/* Subroutine */ int ssysvx_(char *fact, char *uplo, integer *n, integer *
nrhs, real *a, integer *lda, real *af, integer *ldaf, integer *ipiv,
real *b, integer *ldb, real *x, integer *ldx, real *rcond, real *ferr,
real *berr, real *work, integer *lwork, integer *iwork, integer *
info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2;
/* Local variables */
integer nb;
real anorm;
logical nofact;
integer lwkopt;
logical lquery;
/* -- LAPACK driver routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* SSYSVX uses the diagonal pivoting factorization to compute the */
/* solution to a real system of linear equations A * X = B, */
/* where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A. */
/* The form of the factorization is */
/* A = U * D * U**T, if UPLO = 'U', or */
/* A = L * D * L**T, if UPLO = 'L', */
/* where U (or L) is a product of permutation and unit upper (lower) */
/* triangular matrices, and D is symmetric and block diagonal with */
/* 1-by-1 and 2-by-2 diagonal blocks. */
/* 2. If some D(i,i)=0, so that D 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. */
/* 3. The system of equations is solved for X using the factored form */
/* of A. */
/* 4. Iterative refinement is applied to improve the computed solution */
/* matrix and calculate error bounds and backward error estimates */
/* for it. */
/* Arguments */
/* ========= */
/* FACT (input) CHARACTER*1 */
/* Specifies whether or not the factored form of A has been */
/* supplied on entry. */
/* = 'F': On entry, AF and IPIV contain the factored form of */
/* A. AF and IPIV will not be modified. */
/* = 'N': The matrix A will be copied to AF and factored. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* A (input) REAL array, dimension (LDA,N) */
/* The symmetric matrix A. If UPLO = 'U', the leading N-by-N */
/* upper triangular part of A contains the upper triangular part */
/* of the matrix A, and the strictly lower triangular part of A */
/* is not referenced. If UPLO = 'L', the leading N-by-N lower */
/* triangular part of A contains the lower triangular part of */
/* the matrix A, and the strictly upper triangular part of A is */
/* not referenced. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input or output) REAL array, dimension (LDAF,N) */
/* If FACT = 'F', then AF is an input argument and on entry */
/* contains the block diagonal matrix D and the multipliers used */
/* to obtain the factor U or L from the factorization */
/* A = U*D*U**T or A = L*D*L**T as computed by SSYTRF. */
/* If FACT = 'N', then AF is an output argument and on exit */
/* returns the block diagonal matrix D and the multipliers used */
/* to obtain the factor U or L from the factorization */
/* A = U*D*U**T or A = L*D*L**T. */
/* 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 details of the interchanges and the block structure */
/* of D, as determined by SSYTRF. */
/* If IPIV(k) > 0, then rows and columns k and IPIV(k) were */
/* interchanged and D(k,k) is a 1-by-1 diagonal block. */
/* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and */
/* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) */
/* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = */
/* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were */
/* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. */
/* If FACT = 'N', then IPIV is an output argument and on exit */
/* contains details of the interchanges and the block structure */
/* of D, as determined by SSYTRF. */
/* B (input) REAL array, dimension (LDB,NRHS) */
/* The N-by-NRHS right hand side matrix 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. */
/* 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. 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 (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The length of WORK. LWORK >= max(1,3*N), and for best */
/* performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where */
/* NB is the optimal blocksize for SSYTRF. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace) INTEGER array, dimension (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: D(i,i) is exactly zero. The factorization */
/* has been completed but the factor D is exactly */
/* singular, so the solution and error bounds could */
/* not be computed. RCOND = 0 is returned. */
/* = N+1: D 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. */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--iwork;
/* Function Body */
*info = 0;
nofact = lsame_(fact, "N");
lquery = *lwork == -1;
if (! nofact && ! lsame_(fact, "F")) {
*info = -1;
} else if (! lsame_(uplo, "U") && ! lsame_(uplo,
"L")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*nrhs < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else if (*ldaf < max(1,*n)) {
*info = -8;
} else if (*ldb < max(1,*n)) {
*info = -11;
} else if (*ldx < max(1,*n)) {
*info = -13;
} else /* if(complicated condition) */ {
/* Computing MAX */
i__1 = 1, i__2 = *n * 3;
if (*lwork < max(i__1,i__2) && ! lquery) {
*info = -18;
}
}
if (*info == 0) {
/* Computing MAX */
i__1 = 1, i__2 = *n * 3;
lwkopt = max(i__1,i__2);
if (nofact) {
nb = ilaenv_(&c__1, "SSYTRF", uplo, n, &c_n1, &c_n1, &c_n1);
/* Computing MAX */
i__1 = lwkopt, i__2 = *n * nb;
lwkopt = max(i__1,i__2);
}
work[1] = (real) lwkopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSYSVX", &i__1);
return 0;
} else if (lquery) {
return 0;
}
if (nofact) {
/* Compute the factorization A = U*D*U' or A = L*D*L'. */
slacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf);
ssytrf_(uplo, n, &af[af_offset], ldaf, &ipiv[1], &work[1], lwork,
info);
/* Return if INFO is non-zero. */
if (*info > 0) {
*rcond = 0.f;
return 0;
}
}
/* Compute the norm of the matrix A. */
anorm = slansy_("I", uplo, n, &a[a_offset], lda, &work[1]);
/* Compute the reciprocal of the condition number of A. */
ssycon_(uplo, n, &af[af_offset], ldaf, &ipiv[1], &anorm, rcond, &work[1],
&iwork[1], info);
/* Compute the solution vectors X. */
slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
ssytrs_(uplo, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx,
info);
/* Use iterative refinement to improve the computed solutions and */
/* compute error bounds and backward error estimates for them. */
ssyrfs_(uplo, 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);
/* Set INFO = N+1 if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon")) {
*info = *n + 1;
}
work[1] = (real) lwkopt;
return 0;
/* End of SSYSVX */
} /* ssysvx_ */
/* Subroutine */ int ssycon_(char *uplo, integer *n, real *a, integer *lda,
integer *ipiv, real *anorm, real *rcond, real *work, integer *iwork,
integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
March 31, 1993
Purpose
=======
SSYCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by SSYTRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) REAL array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF.
ANORM (input) REAL
The 1-norm of the original matrix A.
RCOND (output) REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK (workspace) REAL array, dimension (2*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1;
/* Local variables */
static integer kase, i__;
extern logical lsame_(char *, char *);
static logical upper;
extern /* Subroutine */ int xerbla_(char *, integer *), slacon_(
integer *, real *, real *, integer *, real *, integer *);
static real ainvnm;
extern /* Subroutine */ int ssytrs_(char *, integer *, integer *, real *,
integer *, integer *, real *, integer *, integer *);
#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;
--ipiv;
--work;
--iwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*anorm < 0.f) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSYCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0) {
*rcond = 1.f;
return 0;
} else if (*anorm <= 0.f) {
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper) {
/* Upper triangular storage: examine D from bottom to top */
for (i__ = *n; i__ >= 1; --i__) {
if (ipiv[i__] > 0 && a_ref(i__, i__) == 0.f) {
return 0;
}
/* L10: */
}
} else {
/* Lower triangular storage: examine D from top to bottom. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (ipiv[i__] > 0 && a_ref(i__, i__) == 0.f) {
return 0;
}
/* L20: */
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
slacon_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase);
if (kase != 0) {
/* Multiply by inv(L*D*L') or inv(U*D*U'). */
ssytrs_(uplo, n, &c__1, &a[a_offset], lda, &ipiv[1], &work[1], n,
info);
goto L30;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f) {
*rcond = 1.f / ainvnm / *anorm;
}
return 0;
/* End of SSYCON */
} /* ssycon_ */
/* Subroutine */ int ssyrfs_(char *uplo, integer *n, integer *nrhs, real *a,
integer *lda, real *af, integer *ldaf, integer *ipiv, real *b,
integer *ldb, real *x, integer *ldx, real *ferr, real *berr, real *
work, integer *iwork, integer *info)
{
/* -- LAPACK routine (version 2.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
SSYRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite, and
provides error bounds and backward error estimates for the solution.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A (input) REAL array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If UPLO = 'L', the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input) REAL array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**T or
A = L*D*L**T as computed by SSYTRF.
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF.
B (input) REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (input/output) REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SSYTRS.
On exit, the improved solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK (workspace) REAL array, dimension (3*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters
===================
ITMAX is the maximum number of steps of iterative refinement.
=====================================================================
Test the input parameters.
