LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ dgesvx()

subroutine dgesvx ( character  FACT,
character  TRANS,
integer  N,
integer  NRHS,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( ldaf, * )  AF,
integer  LDAF,
integer, dimension( * )  IPIV,
character  EQUED,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
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 
)

DGESVX computes the solution to system of linear equations A * X = B for GE matrices

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

Purpose:
 DGESVX uses the LU factorization to compute the solution to a real
 system of linear equations
    A * X = B,
 where A is an N-by-N 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:
       TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
       TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
       TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
    or diag(C)*B (if TRANS = 'T' or 'C').

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

 3. If some U(i,i)=0, so that U 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.

 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A.
                  If EQUED is not 'N', the matrix A has been
                  equilibrated with scaling factors given by R and C.
                  A, AF, and IPIV are not 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]TRANS
          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Transpose)
[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 N-by-N matrix A.  If FACT = 'F' and EQUED is
          not 'N', then A must have been equilibrated by the scaling
          factors in R and/or C.  A is not modified if FACT = 'F' or
          'N', or if FACT = 'E' and EQUED = 'N' on exit.

          On exit, if EQUED .ne. 'N', A is scaled as follows:
          EQUED = 'R':  A := diag(R) * A
          EQUED = 'C':  A := A * diag(C)
          EQUED = 'B':  A := diag(R) * A * diag(C).
[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 factors L and U from the factorization
          A = P*L*U as computed by DGETRF.  If EQUED .ne. 'N', then
          AF is the factored form of the equilibrated matrix A.

          If FACT = 'N', then AF is an output argument and on exit
          returns the factors L and U from the factorization A = P*L*U
          of the original matrix A.

          If FACT = 'E', then AF is an output argument and on exit
          returns the factors L and U from the factorization A = P*L*U
          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]IPIV
          IPIV is INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the factorization A = P*L*U
          as computed by DGETRF; row i of the matrix was interchanged
          with row IPIV(i).

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = P*L*U
          of the original matrix A.

          If FACT = 'E', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = P*L*U
          of the equilibrated matrix A.
[in,out]EQUED
          EQUED is CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'R':  Row equilibration, i.e., A has been premultiplied by
                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.
[in,out]R
          R is DOUBLE PRECISION array, dimension (N)
          The row scale factors for A.  If EQUED = 'R' or 'B', A is
          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
          is not accessed.  R is an input argument if FACT = 'F';
          otherwise, R is an output argument.  If FACT = 'F' and
          EQUED = 'R' or 'B', each element of R must be positive.
[in,out]C
          C is DOUBLE PRECISION array, dimension (N)
          The column scale factors for A.  If EQUED = 'C' or 'B', A is
          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
          is not accessed.  C is an input argument if FACT = 'F';
          otherwise, C is an output argument.  If FACT = 'F' and
          EQUED = 'C' or 'B', each element of C 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
          diag(R)*B;
          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
          overwritten by diag(C)*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 A and B are
          modified on exit if EQUED .ne. 'N', and the solution to the
          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
          and EQUED = 'R' or 'B'.
[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 (4*N)
          On exit, WORK(1) contains the reciprocal pivot growth
          factor norm(A)/norm(U). The "max absolute element" norm is
          used. If WORK(1) is much less than 1, then the stability
          of the LU factorization of the (equilibrated) matrix A
          could be poor. This also means that the solution X, condition
          estimator RCOND, and forward error bound FERR could be
          unreliable. If factorization fails with 0<INFO<=N, then
          WORK(1) contains the reciprocal pivot growth factor for the
          leading INFO columns of A.
[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:  U(i,i) is exactly zero.  The factorization has
                       been completed, but the factor U is exactly
                       singular, so the solution and error bounds
                       could not be 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 346 of file dgesvx.f.

