3 = *m - j - jb + 1;
i__4 = *n - j - jb + 1;
sgemm_("No transpose", "No transpose", &i__3, &i__4, &jb,
&c_b19, &a[j + jb + j * a_dim1], lda, &a[j + (j +
jb) * a_dim1], lda, &c_b16, &a[j + jb + (j + jb) *
a_dim1], lda);
}
}
/* L20: */
}
}
return 0;
/* End of SGETRF */
} /* sgetrf_ */
/* Subroutine */
int slatdf_(integer *ijob, integer *n, real *z__, integer * ldz, real *rhs, real *rdsum, real *rdscal, integer *ipiv, integer * jpiv)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j, k;
real bm, bp, xm[8], xp[8];
integer info;
real temp;
extern real sdot_(integer *, real *, integer *, real *, integer *);
real work[32];
extern /* Subroutine */
int sscal_(integer *, real *, real *, integer *);
real pmone;
extern real sasum_(integer *, real *, integer *);
real sminu;
integer iwork[8];
extern /* Subroutine */
int scopy_(integer *, real *, integer *, real *, integer *), saxpy_(integer *, real *, real *, integer *, real *, integer *);
real splus;
extern /* Subroutine */
int sgesc2_(integer *, real *, integer *, real *, integer *, integer *, real *), sgecon_(char *, integer *, real *, integer *, real *, real *, real *, integer *, integer *), slassq_(integer *, real *, integer *, real *, real *), slaswp_( integer *, real *, integer *, integer *, integer *, integer *, integer *);
/* -- LAPACK auxiliary routine (version 3.4.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* September 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--rhs;
--ipiv;
--jpiv;
/* Function Body */
if (*ijob != 2)
{
/* Apply permutations IPIV to RHS */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], ldz, &c__1, &i__1, &ipiv[1], &c__1);
/* Solve for L-part choosing RHS either to +1 or -1. */
pmone = -1.f;
i__1 = *n - 1;
for (j = 1;
j <= i__1;
++j)
{
bp = rhs[j] + 1.f;
bm = rhs[j] - 1.f;
splus = 1.f;
/* Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and */
/* SMIN computed more efficiently than in BSOLVE [1]. */
i__2 = *n - j;
splus += sdot_(&i__2, &z__[j + 1 + j * z_dim1], &c__1, &z__[j + 1 + j * z_dim1], &c__1);
i__2 = *n - j;
sminu = sdot_(&i__2, &z__[j + 1 + j * z_dim1], &c__1, &rhs[j + 1], &c__1);
splus *= rhs[j];
if (splus > sminu)
{
rhs[j] = bp;
}
else if (sminu > splus)
{
rhs[j] = bm;
}
else
{
/* In this case the updating sums are equal and we can */
/* choose RHS(J) +1 or -1. The first time this happens */
/* we choose -1, thereafter +1. This is a simple way to */
/* get good estimates of matrices like Byers well-known */
/* example (see [1]). (Not done in BSOLVE.) */
rhs[j] += pmone;
pmone = 1.f;
}
/* Compute the remaining r.h.s. */
temp = -rhs[j];
i__2 = *n - j;
saxpy_(&i__2, &temp, &z__[j + 1 + j * z_dim1], &c__1, &rhs[j + 1], &c__1);
/* L10: */
}
/* Solve for U-part, look-ahead for RHS(N) = +-1. This is not done */
/* in BSOLVE and will hopefully give us a better estimate because */
/* any ill-conditioning of the original matrix is transfered to U */
/* and not to L. U(N, N) is an approximation to sigma_min(LU). */
i__1 = *n - 1;
scopy_(&i__1, &rhs[1], &c__1, xp, &c__1);
xp[*n - 1] = rhs[*n] + 1.f;
rhs[*n] += -1.f;
splus = 0.f;
sminu = 0.f;
for (i__ = *n;
i__ >= 1;
--i__)
{
temp = 1.f / z__[i__ + i__ * z_dim1];
xp[i__ - 1] *= temp;
rhs[i__] *= temp;
i__1 = *n;
for (k = i__ + 1;
k <= i__1;
++k)
{
xp[i__ - 1] -= xp[k - 1] * (z__[i__ + k * z_dim1] * temp);
rhs[i__] -= rhs[k] * (z__[i__ + k * z_dim1] * temp);
/* L20: */
}
splus += (r__1 = xp[i__ - 1], abs(r__1));
sminu += (r__1 = rhs[i__], abs(r__1));
/* L30: */
}
if (splus > sminu)
{
scopy_(n, xp, &c__1, &rhs[1], &c__1);
}
/* Apply the permutations JPIV to the computed solution (RHS) */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], ldz, &c__1, &i__1, &jpiv[1], &c_n1);
/* Compute the sum of squares */
