LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ csysvx()

subroutine csysvx ( character  FACT,
character  UPLO,
integer  N,
integer  NRHS,
complex, dimension( lda, * )  A,
integer  LDA,
complex, dimension( ldaf, * )  AF,
integer  LDAF,
integer, dimension( * )  IPIV,
complex, dimension( ldb, * )  B,
integer  LDB,
complex, dimension( ldx, * )  X,
integer  LDX,
real  RCOND,
real, dimension( * )  FERR,
real, dimension( * )  BERR,
complex, dimension( * )  WORK,
integer  LWORK,
real, dimension( * )  RWORK,
integer  INFO 
)

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

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Purpose:
 CSYSVX 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 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 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 CSYTRF.

          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 CSYTRF.
          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 CSYTRF.
[in]B
          B is COMPLEX 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 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 REAL
          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 REAL 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 REAL 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 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 CSYTRF.

          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 REAL 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 csysvx.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  REAL RCOND
294 * ..
295 * .. Array Arguments ..
296  INTEGER IPIV( * )
297  REAL BERR( * ), FERR( * ), RWORK( * )
298  COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
299  $ WORK( * ), X( LDX, * )
300 * ..
301 *
302 * =====================================================================
303 *
304 * .. Parameters ..
305  REAL ZERO
306  parameter( zero = 0.0e+0 )
307 * ..
308 * .. Local Scalars ..
309  LOGICAL LQUERY, NOFACT
310  INTEGER LWKOPT, NB
311  REAL ANORM
312 * ..
313 * .. External Functions ..
314  LOGICAL LSAME
315  INTEGER ILAENV
316  REAL CLANSY, SLAMCH
317  EXTERNAL ilaenv, lsame, clansy, slamch
318 * ..
319 * .. External Subroutines ..
320  EXTERNAL clacpy, csycon, csyrfs, csytrf, csytrs, xerbla
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, 'CSYTRF', 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( 'CSYSVX', -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 clacpy( uplo, n, n, a, lda, af, ldaf )
374  CALL csytrf( 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 = clansy( 'I', uplo, n, a, lda, rwork )
387 *
388 * Compute the reciprocal of the condition number of A.
389 *
390  CALL csycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work, info )
391 *
392 * Compute the solution vectors X.
393 *
394  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
395  CALL csytrs( 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 csyrfs( 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.slamch( 'Epsilon' ) )
406  $ info = n + 1
407 *
408  work( 1 ) = lwkopt
409 *
410  RETURN
411 *
412 * End of CSYSVX
413 *
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 clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
real function clansy(NORM, UPLO, N, A, LDA, WORK)
CLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clansy.f:123
subroutine csyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CSYRFS
Definition: csyrfs.f:192
subroutine csycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
CSYCON
Definition: csycon.f:125
subroutine csytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS
Definition: csytrs.f:120
subroutine csytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF
Definition: csytrf.f:182
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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