LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine zsysvx ( character FACT, character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO )

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

Purpose:
``` ZSYSVX uses the diagonal pivoting factorization to compute the
solution to a complex 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. 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 COMPLEX*16 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 COMPLEX*16 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 ZSYTRF. 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 ZSYTRF. 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 ZSYTRF.``` [in] B ``` B is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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(1,2*N), and for best performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where NB is the optimal blocksize for ZSYTRF. 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] RWORK ` RWORK is DOUBLE PRECISION 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.```
Date
April 2012

Definition at line 287 of file zsysvx.f.

287 *
288 * -- LAPACK driver routine (version 3.4.1) --
289 * -- LAPACK is a software package provided by Univ. of Tennessee, --
290 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
291 * April 2012
292 *
293 * .. Scalar Arguments ..
294  CHARACTER fact, uplo
295  INTEGER info, lda, ldaf, ldb, ldx, lwork, n, nrhs
296  DOUBLE PRECISION rcond
297 * ..
298 * .. Array Arguments ..
299  INTEGER ipiv( * )
300  DOUBLE PRECISION berr( * ), ferr( * ), rwork( * )
301  COMPLEX*16 a( lda, * ), af( ldaf, * ), b( ldb, * ),
302  \$ work( * ), x( ldx, * )
303 * ..
304 *
305 * =====================================================================
306 *
307 * .. Parameters ..
308  DOUBLE PRECISION zero
309  parameter ( zero = 0.0d+0 )
310 * ..
311 * .. Local Scalars ..
312  LOGICAL lquery, nofact
313  INTEGER lwkopt, nb
314  DOUBLE PRECISION anorm
315 * ..
316 * .. External Functions ..
317  LOGICAL lsame
318  INTEGER ilaenv
319  DOUBLE PRECISION dlamch, zlansy
320  EXTERNAL lsame, ilaenv, dlamch, zlansy
321 * ..
322 * .. External Subroutines ..
323  EXTERNAL xerbla, zlacpy, zsycon, zsyrfs, zsytrf, zsytrs
324 * ..
325 * .. Intrinsic Functions ..
326  INTRINSIC max
327 * ..
328 * .. Executable Statements ..
329 *
330 * Test the input parameters.
331 *
332  info = 0
333  nofact = lsame( fact, 'N' )
334  lquery = ( lwork.EQ.-1 )
335  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
336  info = -1
337  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
338  \$ THEN
339  info = -2
340  ELSE IF( n.LT.0 ) THEN
341  info = -3
342  ELSE IF( nrhs.LT.0 ) THEN
343  info = -4
344  ELSE IF( lda.LT.max( 1, n ) ) THEN
345  info = -6
346  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
347  info = -8
348  ELSE IF( ldb.LT.max( 1, n ) ) THEN
349  info = -11
350  ELSE IF( ldx.LT.max( 1, n ) ) THEN
351  info = -13
352  ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
353  info = -18
354  END IF
355 *
356  IF( info.EQ.0 ) THEN
357  lwkopt = max( 1, 2*n )
358  IF( nofact ) THEN
359  nb = ilaenv( 1, 'ZSYTRF', uplo, n, -1, -1, -1 )
360  lwkopt = max( lwkopt, n*nb )
361  END IF
362  work( 1 ) = lwkopt
363  END IF
364 *
365  IF( info.NE.0 ) THEN
366  CALL xerbla( 'ZSYSVX', -info )
367  RETURN
368  ELSE IF( lquery ) THEN
369  RETURN
370  END IF
371 *
372  IF( nofact ) THEN
373 *
374 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
375 *
376  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
377  CALL zsytrf( uplo, n, af, ldaf, ipiv, work, lwork, info )
378 *
379 * Return if INFO is non-zero.
380 *
381  IF( info.GT.0 )THEN
382  rcond = zero
383  RETURN
384  END IF
385  END IF
386 *
387 * Compute the norm of the matrix A.
388 *
389  anorm = zlansy( 'I', uplo, n, a, lda, rwork )
390 *
391 * Compute the reciprocal of the condition number of A.
392 *
393  CALL zsycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work, info )
394 *
395 * Compute the solution vectors X.
396 *
397  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
398  CALL zsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
399 *
400 * Use iterative refinement to improve the computed solutions and
401 * compute error bounds and backward error estimates for them.
402 *
403  CALL zsyrfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
404  \$ ldx, ferr, berr, work, rwork, info )
405 *
406 * Set INFO = N+1 if the matrix is singular to working precision.
407 *
408  IF( rcond.LT.dlamch( 'Epsilon' ) )
409  \$ info = n + 1
410 *
411  work( 1 ) = lwkopt
412 *
413  RETURN
414 *
415 * End of ZSYSVX
416 *
subroutine zsycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
ZSYCON
Definition: zsycon.f:127
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zsytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
ZSYTRF
Definition: zsytrf.f:184
double precision function zlansy(NORM, UPLO, N, A, LDA, WORK)
ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.
Definition: zlansy.f:125
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
subroutine zsyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZSYRFS
Definition: zsyrfs.f:194
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:122
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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