slassq_(n, &rhs[1], &c__1, rdscal, rdsum);
}
else
{
/* IJOB = 2, Compute approximate nullvector XM of Z */
sgecon_("I", n, &z__[z_offset], ldz, &c_b23, &temp, work, iwork, & info);
scopy_(n, &work[*n], &c__1, xm, &c__1);
/* Compute RHS */
i__1 = *n - 1;
slaswp_(&c__1, xm, ldz, &c__1, &i__1, &ipiv[1], &c_n1);
temp = 1.f / sqrt(sdot_(n, xm, &c__1, xm, &c__1));
sscal_(n, &temp, xm, &c__1);
scopy_(n, xm, &c__1, xp, &c__1);
saxpy_(n, &c_b23, &rhs[1], &c__1, xp, &c__1);
saxpy_(n, &c_b37, xm, &c__1, &rhs[1], &c__1);
sgesc2_(n, &z__[z_offset], ldz, &rhs[1], &ipiv[1], &jpiv[1], &temp);
sgesc2_(n, &z__[z_offset], ldz, xp, &ipiv[1], &jpiv[1], &temp);
if (sasum_(n, xp, &c__1) > sasum_(n, &rhs[1], &c__1))
{
scopy_(n, xp, &c__1, &rhs[1], &c__1);
}
/* Compute the sum of squares */
slassq_(n, &rhs[1], &c__1, rdscal, rdsum);
}
return 0;
/* End of SLATDF */
}
/* Subroutine */ int sgesc2_(integer *n, real *a, integer *lda, real *rhs,
integer *ipiv, integer *jpiv, real *scale)
{
/* -- LAPACK auxiliary 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
=======
SGESC2 solves a system of linear equations
A * X = scale* RHS
with a general N-by-N matrix A using the LU factorization with
complete pivoting computed by SGETC2.
Arguments
=========
N (input) INTEGER
The order of the matrix A.
A (input) REAL array, dimension (LDA,N)
On entry, the LU part of the factorization of the n-by-n
matrix A computed by SGETC2: A = P * L * U * Q
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1, N).
RHS (input/output) REAL array, dimension (N).
On entry, the right hand side vector b.
On exit, the solution vector X.
IPIV (iput) INTEGER array, dimension (N).
The pivot indices; for 1 <= i <= N, row i of the
matrix has been interchanged with row IPIV(i).
JPIV (iput) INTEGER array, dimension (N).
The pivot indices; for 1 <= j <= N, column j of the
matrix has been interchanged with column JPIV(j).
SCALE (output) REAL
On exit, SCALE contains the scale factor. SCALE is chosen
0 <= SCALE <= 1 to prevent owerflow in the solution.
Further Details
===============
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
=====================================================================
Set constant to control owerflow
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
real r__1, r__2;
/* Local variables */
static real temp;
static integer i__, j;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *),
slabad_(real *, real *);
extern doublereal slamch_(char *);
static real bignum;
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slaswp_(integer *, real *, integer *, integer
*, integer *, integer *, integer *);
static real smlnum, eps;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--rhs;
--ipiv;
--jpiv;
/* Function Body */
eps = slamch_("P");
smlnum = slamch_("S") / eps;
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
/* Apply permutations IPIV to RHS */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], lda, &c__1, &i__1, &ipiv[1], &c__1);
/* Solve for L part */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
rhs[j] -= a_ref(j, i__) * rhs[i__];
/* L10: */
}
/* L20: */
}
/* Solve for U part */
*scale = 1.f;
/* Check for scaling */
i__ = isamax_(n, &rhs[1], &c__1);
if (smlnum * 2.f * (r__1 = rhs[i__], dabs(r__1)) > (r__2 = a_ref(*n, *n),
dabs(r__2))) {
temp = .5f / (r__1 = rhs[i__], dabs(r__1));
sscal_(n, &temp, &rhs[1], &c__1);
*scale *= temp;
}
for (i__ = *n; i__ >= 1; --i__) {
temp = 1.f / a_ref(i__, i__);
rhs[i__] *= temp;
i__1 = *n;
for (j = i__ + 1; j <= i__1; ++j) {
rhs[i__] -= rhs[j] * (a_ref(i__, j) * temp);
/* L30: */
}
/* L40: */
}
/* Apply permutations JPIV to the solution (RHS) */
i__1 = *n - 1;
slaswp_(&c__1, &rhs[1], lda, &c__1, &i__1, &jpiv[1], &c_n1);
return 0;
/* End of SGESC2 */
} /* sgesc2_ */