Parameter adjustments
Function Body */
/* Table of constant values */
static integer c__1 = 1;
static real c_b12 = -1.f;
static real c_b14 = 1.f;
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2, i__3;
real r__1, r__2, r__3;
/* Local variables */
static integer kase;
static real safe1, safe2;
static integer i, j, k;
static real s;
extern logical lsame_(char *, char *);
static integer count;
static logical upper;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), saxpy_(integer *, real *, real *, integer *, real *,
integer *), ssymv_(char *, integer *, real *, real *, integer *,
real *, integer *, real *, real *, integer *);
static real xk;
extern doublereal slamch_(char *);
static integer nz;
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *), slacon_(
integer *, real *, real *, integer *, real *, integer *);
static real lstres;
extern /* Subroutine */ int ssytrs_(char *, integer *, integer *, real *,
integer *, integer *, real *, integer *, integer *);
static real eps;
#define IPIV(I) ipiv[(I)-1]
#define FERR(I) ferr[(I)-1]
#define BERR(I) berr[(I)-1]
#define WORK(I) work[(I)-1]
#define IWORK(I) iwork[(I)-1]
#define A(I,J) a[(I)-1 + ((J)-1)* ( *lda)]
#define AF(I,J) af[(I)-1 + ((J)-1)* ( *ldaf)]
#define B(I,J) b[(I)-1 + ((J)-1)* ( *ldb)]
#define X(I,J) x[(I)-1 + ((J)-1)* ( *ldx)]
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldaf < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -10;
} else if (*ldx < max(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSYRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= *nrhs; ++j) {
FERR(j) = 0.f;
BERR(j) = 0.f;
/* L10: */
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= *nrhs; ++j) {
count = 1;
lstres = 3.f;
L20:
/* Loop until stopping criterion is satisfied.
Compute residual R = B - A * X */
scopy_(n, &B(1,j), &c__1, &WORK(*n + 1), &c__1);
ssymv_(uplo, n, &c_b12, &A(1,1), lda, &X(1,j), &c__1,
&c_b14, &WORK(*n + 1), &c__1);
/* Compute componentwise relative backward error from formula
max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
where abs(Z) is the componentwise absolute value of the matr
ix
or vector Z. If the i-th component of the denominator is le
ss
than SAFE2, then SAFE1 is added to the i-th components of th
e
numerator and denominator before dividing. */
i__2 = *n;
for (i = 1; i <= *n; ++i) {
WORK(i) = (r__1 = B(i,j), dabs(r__1));
/* L30: */
}
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
i__2 = *n;
for (k = 1; k <= *n; ++k) {
s = 0.f;
xk = (r__1 = X(k,j), dabs(r__1));
i__3 = k - 1;
for (i = 1; i <= k-1; ++i) {
WORK(i) += (r__1 = A(i,k), dabs(r__1)) * xk;
s += (r__1 = A(i,k), dabs(r__1)) * (r__2 = X(i,j), dabs(r__2));
/* L40: */
}
WORK(k) = WORK(k) + (r__1 = A(k,k), dabs(r__1)) *
xk + s;
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= *n; ++k) {
s = 0.f;
xk = (r__1 = X(k,j), dabs(r__1));
WORK(k) += (r__1 = A(k,k), dabs(r__1)) * xk;
i__3 = *n;
for (i = k + 1; i <= *n; ++i) {
WORK(i) += (r__1 = A(i,k), dabs(r__1)) * xk;
s += (r__1 = A(i,k), dabs(r__1)) * (r__2 = X(i,j), dabs(r__2));
/* L60: */
}
WORK(k) += s;
/* L70: */
}
}
s = 0.f;
i__2 = *n;
for (i = 1; i <= *n; ++i) {
if (WORK(i) > safe2) {
/* Computing MAX */
r__2 = s, r__3 = (r__1 = WORK(*n + i), dabs(r__1)) / WORK(i);
s = dmax(r__2,r__3);
} else {
/* Computing MAX */
r__2 = s, r__3 = ((r__1 = WORK(*n + i), dabs(r__1)) + safe1) /
(WORK(i) + safe1);
s = dmax(r__2,r__3);
}
/* L80: */
}
BERR(j) = s;
/* Test stopping criterion. Continue iterating if
1) The residual BERR(J) is larger than machine epsilon, a
nd
2) BERR(J) decreased by at least a factor of 2 during the
last iteration, and
3) At most ITMAX iterations tried. */
if (BERR(j) > eps && BERR(j) * 2.f <= lstres && count <= 5) {
/* Update solution and try again. */
ssytrs_(uplo, n, &c__1, &AF(1,1), ldaf, &IPIV(1), &WORK(*n
+ 1), n, info);
saxpy_(n, &c_b14, &WORK(*n + 1), &c__1, &X(1,j), &c__1)
;
lstres = BERR(j);
++count;
goto L20;
}
/* Bound error from formula
norm(X - XTRUE) / norm(X) .le. FERR =
norm( abs(inv(A))*
( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
where
norm(Z) is the magnitude of the largest component of Z
inv(A) is the inverse of A
abs(Z) is the componentwise absolute value of the matrix o
r
vector Z
NZ is the maximum number of nonzeros in any row of A, plus
1
EPS is machine epsilon
The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
is incremented by SAFE1 if the i-th component of
abs(A)*abs(X) + abs(B) is less than SAFE2.
Use SLACON to estimate the infinity-norm of the matrix
inv(A) * diag(W),
where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i = 1; i <= *n; ++i) {
if (WORK(i) > safe2) {
WORK(i) = (r__1 = WORK(*n + i), dabs(r__1)) + nz * eps * WORK(
i);
} else {
WORK(i) = (r__1 = WORK(*n + i), dabs(r__1)) + nz * eps * WORK(
i) + safe1;
}
/* L90: */
}
kase = 0;
L100:
slacon_(n, &WORK((*n << 1) + 1), &WORK(*n + 1), &IWORK(1), &FERR(j), &
kase);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
ssytrs_(uplo, n, &c__1, &AF(1,1), ldaf, &IPIV(1), &WORK(
*n + 1), n, info);
i__2 = *n;
for (i = 1; i <= *n; ++i) {
WORK(*n + i) = WORK(i) * WORK(*n + i);
/* L110: */
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i = 1; i <= *n; ++i) {
WORK(*n + i) = WORK(i) * WORK(*n + i);
/* L120: */
}
ssytrs_(uplo, n, &c__1, &AF(1,1), ldaf, &IPIV(1), &WORK(
*n + 1), n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i = 1; i <= *n; ++i) {
/* Computing MAX */
r__2 = lstres, r__3 = (r__1 = X(i,j), dabs(r__1));
lstres = dmax(r__2,r__3);
/* L130: */
}
if (lstres != 0.f) {
FERR(j) /= lstres;
}
/* L140: */
}
return 0;
/* End of SSYRFS */
} /* ssyrfs_ */
/* Subroutine */
int ssysvx_(char *fact, char *uplo, integer *n, integer * nrhs, real *a, integer *lda, real *af, integer *ldaf, integer *ipiv, real *b, integer *ldb, real *x, integer *ldx, real *rcond, real *ferr, real *berr, real *work, integer *lwork, integer *iwork, integer * info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, x_offset, i__1, i__2;
/* Local variables */
integer nb;
extern logical lsame_(char *, char *);
real anorm;
extern real slamch_(char *);
logical nofact;
extern /* Subroutine */
int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *);
extern /* Subroutine */
int slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *);
extern real slansy_(char *, char *, integer *, real *, integer *, real *);
extern /* Subroutine */
int ssycon_(char *, integer *, real *, integer *, integer *, real *, real *, real *, integer *, integer *);
integer lwkopt;
logical lquery;
extern /* Subroutine */
int ssyrfs_(char *, integer *, integer *, real *, integer *, real *, integer *, integer *, real *, integer *, real * , integer *, real *, real *, real *, integer *, integer *) , ssytrf_(char *, integer *, real *, integer *, integer *, real *, integer *, integer *), ssytrs_(char *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *);
/* -- LAPACK driver routine (version 3.4.1) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* April 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--iwork;
/* Function Body */
*info = 0;
nofact = lsame_(fact, "N");
lquery = *lwork == -1;
if (! nofact && ! lsame_(fact, "F"))
{
*info = -1;
}
else if (! lsame_(uplo, "U") && ! lsame_(uplo, "L"))
{
*info = -2;
}
else if (*n < 0)
{
*info = -3;
}
else if (*nrhs < 0)
{
*info = -4;
}
else if (*lda < max(1,*n))
{
*info = -6;
}
else if (*ldaf < max(1,*n))
{
*info = -8;
}
else if (*ldb < max(1,*n))
{
*info = -11;
}
else if (*ldx < max(1,*n))
{
*info = -13;
}
else /* if(complicated condition) */
{
/* Computing MAX */
i__1 = 1;
i__2 = *n * 3; // , expr subst
if (*lwork < max(i__1,i__2) && ! lquery)
{
*info = -18;
}
}
if (*info == 0)
{
/* Computing MAX */
i__1 = 1;
i__2 = *n * 3; // , expr subst
lwkopt = max(i__1,i__2);
if (nofact)
{
nb = ilaenv_(&c__1, "SSYTRF", uplo, n, &c_n1, &c_n1, &c_n1);
/* Computing MAX */
i__1 = lwkopt;
i__2 = *n * nb; // , expr subst
lwkopt = max(i__1,i__2);
}
work[1] = (real) lwkopt;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("SSYSVX", &i__1);
return 0;
}
else if (lquery)
{
return 0;
}
if (nofact)
{
/* Compute the factorization A = U*D*U**T or A = L*D*L**T. */
slacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf);
ssytrf_(uplo, n, &af[af_offset], ldaf, &ipiv[1], &work[1], lwork, info);
/* Return if INFO is non-zero. */
if (*info > 0)
{
*rcond = 0.f;
return 0;
}
}
/* Compute the norm of the matrix A. */
anorm = slansy_("I", uplo, n, &a[a_offset], lda, &work[1]);
/* Compute the reciprocal of the condition number of A. */
ssycon_(uplo, n, &af[af_offset], ldaf, &ipiv[1], &anorm, rcond, &work[1], &iwork[1], info);
/* Compute the solution vectors X. */
slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
ssytrs_(uplo, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx, info);
/* Use iterative refinement to improve the computed solutions and */
/* compute error bounds and backward error estimates for them. */
ssyrfs_(uplo, 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);
/* Set INFO = N+1 if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon"))
{
*info = *n + 1;
}
work[1] = (real) lwkopt;
return 0;
/* End of SSYSVX */
}
/* Subroutine */ int ssysv_(char *uplo, integer *n, integer *nrhs, real *a,
integer *lda, integer *ipiv, real *b, integer *ldb, real *work,
integer *lwork, 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
=======
SSYSV computes the solution to a real system of linear equations
A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.
The diagonal pivoting method is used to factor A as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then
used to solve the system of equations A * X = B.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the block diagonal matrix D and the
multipliers used to obtain the factor U or L from the
factorization A = U*D*U**T or A = L*D*L**T as computed by
SSYTRF.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D, as
determined by SSYTRF. If IPIV(k) > 0, then rows and columns
k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
then rows and columns k-1 and -IPIV(k) were interchanged and
D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
-IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
diagonal block.
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of WORK. LWORK >= 1, and for best performance
LWORK >= N*NB, where NB is the optimal blocksize for
SSYTRF.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, so the solution could not be computed.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1;
/* Local variables */
extern logical lsame_(char *, char *);
static integer nb;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer lwkopt;
static logical lquery;
extern /* Subroutine */ int ssytrf_(char *, integer *, real *, integer *,
integer *, real *, integer *, integer *), ssytrs_(char *,
integer *, integer *, real *, integer *, integer *, real *,
integer *, integer *);
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--work;
/* Function Body */
*info = 0;
lquery = *lwork == -1;
if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -8;
} else if (*lwork < 1 && ! lquery) {
*info = -10;
}
if (*info == 0) {
nb = ilaenv_(&c__1, "SSYTRF", uplo, n, &c_n1, &c_n1, &c_n1, (ftnlen)6,
(ftnlen)1);
lwkopt = *n * nb;
work[1] = (real) lwkopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSYSV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Compute the factorization A = U*D*U' or A = L*D*L'. */
ssytrf_(uplo, n, &a[a_offset], lda, &ipiv[1], &work[1], lwork, info);
if (*info == 0) {
/* Solve the system A*X = B, overwriting B with X. */
ssytrs_(uplo, n, nrhs, &a[a_offset], lda, &ipiv[1], &b[b_offset], ldb,
info);
}
work[1] = (real) lwkopt;
return 0;
/* End of SSYSV */
} /* ssysv_ */
/* Subroutine */
int ssycon_(char *uplo, integer *n, real *a, integer *lda, integer *ipiv, real *anorm, real *rcond, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1;
/* Local variables */
integer i__, kase;
extern logical lsame_(char *, char *);
integer isave[3];
logical upper;
extern /* Subroutine */
int slacn2_(integer *, real *, real *, integer *, real *, integer *, integer *), xerbla_(char *, integer *);
real ainvnm;
extern /* Subroutine */
int ssytrs_(char *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *);
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
--work;
--iwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L"))
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*n))
{
*info = -4;
}
else if (*anorm < 0.f)
{
*info = -6;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("SSYCON", &i__1);
return 0;
}
/* Quick return if possible */
*rcond = 0.f;
if (*n == 0)
{
*rcond = 1.f;
return 0;
}
else if (*anorm <= 0.f)
{
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper)
{
/* Upper triangular storage: examine D from bottom to top */
for (i__ = *n;
i__ >= 1;
--i__)
{
if (ipiv[i__] > 0 && a[i__ + i__ * a_dim1] == 0.f)
{
return 0;
}
/* L10: */
}
}
else
{
/* Lower triangular storage: examine D from top to bottom. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
if (ipiv[i__] > 0 && a[i__ + i__ * a_dim1] == 0.f)
{
return 0;
}
/* L20: */
}
}
/* Estimate the 1-norm of the inverse. */
kase = 0;
L30:
slacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0)
{
/* Multiply by inv(L*D*L**T) or inv(U*D*U**T). */
ssytrs_(uplo, n, &c__1, &a[a_offset], lda, &ipiv[1], &work[1], n, info);
goto L30;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f)
{
*rcond = 1.f / ainvnm / *anorm;
}
return 0;
/* End of SSYCON */
}
/* Subroutine */ int ssyrfs_(char *uplo, integer *n, integer *nrhs, real *a,
integer *lda, real *af, integer *ldaf, integer *ipiv, real *b,
integer *ldb, real *x, integer *ldx, real *ferr, real *berr, real *
work, integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2, i__3;
real r__1, r__2, r__3;
/* Local variables */
integer i__, j, k;
real s, xk;
integer nz;
real eps;
integer kase;
real safe1, safe2;
integer isave[3], count;
logical upper;
real safmin;
real lstres;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH. */
/* Purpose */
/* ======= */
/* SSYRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is symmetric indefinite, and */
/* provides error bounds and backward error estimates for the solution. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrices B and X. NRHS >= 0. */
/* A (input) REAL array, dimension (LDA,N) */
/* The symmetric matrix A. If UPLO = 'U', the leading N-by-N */
/* upper triangular part of A contains the upper triangular part */
/* of the matrix A, and the strictly lower triangular part of A */
/* is not referenced. If UPLO = 'L', the leading N-by-N lower */
/* triangular part of A contains the lower triangular part of */
/* the matrix A, and the strictly upper triangular part of A is */
/* not referenced. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) REAL array, dimension (LDAF,N) */
/* The factored form of the matrix A. AF contains the block */
/* diagonal matrix D and the multipliers used to obtain the */
/* factor U or L from the factorization A = U*D*U**T or */
/* A = L*D*L**T as computed by SSYTRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by SSYTRF. */
/* B (input) REAL array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) REAL array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by SSYTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) REAL array, dimension (NRHS) */
/* The estimated forward error bound for each solution vector */
/* X(j) (the j-th column of the solution matrix X). */
/* If XTRUE is the true solution corresponding to X(j), FERR(j) */
/* is an estimated upper bound for the magnitude of the largest */
/* element in (X(j) - XTRUE) divided by the magnitude of the */
/* largest element in X(j). The estimate is as reliable as */
/* the estimate for RCOND, and is almost always a slight */
/* overestimate of the true error. */
/* BERR (output) REAL array, dimension (NRHS) */
/* The componentwise relative backward error of each solution */
/* vector X(j) (i.e., the smallest relative change in */
/* any element of A or B that makes X(j) an exact solution). */
/* WORK (workspace) REAL array, dimension (3*N) */
/* IWORK (workspace) INTEGER array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Internal Parameters */
/* =================== */
/* ITMAX is the maximum number of steps of iterative refinement. */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--iwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldaf < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -10;
} else if (*ldx < max(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSYRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.f;
berr[j] = 0.f;
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
count = 1;
lstres = 3.f;
L20:
/* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - A * X */
scopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
ssymv_(uplo, n, &c_b12, &a[a_offset], lda, &x[j * x_dim1 + 1], &c__1,
&c_b14, &work[*n + 1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. If the i-th component of the denominator is less */
/* than SAFE2, then SAFE1 is added to the i-th components of the */
/* numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[i__] = (r__1 = b[i__ + j * b_dim1], dabs(r__1));
}
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
xk = (r__1 = x[k + j * x_dim1], dabs(r__1));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
