 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine cgesvx ( character FACT, character TRANS, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R, real, dimension( * ) C, 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 )

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

Purpose:
``` CGESVX uses the LU factorization to compute the solution to a complex
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 (Conjugate 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 COMPLEX 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 COMPLEX 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 CGETRF. 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 CGETRF; 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 COMPLEX 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 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 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 COMPLEX array, dimension (2*N)` [out] RWORK ``` RWORK is REAL array, dimension (2*N) On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If RWORK(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.```
Date
April 2012

Definition at line 352 of file cgesvx.f.

352 *
353 * -- LAPACK driver routine (version 3.4.1) --
354 * -- LAPACK is a software package provided by Univ. of Tennessee, --
355 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
356 * April 2012
357 *
358 * .. Scalar Arguments ..
359  CHARACTER equed, fact, trans
360  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs
361  REAL rcond
362 * ..
363 * .. Array Arguments ..
364  INTEGER ipiv( * )
365  REAL berr( * ), c( * ), ferr( * ), r( * ),
366  \$ rwork( * )
367  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
368  \$ work( * ), x( ldx, * )
369 * ..
370 *
371 * =====================================================================
372 *
373 * .. Parameters ..
374  REAL zero, one
375  parameter ( zero = 0.0e+0, one = 1.0e+0 )
376 * ..
377 * .. Local Scalars ..
378  LOGICAL colequ, equil, nofact, notran, rowequ
379  CHARACTER norm
380  INTEGER i, infequ, j
381  REAL amax, anorm, bignum, colcnd, rcmax, rcmin,
382  \$ rowcnd, rpvgrw, smlnum
383 * ..
384 * .. External Functions ..
385  LOGICAL lsame
386  REAL clange, clantr, slamch
387  EXTERNAL lsame, clange, clantr, slamch
388 * ..
389 * .. External Subroutines ..
390  EXTERNAL cgecon, cgeequ, cgerfs, cgetrf, cgetrs, clacpy,
391  \$ claqge, xerbla
392 * ..
393 * .. Intrinsic Functions ..
394  INTRINSIC max, min
395 * ..
396 * .. Executable Statements ..
397 *
398  info = 0
399  nofact = lsame( fact, 'N' )
400  equil = lsame( fact, 'E' )
401  notran = lsame( trans, 'N' )
402  IF( nofact .OR. equil ) THEN
403  equed = 'N'
404  rowequ = .false.
405  colequ = .false.
406  ELSE
407  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
408  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
409  smlnum = slamch( 'Safe minimum' )
410  bignum = one / smlnum
411  END IF
412 *
413 * Test the input parameters.
414 *
415  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
416  \$ THEN
417  info = -1
418  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
419  \$ lsame( trans, 'C' ) ) THEN
420  info = -2
421  ELSE IF( n.LT.0 ) THEN
422  info = -3
423  ELSE IF( nrhs.LT.0 ) THEN
424  info = -4
425  ELSE IF( lda.LT.max( 1, n ) ) THEN
426  info = -6
427  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
428  info = -8
429  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
430  \$ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
431  info = -10
432  ELSE
433  IF( rowequ ) THEN
434  rcmin = bignum
435  rcmax = zero
436  DO 10 j = 1, n
437  rcmin = min( rcmin, r( j ) )
438  rcmax = max( rcmax, r( j ) )
439  10 CONTINUE
440  IF( rcmin.LE.zero ) THEN
441  info = -11
442  ELSE IF( n.GT.0 ) THEN
443  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
444  ELSE
445  rowcnd = one
446  END IF
447  END IF
448  IF( colequ .AND. info.EQ.0 ) THEN
449  rcmin = bignum
450  rcmax = zero
451  DO 20 j = 1, n
452  rcmin = min( rcmin, c( j ) )
453  rcmax = max( rcmax, c( j ) )
454  20 CONTINUE
455  IF( rcmin.LE.zero ) THEN
456  info = -12
457  ELSE IF( n.GT.0 ) THEN
458  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
459  ELSE
460  colcnd = one
461  END IF
462  END IF
463  IF( info.EQ.0 ) THEN
464  IF( ldb.LT.max( 1, n ) ) THEN
465  info = -14
466  ELSE IF( ldx.LT.max( 1, n ) ) THEN
467  info = -16
468  END IF
469  END IF
470  END IF
471 *
472  IF( info.NE.0 ) THEN
473  CALL xerbla( 'CGESVX', -info )
474  RETURN
475  END IF
476 *
477  IF( equil ) THEN
478 *
479 * Compute row and column scalings to equilibrate the matrix A.
480 *
481  CALL cgeequ( n, n, a, lda, r, c, rowcnd, colcnd, amax, infequ )
482  IF( infequ.EQ.0 ) THEN
483 *
484 * Equilibrate the matrix.