349 *
350 * -- LAPACK driver routine --
351 * -- LAPACK is a software package provided by Univ. of Tennessee, --
352 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
353 *
354 * .. Scalar Arguments ..
355  CHARACTER EQUED, FACT, TRANS
356  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
357  DOUBLE PRECISION RCOND
358 * ..
359 * .. Array Arguments ..
360  INTEGER IPIV( * ), IWORK( * )
361  DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
362  $ BERR( * ), C( * ), FERR( * ), R( * ),
363  $ WORK( * ), X( LDX, * )
364 * ..
365 *
366 * =====================================================================
367 *
368 * .. Parameters ..
369  DOUBLE PRECISION ZERO, ONE
370  parameter( zero = 0.0d+0, one = 1.0d+0 )
371 * ..
372 * .. Local Scalars ..
373  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
374  CHARACTER NORM
375  INTEGER I, INFEQU, J
376  DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
377  $ ROWCND, RPVGRW, SMLNUM
378 * ..
379 * .. External Functions ..
380  LOGICAL LSAME
381  DOUBLE PRECISION DLAMCH, DLANGE, DLANTR
382  EXTERNAL lsame, dlamch, dlange, dlantr
383 * ..
384 * .. External Subroutines ..
385  EXTERNAL dgecon, dgeequ, dgerfs, dgetrf, dgetrs, dlacpy,
386  $ dlaqge, xerbla
387 * ..
388 * .. Intrinsic Functions ..
389  INTRINSIC max, min
390 * ..
391 * .. Executable Statements ..
392 *
393  info = 0
394  nofact = lsame( fact, 'N' )
395  equil = lsame( fact, 'E' )
396  notran = lsame( trans, 'N' )
397  IF( nofact .OR. equil ) THEN
398  equed = 'N'
399  rowequ = .false.
400  colequ = .false.
401  ELSE
402  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
403  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
404  smlnum = dlamch( 'Safe minimum' )
405  bignum = one / smlnum
406  END IF
407 *
408 * Test the input parameters.
409 *
410  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
411  $ THEN
412  info = -1
413  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
414  $ lsame( trans, 'C' ) ) THEN
415  info = -2
416  ELSE IF( n.LT.0 ) THEN
417  info = -3
418  ELSE IF( nrhs.LT.0 ) THEN
419  info = -4
420  ELSE IF( lda.LT.max( 1, n ) ) THEN
421  info = -6
422  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
423  info = -8
424  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
425  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
426  info = -10
427  ELSE
428  IF( rowequ ) THEN
429  rcmin = bignum
430  rcmax = zero
431  DO 10 j = 1, n
432  rcmin = min( rcmin, r( j ) )
433  rcmax = max( rcmax, r( j ) )
434  10 CONTINUE
435  IF( rcmin.LE.zero ) THEN
436  info = -11
437  ELSE IF( n.GT.0 ) THEN
438  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
439  ELSE
440  rowcnd = one
441  END IF
442  END IF
443  IF( colequ .AND. info.EQ.0 ) THEN
444  rcmin = bignum
445  rcmax = zero
446  DO 20 j = 1, n
447  rcmin = min( rcmin, c( j ) )
448  rcmax = max( rcmax, c( j ) )
449  20 CONTINUE
450  IF( rcmin.LE.zero ) THEN
451  info = -12
452  ELSE IF( n.GT.0 ) THEN
453  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
454  ELSE
455  colcnd = one
456  END IF
457  END IF
458  IF( info.EQ.0 ) THEN
459  IF( ldb.LT.max( 1, n ) ) THEN
460  info = -14
461  ELSE IF( ldx.LT.max( 1, n ) ) THEN
462  info = -16
463  END IF
464  END IF
465  END IF
466 *
467  IF( info.NE.0 ) THEN
468  CALL xerbla( 'DGESVX', -info )
469  RETURN
470  END IF
471 *
472  IF( equil ) THEN
473 *
474 * Compute row and column scalings to equilibrate the matrix A.
475 *
476  CALL dgeequ( n, n, a, lda, r, c, rowcnd, colcnd, amax, infequ )
477  IF( infequ.EQ.0 ) THEN
478 *
479 * Equilibrate the matrix.
480 *
481  CALL dlaqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
482  $ equed )
483  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