work[i__] += (r__1 = a[i__ + k * a_dim1], dabs(r__1)) *
xk;
s += (r__1 = a[i__ + k * a_dim1], dabs(r__1)) * (r__2 = x[
i__ + j * x_dim1], dabs(r__2));
}
work[k] = work[k] + (r__1 = a[k + k * a_dim1], dabs(r__1)) *
xk + s;
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
xk = (r__1 = x[k + j * x_dim1], dabs(r__1));
work[k] += (r__1 = a[k + k * a_dim1], dabs(r__1)) * xk;
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
work[i__] += (r__1 = a[i__ + k * a_dim1], dabs(r__1)) *
xk;
s += (r__1 = a[i__ + k * a_dim1], dabs(r__1)) * (r__2 = x[
i__ + j * x_dim1], dabs(r__2));
}
work[k] += s;
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
/* Computing MAX */
r__2 = s, r__3 = (r__1 = work[*n + i__], dabs(r__1)) / work[
i__];
s = dmax(r__2,r__3);
} else {
/* Computing MAX */
r__2 = s, r__3 = ((r__1 = work[*n + i__], dabs(r__1)) + safe1)
/ (work[i__] + safe1);
s = dmax(r__2,r__3);
}
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if */
/* 1) The residual BERR(J) is larger than machine epsilon, and */
/* 2) BERR(J) decreased by at least a factor of 2 during the */
/* last iteration, and */
/* 3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2.f <= lstres && count <= 5) {
/* Update solution and try again. */
ssytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[*n
+ 1], n, info);
saxpy_(n, &c_b14, &work[*n + 1], &c__1, &x[j * x_dim1 + 1], &c__1)
;
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(A))* */
/* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(A) is the inverse of A */
/* abs(Z) is the componentwise absolute value of the matrix or */
/* vector Z */
/* NZ is the maximum number of nonzeros in any row of A, plus 1 */
/* EPS is machine epsilon */
/* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(A)*abs(X) + abs(B) is less than SAFE2. */
/* Use SLACN2 to estimate the infinity-norm of the matrix */
/* inv(A) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
work[i__] = (r__1 = work[*n + i__], dabs(r__1)) + nz * eps *
work[i__];
} else {
work[i__] = (r__1 = work[*n + i__], dabs(r__1)) + nz * eps *
work[i__] + safe1;
}
}
kase = 0;
L100:
slacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
ssytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
*n + 1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
}
ssytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
*n + 1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
r__2 = lstres, r__3 = (r__1 = x[i__ + j * x_dim1], dabs(r__1));
lstres = dmax(r__2,r__3);
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
}
return 0;
/* End of SSYRFS */
} /* ssyrfs_ */
/* Subroutine */ int schksy_(logical *dotype, integer *nn, integer *nval,
integer *nnb, integer *nbval, integer *nns, integer *nsval, real *
thresh, logical *tsterr, integer *nmax, real *a, real *afac, real *
ainv, real *b, real *x, real *xact, real *work, real *rwork, integer *
iwork, integer *nout)
{
/* Initialized data */
static integer iseedy[4] = { 1988,1989,1990,1991 };
static char uplos[1*2] = "U" "L";
/* Format strings */
static char fmt_9999[] = "(\002 UPLO = '\002,a1,\002', N =\002,i5,\002, "
"NB =\002,i4,\002, type \002,i2,\002, test \002,i2,\002, ratio "
"=\002,g12.5)";
static char fmt_9998[] = "(\002 UPLO = '\002,a1,\002', N =\002,i5,\002, "
"NRHS=\002,i3,\002, type \002,i2,\002, test(\002,i2,\002) =\002,g"
"12.5)";
static char fmt_9997[] = "(\002 UPLO = '\002,a1,\002', N =\002,i5,\002"
",\002,10x,\002 type \002,i2,\002, test(\002,i2,\002) =\002,g12.5)"
;
/* System generated locals */
integer i__1, i__2, i__3, i__4;
/* Builtin functions */
/* Subroutine */ int s_copy(char *, char *, ftnlen, ftnlen);
integer s_wsfe(cilist *), do_fio(integer *, char *, ftnlen), e_wsfe(void);
/* Local variables */
integer i__, j, k, n, i1, i2, nb, in, kl, ku, nt, lda, inb, ioff, mode,
imat, info;
char path[3], dist[1];
integer irhs, nrhs;
char uplo[1], type__[1];
integer nrun;
extern /* Subroutine */ int alahd_(integer *, char *);
integer nfail, iseed[4];
real rcond;
extern /* Subroutine */ int sget04_(integer *, integer *, real *, integer
*, real *, integer *, real *, real *);
integer nimat;
extern doublereal sget06_(real *, real *);
real anorm;
extern /* Subroutine */ int spot02_(char *, integer *, integer *, real *,
integer *, real *, integer *, real *, integer *, real *, real *);
integer iuplo, izero, nerrs;
extern /* Subroutine */ int spot03_(char *, integer *, real *, integer *,
real *, integer *, real *, integer *, real *, real *, real *), spot05_(char *, integer *, integer *, real *, integer *,
real *, integer *, real *, integer *, real *, integer *, real *,
real *, real *);
integer lwork;
logical zerot;
extern /* Subroutine */ int ssyt01_(char *, integer *, real *, integer *,
real *, integer *, integer *, real *, integer *, real *, real *);
char xtype[1];
extern /* Subroutine */ int slatb4_(char *, integer *, integer *, integer
*, char *, integer *, integer *, real *, integer *, real *, char *
), alaerh_(char *, char *, integer *,
integer *, char *, integer *, integer *, integer *, integer *,
integer *, integer *, integer *, integer *, integer *);
real rcondc;
extern /* Subroutine */ int alasum_(char *, integer *, integer *, integer
*, integer *);
real cndnum;
logical trfcon;
extern /* Subroutine */ int 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 *), xlaenv_(
integer *, integer *), slatms_(integer *, integer *, char *,
integer *, char *, real *, integer *, real *, real *, integer *,
integer *, char *, real *, integer *, real *, integer *);
extern doublereal slansy_(char *, char *, integer *, real *, integer *,
real *);
real result[8];
extern /* Subroutine */ int ssycon_(char *, integer *, real *, integer *,
integer *, real *, real *, real *, integer *, integer *),
serrsy_(char *, integer *), ssyrfs_(char *, integer *,
integer *, real *, integer *, real *, integer *, integer *, real *
, integer *, real *, integer *, real *, real *, real *, integer *,
integer *), ssytrf_(char *, integer *, real *, integer *,
integer *, real *, integer *, integer *), ssytri_(char *,
integer *, real *, integer *, integer *, real *, integer *), ssytrs_(char *, integer *, integer *, real *, integer *,
integer *, real *, integer *, integer *);
/* Fortran I/O blocks */
static cilist io___39 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___42 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___44 = { 0, 0, 0, fmt_9997, 0 };
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SCHKSY tests SSYTRF, -TRI, -TRS, -RFS, and -CON. */
/* 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 dimension N. */
/* NNB (input) INTEGER */
/* The number of values of NB contained in the vector NBVAL. */
/* NBVAL (input) INTEGER array, dimension (NBVAL) */
/* The values of the blocksize NB. */
/* NNS (input) INTEGER */
/* The number of values of NRHS contained in the vector NSVAL. */
/* NSVAL (input) INTEGER array, dimension (NNS) */
/* The values of the number of right hand sides NRHS. */
/* 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) */
/* AINV (workspace) REAL array, dimension (NMAX*NMAX) */
/* B (workspace) REAL array, dimension (NMAX*NSMAX) */
/* where NSMAX is the largest entry in NSVAL. */
/* X (workspace) REAL array, dimension (NMAX*NSMAX) */
/* XACT (workspace) REAL array, dimension (NMAX*NSMAX) */
/* WORK (workspace) REAL array, dimension */
/* (NMAX*max(3,NSMAX)) */
/* RWORK (workspace) REAL array, dimension */
/* (max(NMAX,2*NSMAX)) */
/* 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;
--xact;
--x;
--b;
--ainv;
--afac;
--a;
--nsval;
--nbval;
--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, "SY", (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) {
serrsy_(path, nout);
}
infoc_1.infot = 0;
xlaenv_(&c__2, &c__2);
/* 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 = 10;
if (n <= 0) {
nimat = 1;
}
izero = 0;
i__2 = nimat;
