LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dposvx()

subroutine dposvx ( character  FACT,
character  UPLO,
integer  N,
integer  NRHS,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( ldaf, * )  AF,
integer  LDAF,
character  EQUED,
double precision, dimension( * )  S,
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, dimension( * )  IWORK,
integer  INFO 
)

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

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Purpose:
 DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
 compute the solution to a real system of linear equations
    A * X = B,
 where A is an N-by-N symmetric 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**T* U,  if UPLO = 'U', or
       A = L * L**T,  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 DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric 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 DOUBLE PRECISION 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**T*U or A = L*L**T, 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**T*U or A = L*L**T 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**T*U or A = L*L**T 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 DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (3*N)
[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:  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.
Date
April 2012

Definition at line 309 of file dposvx.f.

309 *
310 * -- LAPACK driver routine (version 3.7.0) --
311 * -- LAPACK is a software package provided by Univ. of Tennessee, --
312 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
313 * April 2012
314 *
315 * .. Scalar Arguments ..
316  CHARACTER equed, fact, uplo
317  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs
318  DOUBLE PRECISION rcond
319 * ..
320 * .. Array Arguments ..
321  INTEGER iwork( * )
322  DOUBLE PRECISION a( lda, * ), af( ldaf, * ), b( ldb, * ),
323  $ berr( * ), ferr( * ), s( * ), work( * ),
324  $ x( ldx, * )
325 * ..
326 *
327 * =====================================================================
328 *
329 * .. Parameters ..
330  DOUBLE PRECISION zero, one
331  parameter( zero = 0.0d+0, one = 1.0d+0 )
332 * ..
333 * .. Local Scalars ..
334  LOGICAL equil, nofact, rcequ
335  INTEGER i, infequ, j
336  DOUBLE PRECISION amax, anorm, bignum, scond, smax, smin, smlnum
337 * ..
338 * .. External Functions ..
339  LOGICAL lsame
340  DOUBLE PRECISION dlamch, dlansy
341  EXTERNAL lsame, dlamch, dlansy
342 * ..
343 * .. External Subroutines ..
344  EXTERNAL dlacpy, dlaqsy, dpocon, dpoequ, dporfs, dpotrf,
345  $ dpotrs, xerbla
346 * ..
347 * .. Intrinsic Functions ..
348  INTRINSIC max, min
349 * ..
350 * .. Executable Statements ..
351 *
352  info = 0
353  nofact = lsame( fact, 'N' )
354  equil = lsame( fact, 'E' )
355  IF( nofact .OR. equil ) THEN
356  equed = 'N'
357  rcequ = .false.
358  ELSE
359  rcequ = lsame( equed, 'Y' )
360  smlnum = dlamch( 'Safe minimum' )
361  bignum = one / smlnum
362  END IF
363 *
364 * Test the input parameters.
365 *
366  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
367  $ THEN
368  info = -1
369  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
370  $ THEN
371  info = -2
372  ELSE IF( n.LT.0 ) THEN
373  info = -3
374  ELSE IF( nrhs.LT.0 ) THEN
375  info = -4
376  ELSE IF( lda.LT.max( 1, n ) ) THEN
377  info = -6
378  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
379  info = -8
380  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
381  $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
382  info = -9
383  ELSE
384  IF( rcequ ) THEN
385  smin = bignum
386  smax = zero
387  DO 10 j = 1, n
388  smin = min( smin, s( j ) )
389  smax = max( smax, s( j ) )
390  10 CONTINUE
391  IF( smin.LE.zero ) THEN
392  info = -10
393  ELSE IF( n.GT.0 ) THEN
394  scond = max( smin, smlnum ) / min( smax, bignum )
395  ELSE
396  scond = one
397  END IF
398  END IF
399  IF( info.EQ.0 ) THEN
400  IF( ldb.LT.max( 1, n ) ) THEN
401  info = -12
402  ELSE IF( ldx.LT.max( 1, n ) ) THEN
403  info = -14
404  END IF
405  END IF
406  END IF
407 *
408  IF( info.NE.0 ) THEN
409  CALL xerbla( 'DPOSVX', -info )
410  RETURN
411  END IF
412 *
413  IF( equil ) THEN
414 *
415 * Compute row and column scalings to equilibrate the matrix A.
416 *
417  CALL dpoequ( n, a, lda, s, scond, amax, infequ )
418  IF( infequ.EQ.0 ) THEN
419 *
420 * Equilibrate the matrix.
421 *
422  CALL dlaqsy( uplo, n, a, lda, s, scond, amax, equed )
423  rcequ = lsame( equed, 'Y' )
424  END IF
425  END IF
426 *
427 * Scale the right hand side.
428 *
429  IF( rcequ ) THEN
430  DO 30 j = 1, nrhs
431  DO 20 i = 1, n
432  b( i, j ) = s( i )*b( i, j )
433  20 CONTINUE
434  30 CONTINUE
435  END IF
436 *
437  IF( nofact .OR. equil ) THEN
438 *
439 * Compute the Cholesky factorization A = U**T *U or A = L*L**T.
440 *
441  CALL dlacpy( uplo, n, n, a, lda, af, ldaf )
442  CALL dpotrf( uplo, n, af, ldaf, info )
443 *
444 * Return if INFO is non-zero.
445 *
446  IF( info.GT.0 )THEN
447  rcond = zero
448  RETURN
449  END IF
450  END IF
451 *
452 * Compute the norm of the matrix A.
453 *
454  anorm = dlansy( '1', uplo, n, a, lda, work )
455 *
456 * Compute the reciprocal of the condition number of A.
457 *
458  CALL dpocon( uplo, n, af, ldaf, anorm, rcond, work, iwork, info )
459 *
460 * Compute the solution matrix X.
461 *
462  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
463  CALL dpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
464 *
465 * Use iterative refinement to improve the computed solution and
466 * compute error bounds and backward error estimates for it.
467 *
468  CALL dporfs( uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx,
469  $ ferr, berr, work, iwork, info )
470 *
471 * Transform the solution matrix X to a solution of the original
472 * system.
473 *
474  IF( rcequ ) THEN
475  DO 50 j = 1, nrhs
476  DO 40 i = 1, n
477  x( i, j ) = s( i )*x( i, j )
478  40 CONTINUE
479  50 CONTINUE
480  DO 60 j = 1, nrhs
481  ferr( j ) = ferr( j ) / scond
482  60 CONTINUE
483  END IF
484 *
485 * Set INFO = N+1 if the matrix is singular to working precision.
486 *
487  IF( rcond.LT.dlamch( 'Epsilon' ) )
488  $ info = n + 1
489 *
490  RETURN
491 *
492 * End of DPOSVX
493 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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, or the element of largest absolute value of a real symmetric matrix.
Definition: dlansy.f:124
subroutine dpotrf(UPLO, N, A, LDA, INFO)
DPOTRF
Definition: dpotrf.f:109
subroutine dporfs(UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DPORFS
Definition: dporfs.f:185
subroutine dlaqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition: dlaqsy.f:135
subroutine dpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
DPOTRS
Definition: dpotrs.f:112
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
DPOCON
Definition: dpocon.f:123
subroutine dpoequ(N, A, LDA, S, SCOND, AMAX, INFO)
DPOEQU
Definition: dpoequ.f:114
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