484  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
485  END IF
486  END IF
487 *
488 * Scale the right hand side.
489 *
490  IF( notran ) THEN
491  IF( rowequ ) THEN
492  DO 40 j = 1, nrhs
493  DO 30 i = 1, n
494  b( i, j ) = r( i )*b( i, j )
495  30 CONTINUE
496  40 CONTINUE
497  END IF
498  ELSE IF( colequ ) THEN
499  DO 60 j = 1, nrhs
500  DO 50 i = 1, n
501  b( i, j ) = c( i )*b( i, j )
502  50 CONTINUE
503  60 CONTINUE
504  END IF
505 *
506  IF( nofact .OR. equil ) THEN
507 *
508 * Compute the LU factorization of A.
509 *
510  CALL dlacpy( 'Full', n, n, a, lda, af, ldaf )
511  CALL dgetrf( n, n, af, ldaf, ipiv, info )
512 *
513 * Return if INFO is non-zero.
514 *
515  IF( info.GT.0 ) THEN
516 *
517 * Compute the reciprocal pivot growth factor of the
518 * leading rank-deficient INFO columns of A.
519 *
520  rpvgrw = dlantr( 'M', 'U', 'N', info, info, af, ldaf,
521  $ work )
522  IF( rpvgrw.EQ.zero ) THEN
523  rpvgrw = one
524  ELSE
525  rpvgrw = dlange( 'M', n, info, a, lda, work ) / rpvgrw
526  END IF
527  work( 1 ) = rpvgrw
528  rcond = zero
529  RETURN
530  END IF
531  END IF
532 *
533 * Compute the norm of the matrix A and the
534 * reciprocal pivot growth factor RPVGRW.
535 *
536  IF( notran ) THEN
537  norm = '1'
538  ELSE
539  norm = 'I'
540  END IF
541  anorm = dlange( norm, n, n, a, lda, work )
542  rpvgrw = dlantr( 'M', 'U', 'N', n, n, af, ldaf, work )
543  IF( rpvgrw.EQ.zero ) THEN
544  rpvgrw = one
545  ELSE
546  rpvgrw = dlange( 'M', n, n, a, lda, work ) / rpvgrw
547  END IF
548 *
549 * Compute the reciprocal of the condition number of A.
550 *
551  CALL dgecon( norm, n, af, ldaf, anorm, rcond, work, iwork, info )
552 *
553 * Compute the solution matrix X.
554 *
555  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
556  CALL dgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
557 *
558 * Use iterative refinement to improve the computed solution and
559 * compute error bounds and backward error estimates for it.
560 *
561  CALL dgerfs( trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
562  $ ldx, ferr, berr, work, iwork, info )
563 *
564 * Transform the solution matrix X to a solution of the original
565 * system.
566 *
567  IF( notran ) THEN
568  IF( colequ ) THEN
569  DO 80 j = 1, nrhs
570  DO 70 i = 1, n
571  x( i, j ) = c( i )*x( i, j )
572  70 CONTINUE
573  80 CONTINUE
574  DO 90 j = 1, nrhs
575  ferr( j ) = ferr( j ) / colcnd
576  90 CONTINUE
577  END IF
578  ELSE IF( rowequ ) THEN
579  DO 110 j = 1, nrhs
580  DO 100 i = 1, n
581  x( i, j ) = r( i )*x( i, j )
582  100 CONTINUE
583  110 CONTINUE
584  DO 120 j = 1, nrhs
585  ferr( j ) = ferr( j ) / rowcnd
586  120 CONTINUE
587  END IF
588 *
589  work( 1 ) = rpvgrw
590 *
591 * Set INFO = N+1 if the matrix is singular to working precision.
592 *
593  IF( rcond.LT.dlamch( 'Epsilon' ) )
594  $ info = n + 1
595  RETURN
596 *
597 * End of DGESVX
598 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dlaqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Definition: dlaqge.f:142
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dgetrf(M, N, A, LDA, IPIV, INFO)
DGETRF
Definition: dgetrf.f:108
subroutine dgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
DGECON
Definition: dgecon.f:124
subroutine dgeequ(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
DGEEQU
Definition: dgeequ.f:139
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:121
subroutine dgerfs(TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DGERFS
Definition: dgerfs.f:185
double precision function dlantr(NORM, UPLO, DIAG, M, N, A, LDA, WORK)
DLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlantr.f:141
Here is the call graph for this function:
Here is the caller graph for this function: