LAPACK  3.9.0 LAPACK: Linear Algebra PACKage

## ◆ cposvx()

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

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

Purpose:
``` CPOSVX 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 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 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 REAL 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 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 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 REAL 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 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 (2*N)` [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: 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.```
Date
April 2012

Definition at line 308 of file cposvx.f.

308 *
309 * -- LAPACK driver routine (version 3.7.0) --
310 * -- LAPACK is a software package provided by Univ. of Tennessee, --
311 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
312 * April 2012
313 *
314 * .. Scalar Arguments ..
315  CHARACTER EQUED, FACT, UPLO
316  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
317  REAL RCOND
318 * ..
319 * .. Array Arguments ..
320  REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
321  COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
322  \$ WORK( * ), X( LDX, * )
323 * ..
324 *
325 * =====================================================================
326 *
327 * .. Parameters ..
328  REAL ZERO, ONE
329  parameter( zero = 0.0e+0, one = 1.0e+0 )
330 * ..
331 * .. Local Scalars ..
332  LOGICAL EQUIL, NOFACT, RCEQU
333  INTEGER I, INFEQU, J
334  REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
335 * ..
336 * .. External Functions ..
337  LOGICAL LSAME
338  REAL CLANHE, SLAMCH
339  EXTERNAL lsame, clanhe, slamch
340 * ..
341 * .. External Subroutines ..
342  EXTERNAL clacpy, claqhe, cpocon, cpoequ, cporfs, cpotrf,
343  \$ cpotrs, xerbla
344 * ..
345 * .. Intrinsic Functions ..
346  INTRINSIC max, min
347 * ..
348 * .. Executable Statements ..
349 *
350  info = 0
351  nofact = lsame( fact, 'N' )
352  equil = lsame( fact, 'E' )
353  IF( nofact .OR. equil ) THEN
354  equed = 'N'
355  rcequ = .false.
356  ELSE
357  rcequ = lsame( equed, 'Y' )
358  smlnum = slamch( 'Safe minimum' )
359  bignum = one / smlnum
360  END IF
361 *
362 * Test the input parameters.
363 *
364  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
365  \$ THEN
366  info = -1
367  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
368  \$ THEN
369  info = -2
370  ELSE IF( n.LT.0 ) THEN
371  info = -3
372  ELSE IF( nrhs.LT.0 ) THEN
373  info = -4
374  ELSE IF( lda.LT.max( 1, n ) ) THEN
375  info = -6
376  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
377  info = -8
378  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
379  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
380  info = -9
381  ELSE
382  IF( rcequ ) THEN
383  smin = bignum
384  smax = zero
385  DO 10 j = 1, n
386  smin = min( smin, s( j ) )
387  smax = max( smax, s( j ) )
388  10 CONTINUE
389  IF( smin.LE.zero ) THEN
390  info = -10
391  ELSE IF( n.GT.0 ) THEN
392  scond = max( smin, smlnum ) / min( smax, bignum )
393  ELSE
394  scond = one
395  END IF
396  END IF
397  IF( info.EQ.0 ) THEN
398  IF( ldb.LT.max( 1, n ) ) THEN
399  info = -12
400  ELSE IF( ldx.LT.max( 1, n ) ) THEN
401  info = -14
402  END IF
403  END IF
404  END IF
405 *
406  IF( info.NE.0 ) THEN
407  CALL xerbla( 'CPOSVX', -info )
408  RETURN
409  END IF
410 *
411  IF( equil ) THEN
412 *
413 * Compute row and column scalings to equilibrate the matrix A.
414 *
415  CALL cpoequ( n, a, lda, s, scond, amax, infequ )
416  IF( infequ.EQ.0 ) THEN
417 *
418 * Equilibrate the matrix.
419 *
420  CALL claqhe( uplo, n, a, lda, s, scond, amax, equed )
421  rcequ = lsame( equed, 'Y' )
422  END IF
423  END IF
424 *
425 * Scale the right hand side.
426 *
427  IF( rcequ ) THEN
428  DO 30 j = 1, nrhs
429  DO 20 i = 1, n
430  b( i, j ) = s( i )*b( i, j )
431  20 CONTINUE
432  30 CONTINUE
433  END IF
434 *
435  IF( nofact .OR. equil ) THEN
436 *
437 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
438 *
439  CALL clacpy( uplo, n, n, a, lda, af, ldaf )
440  CALL cpotrf( uplo, n, af, ldaf, info )
441 *
442 * Return if INFO is non-zero.
443 *
444  IF( info.GT.0 )THEN
445  rcond = zero
446  RETURN
447  END IF
448  END IF
449 *
450 * Compute the norm of the matrix A.
451 *
452  anorm = clanhe( '1', uplo, n, a, lda, rwork )
453 *
454 * Compute the reciprocal of the condition number of A.
455 *
456  CALL cpocon( uplo, n, af, ldaf, anorm, rcond, work, rwork, info )
457 *
458 * Compute the solution matrix X.
459 *
460  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
461  CALL cpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
462 *
463 * Use iterative refinement to improve the computed solution and
464 * compute error bounds and backward error estimates for it.
465 *
466  CALL cporfs( uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx,
467  \$ ferr, berr, work, rwork, info )
468 *
469 * Transform the solution matrix X to a solution of the original
470 * system.
471 *
472  IF( rcequ ) THEN
473  DO 50 j = 1, nrhs
474  DO 40 i = 1, n
475  x( i, j ) = s( i )*x( i, j )
476  40 CONTINUE
477  50 CONTINUE
478  DO 60 j = 1, nrhs
479  ferr( j ) = ferr( j ) / scond
480  60 CONTINUE
481  END IF
482 *
483 * Set INFO = N+1 if the matrix is singular to working precision.
484 *
485  IF( rcond.LT.slamch( 'Epsilon' ) )
486  \$ info = n + 1
487 *
488  RETURN
489 *
490 * End of CPOSVX
491 *
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claqhe
subroutine claqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
CLAQHE scales a Hermitian matrix.
Definition: claqhe.f:136
cpoequ
subroutine cpoequ(N, A, LDA, S, SCOND, AMAX, INFO)
CPOEQU
Definition: cpoequ.f:115
cpotrs
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:112
cpotrf
subroutine cpotrf(UPLO, N, A, LDA, INFO)
CPOTRF
Definition: cpotrf.f:109
clacpy
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
cporfs
subroutine cporfs(UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPORFS
Definition: cporfs.f:185
clanhe
real function clanhe(NORM, UPLO, N, A, LDA, WORK)
CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhe.f:126
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
lsame
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
slamch
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:70
cpocon
subroutine cpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
CPOCON
Definition: cpocon.f:123