 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dsysvx()

 subroutine dsysvx ( character FACT, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO )

DSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Purpose:
``` DSYSVX 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.```
Parameters
 [in] FACT ``` FACT is 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.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.``` [in] A ``` A is DOUBLE PRECISION 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] AF ``` AF is DOUBLE PRECISION 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 DSYTRF. 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.``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in,out] IPIV ``` IPIV is 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 DSYTRF. 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 DSYTRF.``` [in] B ``` B is DOUBLE PRECISION array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] X ``` X is DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] RCOND ``` RCOND is DOUBLE PRECISION 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.``` [out] FERR ``` FERR is DOUBLE PRECISION 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.``` [out] BERR ``` BERR is DOUBLE PRECISION 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).``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is 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 DSYTRF. 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.``` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] INFO ``` INFO is 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.```

Definition at line 281 of file dsysvx.f.

284 *
285 * -- LAPACK driver routine --
286 * -- LAPACK is a software package provided by Univ. of Tennessee, --
287 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
288 *
289 * .. Scalar Arguments ..
290  CHARACTER FACT, UPLO
291  INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
292  DOUBLE PRECISION RCOND
293 * ..
294 * .. Array Arguments ..
295  INTEGER IPIV( * ), IWORK( * )
296  DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
297  \$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
298 * ..
299 *
300 * =====================================================================
301 *
302 * .. Parameters ..
303  DOUBLE PRECISION ZERO
304  parameter( zero = 0.0d+0 )
305 * ..
306 * .. Local Scalars ..
307  LOGICAL LQUERY, NOFACT
308  INTEGER LWKOPT, NB
309  DOUBLE PRECISION ANORM
310 * ..
311 * .. External Functions ..
312  LOGICAL LSAME
313  INTEGER ILAENV
314  DOUBLE PRECISION DLAMCH, DLANSY
315  EXTERNAL lsame, ilaenv, dlamch, dlansy
316 * ..
317 * .. External Subroutines ..
318  EXTERNAL dlacpy, dsycon, dsyrfs, dsytrf, dsytrs, xerbla
319 * ..
320 * .. Intrinsic Functions ..
321  INTRINSIC max
322 * ..
323 * .. Executable Statements ..
324 *
325 * Test the input parameters.
326 *
327  info = 0
328  nofact = lsame( fact, 'N' )
329  lquery = ( lwork.EQ.-1 )
330  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
331  info = -1
332  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
333  \$ THEN
334  info = -2
335  ELSE IF( n.LT.0 ) THEN
336  info = -3
337  ELSE IF( nrhs.LT.0 ) THEN
338  info = -4
339  ELSE IF( lda.LT.max( 1, n ) ) THEN
340  info = -6
341  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
342  info = -8
343  ELSE IF( ldb.LT.max( 1, n ) ) THEN
344  info = -11
345  ELSE IF( ldx.LT.max( 1, n ) ) THEN
346  info = -13
347  ELSE IF( lwork.LT.max( 1, 3*n ) .AND. .NOT.lquery ) THEN
348  info = -18
349  END IF
350 *
351  IF( info.EQ.0 ) THEN
352  lwkopt = max( 1, 3*n )
353  IF( nofact ) THEN
354  nb = ilaenv( 1, 'DSYTRF', uplo, n, -1, -1, -1 )
355  lwkopt = max( lwkopt, n*nb )
356  END IF
357  work( 1 ) = lwkopt
358  END IF
359 *
360  IF( info.NE.0 ) THEN
361  CALL xerbla( 'DSYSVX', -info )
362  RETURN
363  ELSE IF( lquery ) THEN
364  RETURN
365  END IF
366 *
367  IF( nofact ) THEN
368 *
369 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
370 *
371  CALL dlacpy( uplo, n, n, a, lda, af, ldaf )
372  CALL dsytrf( uplo, n, af, ldaf, ipiv, work, lwork, info )
373 *
374 * Return if INFO is non-zero.
375 *
376  IF( info.GT.0 )THEN
377  rcond = zero
378  RETURN
379  END IF
380  END IF
381 *
382 * Compute the norm of the matrix A.
383 *
384  anorm = dlansy( 'I', uplo, n, a, lda, work )
385 *
386 * Compute the reciprocal of the condition number of A.
387 *
388  CALL dsycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work, iwork,
389  \$ info )
390 *
391 * Compute the solution vectors X.
392 *
393  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
394  CALL dsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
395 *
396 * Use iterative refinement to improve the computed solutions and
397 * compute error bounds and backward error estimates for them.
398 *
399  CALL dsyrfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
400  \$ ldx, ferr, berr, work, iwork, info )
401 *
402 * Set INFO = N+1 if the matrix is singular to working precision.
403 *
404  IF( rcond.LT.dlamch( 'Epsilon' ) )
405  \$ info = n + 1
406 *
407  work( 1 ) = lwkopt
408 *
409  RETURN
410 *
411 * End of DSYSVX
412 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
subroutine dsycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DSYCON
Definition: dsycon.f:130
subroutine dsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DSYTRS
Definition: dsytrs.f:120
subroutine dsyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DSYRFS
Definition: dsyrfs.f:191
subroutine dsytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
DSYTRF
Definition: dsytrf.f:182
Here is the call graph for this function:
Here is the caller graph for this function: