LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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
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