for (imat = 1; imat <= i__2; ++imat) {
/* Do the tests only if DOTYPE( IMAT ) is true. */
if (! dotype[imat]) {
goto L170;
}
/* Skip types 3, 4, 5, or 6 if the matrix size is too small. */
zerot = imat >= 3 && imat <= 6;
if (zerot && n < imat - 2) {
goto L170;
}
/* Do first for UPLO = 'U', then for UPLO = 'L' */
for (iuplo = 1; iuplo <= 2; ++iuplo) {
*(unsigned char *)uplo = *(unsigned char *)&uplos[iuplo - 1];
/* Set up parameters with SLATB4 and generate a test matrix */
/* with SLATMS. */
slatb4_(path, &imat, &n, &n, type__, &kl, &ku, &anorm, &mode,
&cndnum, dist);
s_copy(srnamc_1.srnamt, "SLATMS", (ftnlen)32, (ftnlen)6);
slatms_(&n, &n, dist, iseed, type__, &rwork[1], &mode, &
cndnum, &anorm, &kl, &ku, uplo, &a[1], &lda, &work[1],
&info);
/* Check error code from SLATMS. */
if (info != 0) {
alaerh_(path, "SLATMS", &info, &c__0, uplo, &n, &n, &c_n1,
&c_n1, &c_n1, &imat, &nfail, &nerrs, nout);
goto L160;
}
/* For types 3-6, zero one or more rows and columns of */
/* the matrix to test that INFO is returned correctly. */
if (zerot) {
if (imat == 3) {
izero = 1;
} else if (imat == 4) {
izero = n;
} else {
izero = n / 2 + 1;
}
if (imat < 6) {
/* Set row and column IZERO to zero. */
if (iuplo == 1) {
ioff = (izero - 1) * lda;
i__3 = izero - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
a[ioff + i__] = 0.f;
/* L20: */
}
ioff += izero;
i__3 = n;
for (i__ = izero; i__ <= i__3; ++i__) {
a[ioff] = 0.f;
ioff += lda;
/* L30: */
}
} else {
ioff = izero;
i__3 = izero - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
a[ioff] = 0.f;
ioff += lda;
/* L40: */
}
ioff -= izero;
i__3 = n;
for (i__ = izero; i__ <= i__3; ++i__) {
a[ioff + i__] = 0.f;
/* L50: */
}
}
} else {
ioff = 0;
if (iuplo == 1) {
/* Set the first IZERO rows and columns to zero. */
i__3 = n;
for (j = 1; j <= i__3; ++j) {
i2 = min(j,izero);
i__4 = i2;
for (i__ = 1; i__ <= i__4; ++i__) {
a[ioff + i__] = 0.f;
/* L60: */
}
ioff += lda;
/* L70: */
}
} else {
/* Set the last IZERO rows and columns to zero. */
i__3 = n;
for (j = 1; j <= i__3; ++j) {
i1 = max(j,izero);
i__4 = n;
for (i__ = i1; i__ <= i__4; ++i__) {
a[ioff + i__] = 0.f;
/* L80: */
}
ioff += lda;
/* L90: */
}
}
}
} else {
izero = 0;
}
/* Do for each value of NB in NBVAL */
i__3 = *nnb;
for (inb = 1; inb <= i__3; ++inb) {
nb = nbval[inb];
xlaenv_(&c__1, &nb);
/* Compute the L*D*L' or U*D*U' factorization of the */
/* matrix. */
slacpy_(uplo, &n, &n, &a[1], &lda, &afac[1], &lda);
lwork = max(2,nb) * lda;
s_copy(srnamc_1.srnamt, "SSYTRF", (ftnlen)32, (ftnlen)6);
ssytrf_(uplo, &n, &afac[1], &lda, &iwork[1], &ainv[1], &
lwork, &info);
/* Adjust the expected value of INFO to account for */
/* pivoting. */
k = izero;
if (k > 0) {
L100:
if (iwork[k] < 0) {
if (iwork[k] != -k) {
k = -iwork[k];
goto L100;
}
} else if (iwork[k] != k) {
k = iwork[k];
goto L100;
}
}
/* Check error code from SSYTRF. */
if (info != k) {
alaerh_(path, "SSYTRF", &info, &k, uplo, &n, &n, &
c_n1, &c_n1, &nb, &imat, &nfail, &nerrs, nout);
}
if (info != 0) {
trfcon = TRUE_;
} else {
trfcon = FALSE_;
}
/* + TEST 1 */
/* Reconstruct matrix from factors and compute residual. */
ssyt01_(uplo, &n, &a[1], &lda, &afac[1], &lda, &iwork[1],
&ainv[1], &lda, &rwork[1], result);
nt = 1;
/* + TEST 2 */
/* Form the inverse and compute the residual. */
if (inb == 1 && ! trfcon) {
slacpy_(uplo, &n, &n, &afac[1], &lda, &ainv[1], &lda);
s_copy(srnamc_1.srnamt, "SSYTRI", (ftnlen)32, (ftnlen)
6);
ssytri_(uplo, &n, &ainv[1], &lda, &iwork[1], &work[1],
&info);
/* Check error code from SSYTRI. */
if (info != 0) {
alaerh_(path, "SSYTRI", &info, &c_n1, uplo, &n, &
n, &c_n1, &c_n1, &c_n1, &imat, &nfail, &
nerrs, nout);
}
spot03_(uplo, &n, &a[1], &lda, &ainv[1], &lda, &work[
1], &lda, &rwork[1], &rcondc, &result[1]);
nt = 2;
}
/* 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) {
alahd_(nout, path);
}
io___39.ciunit = *nout;
s_wsfe(&io___39);
do_fio(&c__1, uplo, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer))
;
do_fio(&c__1, (char *)&nb, (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;
}
/* L110: */
}
nrun += nt;
/* Skip the other tests if this is not the first block */
/* size. */
if (inb > 1) {
goto L150;
}
/* Do only the condition estimate if INFO is not 0. */
if (trfcon) {
rcondc = 0.f;
goto L140;
}
i__4 = *nns;
for (irhs = 1; irhs <= i__4; ++irhs) {
nrhs = nsval[irhs];
/* + TEST 3 */
/* Solve and compute residual for A * X = B. */
s_copy(srnamc_1.srnamt, "SLARHS", (ftnlen)32, (ftnlen)
6);
slarhs_(path, xtype, uplo, " ", &n, &n, &kl, &ku, &
nrhs, &a[1], &lda, &xact[1], &lda, &b[1], &
lda, iseed, &info);
slacpy_("Full", &n, &nrhs, &b[1], &lda, &x[1], &lda);
s_copy(srnamc_1.srnamt, "SSYTRS", (ftnlen)32, (ftnlen)
6);
ssytrs_(uplo, &n, &nrhs, &afac[1], &lda, &iwork[1], &
x[1], &lda, &info);
/* Check error code from SSYTRS. */
if (info != 0) {
alaerh_(path, "SSYTRS", &info, &c__0, uplo, &n, &
n, &c_n1, &c_n1, &nrhs, &imat, &nfail, &
nerrs, nout);
}
slacpy_("Full", &n, &nrhs, &b[1], &lda, &work[1], &
lda);
spot02_(uplo, &n, &nrhs, &a[1], &lda, &x[1], &lda, &
work[1], &lda, &rwork[1], &result[2]);
/* + TEST 4 */
/* Check solution from generated exact solution. */
sget04_(&n, &nrhs, &x[1], &lda, &xact[1], &lda, &
rcondc, &result[3]);
/* + TESTS 5, 6, and 7 */
/* Use iterative refinement to improve the solution. */
s_copy(srnamc_1.srnamt, "SSYRFS", (ftnlen)32, (ftnlen)
6);
ssyrfs_(uplo, &n, &nrhs, &a[1], &lda, &afac[1], &lda,
&iwork[1], &b[1], &lda, &x[1], &lda, &rwork[1]
, &rwork[nrhs + 1], &work[1], &iwork[n + 1], &
info);
/* Check error code from SSYRFS. */
if (info != 0) {
alaerh_(path, "SSYRFS", &info, &c__0, uplo, &n, &
n, &c_n1, &c_n1, &nrhs, &imat, &nfail, &
nerrs, nout);
}
sget04_(&n, &nrhs, &x[1], &lda, &xact[1], &lda, &
rcondc, &result[4]);
spot05_(uplo, &n, &nrhs, &a[1], &lda, &b[1], &lda, &x[
1], &lda, &xact[1], &lda, &rwork[1], &rwork[
nrhs + 1], &result[5]);
/* Print information about the tests that did not pass */
/* the threshold. */
for (k = 3; k <= 7; ++k) {
if (result[k - 1] >= *thresh) {
if (nfail == 0 && nerrs == 0) {
alahd_(nout, path);
}
io___42.ciunit = *nout;
s_wsfe(&io___42);
do_fio(&c__1, uplo, (ftnlen)1);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&nrhs, (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;
}
/* L120: */
}
nrun += 5;
/* L130: */
}
/* + TEST 8 */
/* Get an estimate of RCOND = 1/CNDNUM. */
L140:
anorm = slansy_("1", uplo, &n, &a[1], &lda, &rwork[1]);
s_copy(srnamc_1.srnamt, "SSYCON", (ftnlen)32, (ftnlen)6);
ssycon_(uplo, &n, &afac[1], &lda, &iwork[1], &anorm, &
rcond, &work[1], &iwork[n + 1], &info);
/* Check error code from SSYCON. */
if (info != 0) {
alaerh_(path, "SSYCON", &info, &c__0, uplo, &n, &n, &
c_n1, &c_n1, &c_n1, &imat, &nfail, &nerrs,
nout);
}
result[7] = sget06_(&rcond, &rcondc);
/* Print information about the tests that did not pass */
/* the threshold. */
if (result[7] >= *thresh) {
if (nfail == 0 && nerrs == 0) {
alahd_(nout, path);
}
io___44.ciunit = *nout;
s_wsfe(&io___44);
do_fio(&c__1, uplo, (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__8, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&result[7], (ftnlen)sizeof(real)
);
e_wsfe();
++nfail;
}
++nrun;
L150:
;
}
L160:
;
}
L170:
;
}
/* L180: */
}
/* Print a summary of the results. */
alasum_(path, nout, &nfail, &nrun, &nerrs);
return 0;
/* End of SCHKSY */
} /* schksy_ */
/* Subroutine */ int sla_syrfsx_extended__(integer *prec_type__, char *uplo,
integer *n, integer *nrhs, real *a, integer *lda, real *af, integer *
ldaf, integer *ipiv, logical *colequ, real *c__, real *b, integer *
ldb, real *y, integer *ldy, real *berr_out__, integer *n_norms__,
real *err_bnds_norm__, real *err_bnds_comp__, real *res, real *ayb,
real *dy, real *y_tail__, real *rcond, integer *ithresh, real *
rthresh, real *dz_ub__, logical *ignore_cwise__, integer *info,
ftnlen uplo_len)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, y_dim1,
y_offset, err_bnds_norm_dim1, err_bnds_norm_offset,
