LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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

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 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.```

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
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