1 *> \brief <b> CGESVXX computes the solution to system of linear equations A * X = B for GE matrices</b>
3 * =========== DOCUMENTATION ===========
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21 * SUBROUTINE CGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
22 * EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
23 * BERR, N_ERR_BNDS, ERR_BNDS_NORM,
24 * ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
27 * .. Scalar Arguments ..
28 * CHARACTER EQUED, FACT, TRANS
29 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
33 * .. Array Arguments ..
35 * COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
36 * $ X( LDX , * ),WORK( * )
37 * REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
38 * $ ERR_BNDS_NORM( NRHS, * ),
39 * $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
48 *> CGESVXX uses the LU factorization to compute the solution to a
49 *> complex system of linear equations A * X = B, where A is an
50 *> N-by-N matrix and X and B are N-by-NRHS matrices.
52 *> If requested, both normwise and maximum componentwise error bounds
53 *> are returned. CGESVXX will return a solution with a tiny
54 *> guaranteed error (O(eps) where eps is the working machine
55 *> precision) unless the matrix is very ill-conditioned, in which
56 *> case a warning is returned. Relevant condition numbers also are
57 *> calculated and returned.
59 *> CGESVXX accepts user-provided factorizations and equilibration
60 *> factors; see the definitions of the FACT and EQUED options.
61 *> Solving with refinement and using a factorization from a previous
62 *> CGESVXX call will also produce a solution with either O(eps)
63 *> errors or warnings, but we cannot make that claim for general
64 *> user-provided factorizations and equilibration factors if they
65 *> differ from what CGESVXX would itself produce.
73 *> The following steps are performed:
75 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
78 *> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
79 *> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
80 *> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
82 *> Whether or not the system will be equilibrated depends on the
83 *> scaling of the matrix A, but if equilibration is used, A is
84 *> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
85 *> or diag(C)*B (if TRANS = 'T' or 'C').
87 *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
88 *> the matrix A (after equilibration if FACT = 'E') as
92 *> where P is a permutation matrix, L is a unit lower triangular
93 *> matrix, and U is upper triangular.
95 *> 3. If some U(i,i)=0, so that U is exactly singular, then the
96 *> routine returns with INFO = i. Otherwise, the factored form of A
97 *> is used to estimate the condition number of the matrix A (see
98 *> argument RCOND). If the reciprocal of the condition number is less
99 *> than machine precision, the routine still goes on to solve for X
100 *> and compute error bounds as described below.
102 *> 4. The system of equations is solved for X using the factored form
105 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
106 *> the routine will use iterative refinement to try to get a small
107 *> error and error bounds. Refinement calculates the residual to at
108 *> least twice the working precision.
110 *> 6. If equilibration was used, the matrix X is premultiplied by
111 *> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
112 *> that it solves the original system before equilibration.
119 *> Some optional parameters are bundled in the PARAMS array. These
120 *> settings determine how refinement is performed, but often the
121 *> defaults are acceptable. If the defaults are acceptable, users
122 *> can pass NPARAMS = 0 which prevents the source code from accessing
123 *> the PARAMS argument.
128 *> FACT is CHARACTER*1
129 *> Specifies whether or not the factored form of the matrix A is
130 *> supplied on entry, and if not, whether the matrix A should be
131 *> equilibrated before it is factored.
132 *> = 'F': On entry, AF and IPIV contain the factored form of A.
133 *> If EQUED is not 'N', the matrix A has been
134 *> equilibrated with scaling factors given by R and C.
135 *> A, AF, and IPIV are not modified.
136 *> = 'N': The matrix A will be copied to AF and factored.
137 *> = 'E': The matrix A will be equilibrated if necessary, then
138 *> copied to AF and factored.