err_bnds_comp_dim1, err_bnds_comp_offset, i__1, i__2, i__3;
real r__1, r__2;
/* Local variables */
real dxratmax, dzratmax;
integer i__, j;
logical incr_prec__;
extern /* Subroutine */ int sla_syamv__(integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *);
real prev_dz_z__, yk, final_dx_x__, final_dz_z__;
extern /* Subroutine */ int sla_wwaddw__(integer *, real *, real *, real *
);
real prevnormdx;
integer cnt;
real dyk, eps, incr_thresh__, dx_x__, dz_z__, ymin;
extern /* Subroutine */ int sla_lin_berr__(integer *, integer *, integer *
, real *, real *, real *);
integer y_prec_state__, uplo2;
extern /* Subroutine */ int blas_ssymv_x__(integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *,
integer *);
extern logical lsame_(char *, char *);
real dxrat, dzrat;
extern /* Subroutine */ int blas_ssymv2_x__(integer *, integer *, real *,
real *, integer *, real *, real *, integer *, real *, real *,
integer *, integer *), scopy_(integer *, real *, integer *, real *
, integer *);
real normx, normy;
extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *,
real *, integer *), ssymv_(char *, integer *, real *, real *,
integer *, real *, integer *, real *, real *, integer *);
extern doublereal slamch_(char *);
real normdx;
extern /* Subroutine */ int ssytrs_(char *, integer *, integer *, real *,
integer *, integer *, real *, integer *, integer *);
real hugeval;
extern integer ilauplo_(char *);
integer x_state__, z_state__;
/* -- LAPACK routine (version 3.2.1) -- */
/* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/* -- Jason Riedy of Univ. of California Berkeley. -- */
/* -- April 2009 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley and NAG Ltd. -- */
/* .. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLA_SYRFSX_EXTENDED improves the computed solution to a system of */
/* linear equations by performing extra-precise iterative refinement */
/* and provides error bounds and backward error estimates for the solution. */
/* This subroutine is called by SSYRFSX to perform iterative refinement. */
/* In addition to normwise error bound, the code provides maximum */
/* componentwise error bound if possible. See comments for ERR_BNDS_NORM */
/* and ERR_BNDS_COMP for details of the error bounds. Note that this */
/* subroutine is only resonsible for setting the second fields of */
/* ERR_BNDS_NORM and ERR_BNDS_COMP. */
/* Arguments */
/* ========= */
/* PREC_TYPE (input) INTEGER */
/* Specifies the intermediate precision to be used in refinement. */
/* The value is defined by ILAPREC(P) where P is a CHARACTER and */
/* P = 'S': Single */
/* = 'D': Double */
/* = 'I': Indigenous */
/* = 'X', 'E': Extra */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right-hand-sides, i.e., the number of columns of the */
/* matrix B. */
/* A (input) REAL array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) REAL array, dimension (LDAF,N) */
/* The block diagonal matrix D and the multipliers used to */
/* obtain the factor U or L as computed by SSYTRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by SSYTRF. */
/* COLEQU (input) LOGICAL */
/* If .TRUE. then column equilibration was done to A before calling */
/* this routine. This is needed to compute the solution and error */
/* bounds correctly. */
/* C (input) REAL array, dimension (N) */
/* The column scale factors for A. If COLEQU = .FALSE., C */
/* is not accessed. If C is input, each element of C should be a power */
/* of the radix to ensure a reliable solution and error estimates. */
/* Scaling by powers of the radix does not cause rounding errors unless */
/* the result underflows or overflows. Rounding errors during scaling */
/* lead to refining with a matrix that is not equivalent to the */
/* input matrix, producing error estimates that may not be */
/* reliable. */
/* B (input) REAL array, dimension (LDB,NRHS) */
/* The right-hand-side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* Y (input/output) REAL array, dimension (LDY,NRHS) */
/* On entry, the solution matrix X, as computed by SSYTRS. */
/* On exit, the improved solution matrix Y. */
/* LDY (input) INTEGER */
/* The leading dimension of the array Y. LDY >= max(1,N). */
/* BERR_OUT (output) REAL array, dimension (NRHS) */
/* On exit, BERR_OUT(j) contains the componentwise relative backward */
/* error for right-hand-side j from the formula */
/* max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. This is computed by SLA_LIN_BERR. */
/* N_NORMS (input) INTEGER */
/* Determines which error bounds to return (see ERR_BNDS_NORM */
/* and ERR_BNDS_COMP). */
/* If N_NORMS >= 1 return normwise error bounds. */
/* If N_NORMS >= 2 return componentwise error bounds. */
/* ERR_BNDS_NORM (input/output) REAL array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* normwise relative error, which is defined as follows: */
/* Normwise relative error in the ith solution vector: */
/* max_j (abs(XTRUE(j,i) - X(j,i))) */
/* ------------------------------ */
/* max_j abs(X(j,i)) */
/* The array is indexed by the type of error information as described */
/* below. There currently are up to three pieces of information */
/* returned. */
/* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_NORM(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * slamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * slamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated normwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * slamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*A, where S scales each row by a power of the */
/* radix so all absolute row sums of Z are approximately 1. */
/* This subroutine is only responsible for setting the second field */
/* above. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* ERR_BNDS_COMP (input/output) REAL array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* componentwise relative error, which is defined as follows: */
/* Componentwise relative error in the ith solution vector: */
/* abs(XTRUE(j,i) - X(j,i)) */
/* max_j ---------------------- */
/* abs(X(j,i)) */
/* The array is indexed by the right-hand side i (on which the */
/* componentwise relative error depends), and the type of error */
/* information as described below. There currently are up to three */
/* pieces of information returned for each right-hand side. If */
/* componentwise accuracy is not requested (PARAMS(3) = 0.0), then */
/* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most */
/* the first (:,N_ERR_BNDS) entries are returned. */
/* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_COMP(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * slamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * slamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated componentwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * slamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*(A*diag(x)), where x is the solution for the */
/* current right-hand side and S scales each row of */
/* A*diag(x) by a power of the radix so all absolute row */
/* sums of Z are approximately 1. */
/* This subroutine is only responsible for setting the second field */
/* above. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* RES (input) REAL array, dimension (N) */
/* Workspace to hold the intermediate residual. */
/* AYB (input) REAL array, dimension (N) */
/* Workspace. This can be the same workspace passed for Y_TAIL. */
/* DY (input) REAL array, dimension (N) */
/* Workspace to hold the intermediate solution. */
/* Y_TAIL (input) REAL array, dimension (N) */
/* Workspace to hold the trailing bits of the intermediate solution. */
/* RCOND (input) REAL */
/* Reciprocal scaled condition number. This is an estimate of the */
/* reciprocal Skeel condition number of the matrix A after */
/* equilibration (if done). If this is less than the machine */