485 *
486  CALL claqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
487  \$ equed )
488  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
489  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
490  END IF
491  END IF
492 *
493 * Scale the right hand side.
494 *
495  IF( notran ) THEN
496  IF( rowequ ) THEN
497  DO 40 j = 1, nrhs
498  DO 30 i = 1, n
499  b( i, j ) = r( i )*b( i, j )
500  30 CONTINUE
501  40 CONTINUE
502  END IF
503  ELSE IF( colequ ) THEN
504  DO 60 j = 1, nrhs
505  DO 50 i = 1, n
506  b( i, j ) = c( i )*b( i, j )
507  50 CONTINUE
508  60 CONTINUE
509  END IF
510 *
511  IF( nofact .OR. equil ) THEN
512 *
513 * Compute the LU factorization of A.
514 *
515  CALL clacpy( 'Full', n, n, a, lda, af, ldaf )
516  CALL cgetrf( n, n, af, ldaf, ipiv, info )
517 *
518 * Return if INFO is non-zero.
519 *
520  IF( info.GT.0 ) THEN
521 *
522 * Compute the reciprocal pivot growth factor of the
523 * leading rank-deficient INFO columns of A.
524 *
525  rpvgrw = clantr( 'M', 'U', 'N', info, info, af, ldaf,
526  \$ rwork )
527  IF( rpvgrw.EQ.zero ) THEN
528  rpvgrw = one
529  ELSE
530  rpvgrw = clange( 'M', n, info, a, lda, rwork ) /
531  \$ rpvgrw
532  END IF
533  rwork( 1 ) = rpvgrw
534  rcond = zero
535  RETURN
536  END IF
537  END IF
538 *
539 * Compute the norm of the matrix A and the
540 * reciprocal pivot growth factor RPVGRW.
541 *
542  IF( notran ) THEN
543  norm = '1'
544  ELSE
545  norm = 'I'
546  END IF
547  anorm = clange( norm, n, n, a, lda, rwork )
548  rpvgrw = clantr( 'M', 'U', 'N', n, n, af, ldaf, rwork )
549  IF( rpvgrw.EQ.zero ) THEN
550  rpvgrw = one
551  ELSE
552  rpvgrw = clange( 'M', n, n, a, lda, rwork ) / rpvgrw
553  END IF
554 *
555 * Compute the reciprocal of the condition number of A.
556 *
557  CALL cgecon( norm, n, af, ldaf, anorm, rcond, work, rwork, info )
558 *
559 * Compute the solution matrix X.
560 *
561  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
562  CALL cgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
563 *
564 * Use iterative refinement to improve the computed solution and
565 * compute error bounds and backward error estimates for it.
566 *
567  CALL cgerfs( trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
568  \$ ldx, ferr, berr, work, rwork, info )
569 *
570 * Transform the solution matrix X to a solution of the original
571 * system.
572 *
573  IF( notran ) THEN
574  IF( colequ ) THEN
575  DO 80 j = 1, nrhs
576  DO 70 i = 1, n
577  x( i, j ) = c( i )*x( i, j )
578  70 CONTINUE
579  80 CONTINUE
580  DO 90 j = 1, nrhs
581  ferr( j ) = ferr( j ) / colcnd
582  90 CONTINUE
583  END IF
584  ELSE IF( rowequ ) THEN
585  DO 110 j = 1, nrhs
586  DO 100 i = 1, n
587  x( i, j ) = r( i )*x( i, j )
588  100 CONTINUE
589  110 CONTINUE
590  DO 120 j = 1, nrhs
591  ferr( j ) = ferr( j ) / rowcnd
592  120 CONTINUE
593  END IF
594 *
595 * Set INFO = N+1 if the matrix is singular to working precision.
596 *
597  IF( rcond.LT.slamch( 'Epsilon' ) )
598  \$ info = n + 1
599 *
600  rwork( 1 ) = rpvgrw
601  RETURN
602 *
603 * End of CGESVX
604 *
subroutine cgerfs(TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CGERFS
Definition: cgerfs.f:188
subroutine cgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CGETRS
Definition: cgetrs.f:123
real function clantr(NORM, UPLO, DIAG, M, N, A, LDA, WORK)
CLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.
Definition: clantr.f:144
real function clange(NORM, M, N, A, LDA, WORK)
CLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: clange.f:117
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine claqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
CLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ...
Definition: claqge.f:145
subroutine cgeequ(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
CGEEQU
Definition: cgeequ.f:142
subroutine cgetrf(M, N, A, LDA, IPIV, INFO)
CGETRF
Definition: cgetrf.f:110
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
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine cgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
CGECON
Definition: cgecon.f:126
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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