LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zsysvx()

 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.```

Definition at line 282 of file zsysvx.f.

285 *
286 * -- LAPACK driver routine --
287 * -- LAPACK is a software package provided by Univ. of Tennessee, --
288 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
289 *
290 * .. Scalar Arguments ..
291  CHARACTER FACT, UPLO
292  INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
293  DOUBLE PRECISION RCOND
294 * ..
295 * .. Array Arguments ..
296  INTEGER IPIV( * )
297  DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
298  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
299  \$ WORK( * ), X( LDX, * )
300 * ..
301 *
302 * =====================================================================
303 *
304 * .. Parameters ..
305  DOUBLE PRECISION ZERO
306  parameter( zero = 0.0d+0 )
307 * ..
308 * .. Local Scalars ..
309  LOGICAL LQUERY, NOFACT
310  INTEGER LWKOPT, NB
311  DOUBLE PRECISION ANORM
312 * ..
313 * .. External Functions ..
314  LOGICAL LSAME
315  INTEGER ILAENV
316  DOUBLE PRECISION DLAMCH, ZLANSY
317  EXTERNAL lsame, ilaenv, dlamch, zlansy
318 * ..
319 * .. External Subroutines ..
320  EXTERNAL xerbla, zlacpy, zsycon, zsyrfs, zsytrf, zsytrs
321 * ..
322 * .. Intrinsic Functions ..
323  INTRINSIC max
324 * ..
325 * .. Executable Statements ..
326 *
327 * Test the input parameters.
328 *
329  info = 0
330  nofact = lsame( fact, 'N' )
331  lquery = ( lwork.EQ.-1 )
332  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
333  info = -1
334  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
335  \$ THEN
336  info = -2
337  ELSE IF( n.LT.0 ) THEN
338  info = -3
339  ELSE IF( nrhs.LT.0 ) THEN
340  info = -4
341  ELSE IF( lda.LT.max( 1, n ) ) THEN
342  info = -6
343  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
344  info = -8
345  ELSE IF( ldb.LT.max( 1, n ) ) THEN
346  info = -11
347  ELSE IF( ldx.LT.max( 1, n ) ) THEN
348  info = -13
349  ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
350  info = -18
351  END IF
352 *
353  IF( info.EQ.0 ) THEN
354  lwkopt = max( 1, 2*n )
355  IF( nofact ) THEN
356  nb = ilaenv( 1, 'ZSYTRF', uplo, n, -1, -1, -1 )
357  lwkopt = max( lwkopt, n*nb )
358  END IF
359  work( 1 ) = lwkopt
360  END IF
361 *
362  IF( info.NE.0 ) THEN
363  CALL xerbla( 'ZSYSVX', -info )
364  RETURN
365  ELSE IF( lquery ) THEN
366  RETURN
367  END IF
368 *
369  IF( nofact ) THEN
370 *
371 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
372 *
373  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
374  CALL zsytrf( uplo, n, af, ldaf, ipiv, work, lwork, info )
375 *
376 * Return if INFO is non-zero.
377 *
378  IF( info.GT.0 )THEN
379  rcond = zero
380  RETURN
381  END IF
382  END IF
383 *
384 * Compute the norm of the matrix A.
385 *
386  anorm = zlansy( 'I', uplo, n, a, lda, rwork )
387 *
388 * Compute the reciprocal of the condition number of A.
389 *
390  CALL zsycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work, info )
391 *
392 * Compute the solution vectors X.
393 *
394  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
395  CALL zsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
396 *
397 * Use iterative refinement to improve the computed solutions and
398 * compute error bounds and backward error estimates for them.
399 *
400  CALL zsyrfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
401  \$ ldx, ferr, berr, work, rwork, info )
402 *
403 * Set INFO = N+1 if the matrix is singular to working precision.
404 *
405  IF( rcond.LT.dlamch( 'Epsilon' ) )
406  \$ info = n + 1
407 *
408  work( 1 ) = lwkopt
409 *
410  RETURN
411 *
412 * End of ZSYSVX
413 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
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
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
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,...
Definition: zlansy.f:123
subroutine zsyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZSYRFS
Definition: zsyrfs.f:192
subroutine zsycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
ZSYCON
Definition: zsycon.f:125
subroutine zsytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
ZSYTRF
Definition: zsytrf.f:182
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:120
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