LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ zposvx()

subroutine zposvx ( character  FACT,
character  UPLO,
integer  N,
integer  NRHS,
complex*16, dimension( lda, * )  A,
integer  LDA,
complex*16, dimension( ldaf, * )  AF,
integer  LDAF,
character  EQUED,
double precision, dimension( * )  S,
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,
double precision, dimension( * )  RWORK,
integer  INFO 
)

ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices

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Purpose:
 ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
 compute the solution to a complex system of linear equations
    A * X = B,
 where A is an N-by-N Hermitian positive definite 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 = 'E', real scaling factors are computed to equilibrate
    the system:
       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
    Whether or not the system will be equilibrated depends on the
    scaling of the matrix A, but if equilibration is used, A is
    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
    factor the matrix A (after equilibration if FACT = 'E') as
       A = U**H* U,  if UPLO = 'U', or
       A = L * L**H,  if UPLO = 'L',
    where U is an upper triangular matrix and L is a lower triangular
    matrix.

 3. If the leading i-by-i principal minor is not positive definite,
    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.

 4. The system of equations is solved for X using the factored form
    of A.

 5. Iterative refinement is applied to improve the computed solution
    matrix and calculate error bounds and backward error estimates
    for it.

 6. If equilibration was used, the matrix X is premultiplied by
    diag(S) so that it solves the original system before
    equilibration.
Parameters
[in]FACT
          FACT is CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AF contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  A and AF will not
                  be modified.
          = 'N':  The matrix A will be copied to AF and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then
                  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,out]A
          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, the Hermitian matrix A, except if FACT = 'F' and
          EQUED = 'Y', then A must contain the equilibrated matrix
          diag(S)*A*diag(S).  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.  A is not modified if
          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).
[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 triangular factor U or L from the Cholesky
          factorization A = U**H *U or A = L*L**H, in the same storage
          format as A.  If EQUED .ne. 'N', then AF is the factored form
          of the equilibrated matrix diag(S)*A*diag(S).

          If FACT = 'N', then AF is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**H *U or A = L*L**H of the original
          matrix A.

          If FACT = 'E', then AF is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**H *U or A = L*L**H of the equilibrated
          matrix A (see the description of A for the form of the
          equilibrated matrix).
[in]LDAF
          LDAF is INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).
[in,out]EQUED
          EQUED is CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by
                  diag(S) * A * diag(S).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.
[in,out]S
          S is DOUBLE PRECISION array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is
          an input argument if FACT = 'F'; otherwise, S is an output
          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
          must be positive.
[in,out]B
          B is COMPLEX*16 array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS righthand side matrix B.
          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
          B is overwritten by diag(S) * 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 to
          the original system of equations.  Note that if EQUED = 'Y',
          A and B are modified on exit, and the solution to the
          equilibrated system is inv(diag(S))*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 after equilibration (if done).  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 (2*N)
[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:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U 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 303 of file zposvx.f.

