LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches

◆ 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

Download ZSYSVX + dependencies [TGZ] [ZIP] [TXT]

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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

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*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zsycon(uplo, n, a, lda, ipiv, anorm, rcond, work, info)
ZSYCON
Definition zsycon.f:125
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 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
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
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 dlamch(cmach)
DLAMCH
Definition dlamch.f:69
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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
Here is the call graph for this function:
Here is the caller graph for this function: