LAPACK 3.12.1
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

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

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

Definition at line 279 of file dsysvx.f.

283*
284* -- LAPACK driver routine --
285* -- LAPACK is a software package provided by Univ. of Tennessee, --
286* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
287*
288* .. Scalar Arguments ..
289 CHARACTER FACT, UPLO
290 INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
291 DOUBLE PRECISION RCOND
292* ..
293* .. Array Arguments ..
294 INTEGER IPIV( * ), IWORK( * )
295 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
296 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
297* ..
298*
299* =====================================================================
300*
301* .. Parameters ..
302 DOUBLE PRECISION ZERO
303 parameter( zero = 0.0d+0 )
304* ..
305* .. Local Scalars ..
306 LOGICAL LQUERY, NOFACT
307 INTEGER LWKMIN, LWKOPT, NB
308 DOUBLE PRECISION ANORM
309* ..
310* .. External Functions ..
311 LOGICAL LSAME
312 INTEGER ILAENV
313 DOUBLE PRECISION DLAMCH, DLANSY
314 EXTERNAL lsame, ilaenv, dlamch, dlansy
315* ..
316* .. External Subroutines ..
317 EXTERNAL dlacpy, dsycon, dsyrfs, dsytrf, dsytrs,
318 $ 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 lwkmin = max( 1, 3*n )
331 IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
332 info = -1
333 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
334 $ .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.lwkmin .AND. .NOT.lquery ) THEN
350 info = -18
351 END IF
352*
353 IF( info.EQ.0 ) THEN
354 lwkopt = lwkmin
355 IF( nofact ) THEN
356 nb = ilaenv( 1, 'DSYTRF', 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( 'DSYSVX', -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 dlacpy( uplo, n, n, a, lda, af, ldaf )
374 CALL dsytrf( 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 = dlansy( 'I', uplo, n, a, lda, work )
387*
388* Compute the reciprocal of the condition number of A.
389*
390 CALL dsycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work,
391 $ iwork,
392 $ info )
393*
394* Compute the solution vectors X.
395*
396 CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
397 CALL dsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
398*
399* Use iterative refinement to improve the computed solutions and
400* compute error bounds and backward error estimates for them.
401*
402 CALL dsyrfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
403 $ ldx, ferr, berr, work, iwork, info )
404*
405* Set INFO = N+1 if the matrix is singular to working precision.
406*
407 IF( rcond.LT.dlamch( 'Epsilon' ) )
408 $ info = n + 1
409*
410 work( 1 ) = lwkopt
411*
412 RETURN
413*
414* End of DSYSVX
415*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
dsycon
subroutine dsycon(uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info)
DSYCON
Definition dsycon.f:128
dsyrfs
subroutine dsyrfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DSYRFS
Definition dsyrfs.f:190
dsytrf
subroutine dsytrf(uplo, n, a, lda, ipiv, work, lwork, info)
DSYTRF
Definition dsytrf.f:180
dsytrs
subroutine dsytrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
DSYTRS
Definition dsytrs.f:118
ilaenv
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:160
dlacpy
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:101
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
dlansy
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:120
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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