/* precision (in particular, if it is zero), the matrix is singular */
/* to working precision. Note that the error may still be small even */
/* if this number is very small and the matrix appears ill- */
/* conditioned. */
/* ITHRESH (input) INTEGER */
/* The maximum number of residual computations allowed for */
/* refinement. The default is 10. For 'aggressive' set to 100 to */
/* permit convergence using approximate factorizations or */
/* factorizations other than LU. If the factorization uses a */
/* technique other than Gaussian elimination, the guarantees in */
/* ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. */
/* RTHRESH (input) REAL */
/* Determines when to stop refinement if the error estimate stops */
/* decreasing. Refinement will stop when the next solution no longer */
/* satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is */
/* the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The */
/* default value is 0.5. For 'aggressive' set to 0.9 to permit */
/* convergence on extremely ill-conditioned matrices. See LAWN 165 */
/* for more details. */
/* DZ_UB (input) REAL */
/* Determines when to start considering componentwise convergence. */
/* Componentwise convergence is only considered after each component */
/* of the solution Y is stable, which we definte as the relative */
/* change in each component being less than DZ_UB. The default value */
/* is 0.25, requiring the first bit to be stable. See LAWN 165 for */
/* more details. */
/* IGNORE_CWISE (input) LOGICAL */
/* If .TRUE. then ignore componentwise convergence. Default value */
/* is .FALSE.. */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* < 0: if INFO = -i, the ith argument to SSYTRS had an illegal */
/* value */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Parameters .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
err_bnds_comp_dim1 = *nrhs;
err_bnds_comp_offset = 1 + err_bnds_comp_dim1;
err_bnds_comp__ -= err_bnds_comp_offset;
err_bnds_norm_dim1 = *nrhs;
err_bnds_norm_offset = 1 + err_bnds_norm_dim1;
err_bnds_norm__ -= err_bnds_norm_offset;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
--c__;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
y_dim1 = *ldy;
y_offset = 1 + y_dim1;
y -= y_offset;
--berr_out__;
--res;
--ayb;
--dy;
--y_tail__;
/* Function Body */
if (*info != 0) {
return 0;
}
eps = slamch_("Epsilon");
hugeval = slamch_("Overflow");
/* Force HUGEVAL to Inf */
hugeval *= hugeval;
/* Using HUGEVAL may lead to spurious underflows. */
incr_thresh__ = (real) (*n) * eps;
if (lsame_(uplo, "L")) {
uplo2 = ilauplo_("L");
} else {
uplo2 = ilauplo_("U");
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
y_prec_state__ = 1;
if (y_prec_state__ == 2) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
y_tail__[i__] = 0.f;
}
}
dxrat = 0.f;
dxratmax = 0.f;
dzrat = 0.f;
dzratmax = 0.f;
final_dx_x__ = hugeval;
final_dz_z__ = hugeval;
prevnormdx = hugeval;
prev_dz_z__ = hugeval;
dz_z__ = hugeval;
dx_x__ = hugeval;
x_state__ = 1;
z_state__ = 0;
incr_prec__ = FALSE_;
i__2 = *ithresh;
for (cnt = 1; cnt <= i__2; ++cnt) {
/* Compute residual RES = B_s - op(A_s) * Y, */
/* op(A) = A, A**T, or A**H depending on TRANS (and type). */
scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
if (y_prec_state__ == 0) {
ssymv_(uplo, n, &c_b9, &a[a_offset], lda, &y[j * y_dim1 + 1],
&c__1, &c_b11, &res[1], &c__1);
} else if (y_prec_state__ == 1) {
blas_ssymv_x__(&uplo2, n, &c_b9, &a[a_offset], lda, &y[j *
y_dim1 + 1], &c__1, &c_b11, &res[1], &c__1,
prec_type__);
} else {
blas_ssymv2_x__(&uplo2, n, &c_b9, &a[a_offset], lda, &y[j *
y_dim1 + 1], &y_tail__[1], &c__1, &c_b11, &res[1], &
c__1, prec_type__);
}
/* XXX: RES is no longer needed. */
scopy_(n, &res[1], &c__1, &dy[1], &c__1);
ssytrs_(uplo, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &dy[1], n,
info);
/* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. */
normx = 0.f;
normy = 0.f;
normdx = 0.f;
dz_z__ = 0.f;
ymin = hugeval;
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
yk = (r__1 = y[i__ + j * y_dim1], dabs(r__1));
dyk = (r__1 = dy[i__], dabs(r__1));
if (yk != 0.f) {
/* Computing MAX */
r__1 = dz_z__, r__2 = dyk / yk;
dz_z__ = dmax(r__1,r__2);
} else if (dyk != 0.f) {
dz_z__ = hugeval;
}
ymin = dmin(ymin,yk);
normy = dmax(normy,yk);
if (*colequ) {
/* Computing MAX */
r__1 = normx, r__2 = yk * c__[i__];
normx = dmax(r__1,r__2);
/* Computing MAX */
r__1 = normdx, r__2 = dyk * c__[i__];
normdx = dmax(r__1,r__2);
} else {
normx = normy;
normdx = dmax(normdx,dyk);
}
}
if (normx != 0.f) {
dx_x__ = normdx / normx;
} else if (normdx == 0.f) {
dx_x__ = 0.f;
} else {
dx_x__ = hugeval;
}
dxrat = normdx / prevnormdx;
dzrat = dz_z__ / prev_dz_z__;
/* Check termination criteria. */
if (ymin * *rcond < incr_thresh__ * normy && y_prec_state__ < 2) {
incr_prec__ = TRUE_;
}
if (x_state__ == 3 && dxrat <= *rthresh) {
x_state__ = 1;
}
if (x_state__ == 1) {
if (dx_x__ <= eps) {
x_state__ = 2;
} else if (dxrat > *rthresh) {
if (y_prec_state__ != 2) {
incr_prec__ = TRUE_;
} else {
x_state__ = 3;
}
} else {
if (dxrat > dxratmax) {
dxratmax = dxrat;
}
}
if (x_state__ > 1) {
final_dx_x__ = dx_x__;
}
}
if (z_state__ == 0 && dz_z__ <= *dz_ub__) {
z_state__ = 1;
}
if (z_state__ == 3 && dzrat <= *rthresh) {
z_state__ = 1;
}
if (z_state__ == 1) {
if (dz_z__ <= eps) {
z_state__ = 2;
} else if (dz_z__ > *dz_ub__) {
z_state__ = 0;
dzratmax = 0.f;
final_dz_z__ = hugeval;
} else if (dzrat > *rthresh) {
if (y_prec_state__ != 2) {
incr_prec__ = TRUE_;
} else {
z_state__ = 3;
}
} else {
if (dzrat > dzratmax) {
dzratmax = dzrat;
}
}
if (z_state__ > 1) {
final_dz_z__ = dz_z__;
}
}
if (x_state__ != 1 && (*ignore_cwise__ || z_state__ != 1)) {
goto L666;
}
if (incr_prec__) {
incr_prec__ = FALSE_;
++y_prec_state__;
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
y_tail__[i__] = 0.f;
}
}
prevnormdx = normdx;
prev_dz_z__ = dz_z__;
/* Update soluton. */
if (y_prec_state__ < 2) {
saxpy_(n, &c_b11, &dy[1], &c__1, &y[j * y_dim1 + 1], &c__1);
} else {
sla_wwaddw__(n, &y[j * y_dim1 + 1], &y_tail__[1], &dy[1]);
}
}
/* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT. */
L666:
/* Set final_* when cnt hits ithresh. */
if (x_state__ == 1) {
final_dx_x__ = dx_x__;
}
if (z_state__ == 1) {
final_dz_z__ = dz_z__;
}
/* Compute error bounds. */
if (*n_norms__ >= 1) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = final_dx_x__ / (
1 - dxratmax);
}
if (*n_norms__ >= 2) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = final_dz_z__ / (
1 - dzratmax);
}
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. */
/* Compute residual RES = B_s - op(A_s) * Y, */
/* op(A) = A, A**T, or A**H depending on TRANS (and type). */
scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
ssymv_(uplo, n, &c_b9, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1, &
c_b11, &res[1], &c__1);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
ayb[i__] = (r__1 = b[i__ + j * b_dim1], dabs(r__1));
}
/* Compute abs(op(A_s))*abs(Y) + abs(B_s). */
sla_syamv__(&uplo2, n, &c_b11, &a[a_offset], lda, &y[j * y_dim1 + 1],
&c__1, &c_b11, &ayb[1], &c__1);
sla_lin_berr__(n, n, &c__1, &res[1], &ayb[1], &berr_out__[j]);
/* End of loop for each RHS. */
}
return 0;
} /* sla_syrfsx_extended__ */
/* ===================================================================== */
real sla_syrcond_(char *uplo, integer *n, real *a, integer *lda, real *af, integer *ldaf, integer *ipiv, integer *cmode, real *c__, integer * info, real *work, integer *iwork)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2;
real ret_val, r__1;
/* Local variables */
integer i__, j;
logical up;
real tmp;
integer kase;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */
int slacn2_(integer *, real *, real *, integer *, real *, integer *, integer *);
extern real slamch_(char *);
extern /* Subroutine */
int xerbla_(char *, integer *);
real ainvnm;
char normin[1];
real smlnum;