306 *
307 * -- LAPACK driver routine --
308 * -- LAPACK is a software package provided by Univ. of Tennessee, --
309 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
310 *
311 * .. Scalar Arguments ..
312  CHARACTER EQUED, FACT, UPLO
313  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
314  DOUBLE PRECISION RCOND
315 * ..
316 * .. Array Arguments ..
317  DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
318  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
319  $ WORK( * ), X( LDX, * )
320 * ..
321 *
322 * =====================================================================
323 *
324 * .. Parameters ..
325  DOUBLE PRECISION ZERO, ONE
326  parameter( zero = 0.0d+0, one = 1.0d+0 )
327 * ..
328 * .. Local Scalars ..
329  LOGICAL EQUIL, NOFACT, RCEQU
330  INTEGER I, INFEQU, J
331  DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
332 * ..
333 * .. External Functions ..
334  LOGICAL LSAME
335  DOUBLE PRECISION DLAMCH, ZLANHE
336  EXTERNAL lsame, dlamch, zlanhe
337 * ..
338 * .. External Subroutines ..
339  EXTERNAL xerbla, zlacpy, zlaqhe, zpocon, zpoequ, zporfs,
340  $ zpotrf, zpotrs
341 * ..
342 * .. Intrinsic Functions ..
343  INTRINSIC max, min
344 * ..
345 * .. Executable Statements ..
346 *
347  info = 0
348  nofact = lsame( fact, 'N' )
349  equil = lsame( fact, 'E' )
350  IF( nofact .OR. equil ) THEN
351  equed = 'N'
352  rcequ = .false.
353  ELSE
354  rcequ = lsame( equed, 'Y' )
355  smlnum = dlamch( 'Safe minimum' )
356  bignum = one / smlnum
357  END IF
358 *
359 * Test the input parameters.
360 *
361  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
362  $ THEN
363  info = -1
364  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
365  $ THEN
366  info = -2
367  ELSE IF( n.LT.0 ) THEN
368  info = -3
369  ELSE IF( nrhs.LT.0 ) THEN
370  info = -4
371  ELSE IF( lda.LT.max( 1, n ) ) THEN
372  info = -6
373  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
374  info = -8
375  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
376  $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
377  info = -9
378  ELSE
379  IF( rcequ ) THEN
380  smin = bignum
381  smax = zero
382  DO 10 j = 1, n
383  smin = min( smin, s( j ) )
384  smax = max( smax, s( j ) )
385  10 CONTINUE
386  IF( smin.LE.zero ) THEN
387  info = -10
388  ELSE IF( n.GT.0 ) THEN
389  scond = max( smin, smlnum ) / min( smax, bignum )
390  ELSE
391  scond = one
392  END IF
393  END IF
394  IF( info.EQ.0 ) THEN
395  IF( ldb.LT.max( 1, n ) ) THEN
396  info = -12
397  ELSE IF( ldx.LT.max( 1, n ) ) THEN
398  info = -14
399  END IF
400  END IF
401  END IF
402 *
403  IF( info.NE.0 ) THEN
404  CALL xerbla( 'ZPOSVX', -info )
405  RETURN
406  END IF
407 *
408  IF( equil ) THEN
409 *
410 * Compute row and column scalings to equilibrate the matrix A.
411 *
412  CALL zpoequ( n, a, lda, s, scond, amax, infequ )
413  IF( infequ.EQ.0 ) THEN
414 *
415 * Equilibrate the matrix.
416 *
417  CALL zlaqhe( uplo, n, a, lda, s, scond, amax, equed )
418  rcequ = lsame( equed, 'Y' )
419  END IF
420  END IF
421 *
422 * Scale the right hand side.
423 *
424  IF( rcequ ) THEN
425  DO 30 j = 1, nrhs
426  DO 20 i = 1, n
427  b( i, j ) = s( i )*b( i, j )
428  20 CONTINUE
429  30 CONTINUE
430  END IF
431 *
432  IF( nofact .OR. equil ) THEN
433 *
434 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
435 *
436  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
437  CALL zpotrf( uplo, n, af, ldaf, info )
438 *
439 * Return if INFO is non-zero.
440 *
441  IF( info.GT.0 )THEN
442  rcond = zero
443  RETURN
444  END IF
445  END IF
446 *
447 * Compute the norm of the matrix A.
448 *
449  anorm = zlanhe( '1', uplo, n, a, lda, rwork )
450 *
451 * Compute the reciprocal of the condition number of A.
452 *
453  CALL zpocon( uplo, n, af, ldaf, anorm, rcond, work, rwork, info )
454 *
455 * Compute the solution matrix X.
456 *
457  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
458  CALL zpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
459 *
460 * Use iterative refinement to improve the computed solution and
461 * compute error bounds and backward error estimates for it.
462 *
463  CALL zporfs( uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx,
464  $ ferr, berr, work, rwork, info )
465 *
466 * Transform the solution matrix X to a solution of the original
467 * system.
468 *
469  IF( rcequ ) THEN
470  DO 50 j = 1, nrhs
471  DO 40 i = 1, n
472  x( i, j ) = s( i )*x( i, j )
473  40 CONTINUE
474  50 CONTINUE
475  DO 60 j = 1, nrhs
476  ferr( j ) = ferr( j ) / scond
477  60 CONTINUE
478  END IF
479 *
480 * Set INFO = N+1 if the matrix is singular to working precision.
481 *
482  IF( rcond.LT.dlamch( 'Epsilon' ) )
483  $ info = n + 1
484 *
485  RETURN
486 *
487 * End of ZPOSVX
488 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function zlanhe(NORM, UPLO, N, A, LDA, WORK)
ZLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlanhe.f:124
subroutine zlaqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQHE scales a Hermitian matrix.
Definition: zlaqhe.f:134
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
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:110
subroutine zpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
ZPOCON
Definition: zpocon.f:121
subroutine zpoequ(N, A, LDA, S, SCOND, AMAX, INFO)
ZPOEQU
Definition: zpoequ.f:113
subroutine zporfs(UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZPORFS
Definition: zporfs.f:183
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102
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