 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sgesvx()

 subroutine sgesvx ( character FACT, character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

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

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

Purpose:
``` SGESVX 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 REAL 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 REAL 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 SGETRF. 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 SGETRF; 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 REAL 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 REAL 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 REAL 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 REAL 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 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 REAL 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 sgesvx.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  REAL RCOND
358 * ..
359 * .. Array Arguments ..
360  INTEGER IPIV( * ), IWORK( * )
361  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
362  \$ BERR( * ), C( * ), FERR( * ), R( * ),
363  \$ WORK( * ), X( LDX, * )
364 * ..
365 *
366 * =====================================================================
367 *
368 * .. Parameters ..
369  REAL ZERO, ONE
370  parameter( zero = 0.0e+0, one = 1.0e+0 )
371 * ..
372 * .. Local Scalars ..
373  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
374  CHARACTER NORM
375  INTEGER I, INFEQU, J
376  REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
377  \$ ROWCND, RPVGRW, SMLNUM
378 * ..
379 * .. External Functions ..
380  LOGICAL LSAME
381  REAL SLAMCH, SLANGE, SLANTR
382  EXTERNAL lsame, slamch, slange, slantr
383 * ..
384 * .. External Subroutines ..
385  EXTERNAL sgecon, sgeequ, sgerfs, sgetrf, sgetrs, slacpy,
386  \$ slaqge, 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 = slamch( '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( 'SGESVX', -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 sgeequ( n, n, a, lda, r, c, rowcnd, colcnd, amax, infequ )
477  IF( infequ.EQ.0 ) THEN
478 *
479 * Equilibrate the matrix.
480 *
481  CALL slaqge( 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 slacpy( 'Full', n, n, a, lda, af, ldaf )
511  CALL sgetrf( 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 = slantr( 'M', 'U', 'N', info, info, af, ldaf,
521  \$ work )
522  IF( rpvgrw.EQ.zero ) THEN
523  rpvgrw = one
524  ELSE
525  rpvgrw = slange( '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 = slange( norm, n, n, a, lda, work )
542  rpvgrw = slantr( 'M', 'U', 'N', n, n, af, ldaf, work )
543  IF( rpvgrw.EQ.zero ) THEN
544  rpvgrw = one
545  ELSE
546  rpvgrw = slange( 'M', n, n, a, lda, work ) / rpvgrw
547  END IF
548 *
549 * Compute the reciprocal of the condition number of A.
550 *
551  CALL sgecon( norm, n, af, ldaf, anorm, rcond, work, iwork, info )
552 *
553 * Compute the solution matrix X.
554 *
555  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
556  CALL sgetrs( 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 sgerfs( 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 * Set INFO = N+1 if the matrix is singular to working precision.
590 *
591  IF( rcond.LT.slamch( 'Epsilon' ) )
592  \$ info = n + 1
593 *
594  work( 1 ) = rpvgrw
595  RETURN
596 *
597 * End of SGESVX
598 *
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
subroutine slaqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Definition: slaqge.f:142
subroutine sgeequ(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
SGEEQU
Definition: sgeequ.f:139
subroutine sgerfs(TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGERFS
Definition: sgerfs.f:185
subroutine sgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SGECON
Definition: sgecon.f:124
subroutine sgetrf(M, N, A, LDA, IPIV, INFO)
SGETRF
Definition: sgetrf.f:108
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:121
real function slantr(NORM, UPLO, DIAG, M, N, A, LDA, WORK)
SLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slantr.f:141
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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