extern /* Subroutine */
int ssytrs_(char *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *);
/* -- LAPACK computational routine (version 3.4.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* September 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments */
/* .. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1;
af -= af_offset;
--ipiv;
--c__;
--work;
--iwork;
/* Function Body */
ret_val = 0.f;
*info = 0;
if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*n))
{
*info = -4;
}
else if (*ldaf < max(1,*n))
{
*info = -6;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("SLA_SYRCOND", &i__1);
return ret_val;
}
if (*n == 0)
{
ret_val = 1.f;
return ret_val;
}
up = FALSE_;
if (lsame_(uplo, "U"))
{
up = TRUE_;
}
/* Compute the equilibration matrix R such that */
/* inv(R)*A*C has unit 1-norm. */
if (up)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.f;
if (*cmode == 1)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[j + i__ * a_dim1] * c__[j], f2c_abs(r__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[i__ + j * a_dim1] * c__[j], f2c_abs(r__1));
}
}
else if (*cmode == 0)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[j + i__ * a_dim1], f2c_abs(r__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[i__ + j * a_dim1], f2c_abs(r__1));
}
}
else
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[j + i__ * a_dim1] / c__[j], f2c_abs(r__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[i__ + j * a_dim1] / c__[j], f2c_abs(r__1));
}
}
work[(*n << 1) + i__] = tmp;
}
}
else
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
tmp = 0.f;
if (*cmode == 1)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[i__ + j * a_dim1] * c__[j], f2c_abs(r__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[j + i__ * a_dim1] * c__[j], f2c_abs(r__1));
}
}
else if (*cmode == 0)
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[i__ + j * a_dim1], f2c_abs(r__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[j + i__ * a_dim1], f2c_abs(r__1));
}
}
else
{
i__2 = i__;
for (j = 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[i__ + j * a_dim1] / c__[j], f2c_abs(r__1));
}
i__2 = *n;
for (j = i__ + 1;
j <= i__2;
++j)
{
tmp += (r__1 = a[j + i__ * a_dim1] / c__[j], f2c_abs(r__1));
}
}
work[(*n << 1) + i__] = tmp;
}
}
/* Estimate the norm of inv(op(A)). */
smlnum = slamch_("Safe minimum");
ainvnm = 0.f;
*(unsigned char *)normin = 'N';
kase = 0;
L10:
slacn2_(n, &work[*n + 1], &work[1], &iwork[1], &ainvnm, &kase, isave);
if (kase != 0)
{
if (kase == 2)
{
/* Multiply by R. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] *= work[(*n << 1) + i__];
}
if (up)
{
ssytrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
else
{
ssytrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
/* Multiply by inv(C). */
if (*cmode == 1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] /= c__[i__];
}
}
else if (*cmode == -1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] *= c__[i__];
}
}
}
else
{
/* Multiply by inv(C**T). */
if (*cmode == 1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] /= c__[i__];
}
}
else if (*cmode == -1)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] *= c__[i__];
}
}
if (up)
{
ssytrs_("U", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
else
{
ssytrs_("L", n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[ 1], n, info);
}
/* Multiply by R. */
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
work[i__] *= work[(*n << 1) + i__];
}
}
goto L10;
}
/* Compute the estimate of the reciprocal condition number. */
if (ainvnm != 0.f)
{
ret_val = 1.f / ainvnm;
}
return ret_val;
}
int ssysv_(char *uplo, int *n, int *nrhs, float *a,
int *lda, int *ipiv, float *b, int *ldb, float *work,
int *lwork, int *info)
{
/* System generated locals */
int a_dim1, a_offset, b_dim1, b_offset, i__1;
/* Local variables */
int nb;
extern int lsame_(char *, char *);
extern int xerbla_(char *, int *);
extern int ilaenv_(int *, char *, char *, int *, int *,
int *, int *);
int lwkopt;
int lquery;
extern int ssytrf_(char *, int *, float *, int *,
int *, float *, int *, int *), ssytrs_(char *,
int *, int *, float *, int *, int *, float *,
int *, int *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSYSV computes the solution to a float system of linear equations */
/* A * X = B, */
/* where A is an N-by-N symmetric matrix and X and B are N-by-NRHS */
/* matrices. */
/* The diagonal pivoting method is used to factor A as */
/* A = U * D * U**T, if UPLO = 'U', or */
/* A = L * D * L**T, if UPLO = 'L', */
/* where U (or L) is a product of permutation and unit upper (lower) */
/* triangular matrices, and D is symmetric and block diagonal with */
/* 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then */
/* used to solve the system of equations A * X = B. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The number of linear equations, i.e., the order of the */
/* matrix A. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrix B. NRHS >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the symmetric matrix A. If UPLO = 'U', the leading */
/* N-by-N upper triangular part of A contains the upper */
/* triangular part of the matrix A, and the strictly lower */
/* triangular part of A is not referenced. If UPLO = 'L', the */
/* leading N-by-N lower triangular part of A contains the lower */
/* triangular part of the matrix A, and the strictly upper */
/* triangular part of A is not referenced. */
/* On exit, if INFO = 0, the block diagonal matrix D and the */
/* multipliers used to obtain the factor U or L from the */
/* factorization A = U*D*U**T or A = L*D*L**T as computed by */
/* SSYTRF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,N). */
/* IPIV (output) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D, as */
/* determined by SSYTRF. If IPIV(k) > 0, then rows and columns */
/* k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1 */
/* diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, */
/* then rows and columns k-1 and -IPIV(k) were interchanged and */
/* D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and */
/* IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and */
/* -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 */
/* diagonal block. */
/* B (input/output) REAL array, dimension (LDB,NRHS) */
/* On entry, the N-by-NRHS right hand side matrix B. */
/* On exit, if INFO = 0, the N-by-NRHS solution matrix X. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= MAX(1,N). */
/* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The length of WORK. LWORK >= 1, and for best performance */
/* LWORK >= MAX(1,N*NB), where NB is the optimal blocksize for */
/* SSYTRF. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, D(i,i) is exactly zero. The factorization */
/* has been completed, but the block diagonal matrix D is */
/* exactly singular, so the solution could not be computed. */
/* ===================================================================== */
/* .. 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;
--work;
/* Function Body */
*info = 0;
lquery = *lwork == -1;
if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < MAX(1,*n)) {
*info = -5;
} else if (*ldb < MAX(1,*n)) {
*info = -8;
} else if (*lwork < 1 && ! lquery) {
*info = -10;
}
if (*info == 0) {
if (*n == 0) {
lwkopt = 1;
} else {
nb = ilaenv_(&c__1, "SSYTRF", uplo, n, &c_n1, &c_n1, &c_n1);
lwkopt = *n * nb;
}
work[1] = (float) lwkopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSYSV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Compute the factorization A = U*D*U' or A = L*D*L'. */
ssytrf_(uplo, n, &a[a_offset], lda, &ipiv[1], &work[1], lwork, info);
if (*info == 0) {
/* Solve the system A*X = B, overwriting B with X. */
ssytrs_(uplo, n, nrhs, &a[a_offset], lda, &ipiv[1], &b[b_offset], ldb,
info);
}
work[1] = (float) lwkopt;
return 0;
/* End of SSYSV */
} /* ssysv_ */