143 *> TRANS is CHARACTER*1
144 *> Specifies the form of the system of equations:
145 *> = 'N': A * X = B (No transpose)
146 *> = 'T': A**T * X = B (Transpose)
147 *> = 'C': A**H * X = B (Conjugate Transpose)
153 *> The number of linear equations, i.e., the order of the
160 *> The number of right hand sides, i.e., the number of columns
161 *> of the matrices B and X. NRHS >= 0.
166 *> A is COMPLEX array, dimension (LDA,N)
167 *> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
168 *> not 'N', then A must have been equilibrated by the scaling
169 *> factors in R and/or C. A is not modified if FACT = 'F' or
170 *> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
172 *> On exit, if EQUED .ne. 'N', A is scaled as follows:
173 *> EQUED = 'R': A := diag(R) * A
174 *> EQUED = 'C': A := A * diag(C)
175 *> EQUED = 'B': A := diag(R) * A * diag(C).
181 *> The leading dimension of the array A. LDA >= max(1,N).
186 *> AF is COMPLEX array, dimension (LDAF,N)
187 *> If FACT = 'F', then AF is an input argument and on entry
188 *> contains the factors L and U from the factorization
189 *> A = P*L*U as computed by CGETRF. If EQUED .ne. 'N', then
190 *> AF is the factored form of the equilibrated matrix A.
192 *> If FACT = 'N', then AF is an output argument and on exit
193 *> returns the factors L and U from the factorization A = P*L*U
194 *> of the original matrix A.
196 *> If FACT = 'E', then AF is an output argument and on exit
197 *> returns the factors L and U from the factorization A = P*L*U
198 *> of the equilibrated matrix A (see the description of A for
199 *> the form of the equilibrated matrix).
205 *> The leading dimension of the array AF. LDAF >= max(1,N).
208 *> \param[in,out] IPIV
210 *> IPIV is INTEGER array, dimension (N)
211 *> If FACT = 'F', then IPIV is an input argument and on entry
212 *> contains the pivot indices from the factorization A = P*L*U
213 *> as computed by CGETRF; row i of the matrix was interchanged
216 *> If FACT = 'N', then IPIV is an output argument and on exit
217 *> contains the pivot indices from the factorization A = P*L*U
218 *> of the original matrix A.
220 *> If FACT = 'E', then IPIV is an output argument and on exit
221 *> contains the pivot indices from the factorization A = P*L*U
222 *> of the equilibrated matrix A.
225 *> \param[in,out] EQUED
227 *> EQUED is CHARACTER*1
228 *> Specifies the form of equilibration that was done.
229 *> = 'N': No equilibration (always true if FACT = 'N').
230 *> = 'R': Row equilibration, i.e., A has been premultiplied by
232 *> = 'C': Column equilibration, i.e., A has been postmultiplied
234 *> = 'B': Both row and column equilibration, i.e., A has been
235 *> replaced by diag(R) * A * diag(C).
236 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
242 *> R is REAL array, dimension (N)
243 *> The row scale factors for A. If EQUED = 'R' or 'B', A is
244 *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
245 *> is not accessed. R is an input argument if FACT = 'F';
246 *> otherwise, R is an output argument. If FACT = 'F' and
247 *> EQUED = 'R' or 'B', each element of R must be positive.
248 *> If R is output, each element of R is a power of the radix.
249 *> If R is input, each element of R should be a power of the radix
250 *> to ensure a reliable solution and error estimates. Scaling by
251 *> powers of the radix does not cause rounding errors unless the
252 *> result underflows or overflows. Rounding errors during scaling
253 *> lead to refining with a matrix that is not equivalent to the
254 *> input matrix, producing error estimates that may not be
260 *> C is REAL array, dimension (N)
261 *> The column scale factors for A. If EQUED = 'C' or 'B', A is
262 *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
263 *> is not accessed. C is an input argument if FACT = 'F';
264 *> otherwise, C is an output argument. If FACT = 'F' and
265 *> EQUED = 'C' or 'B', each element of C must be positive.
266 *> If C is output, each element of C is a power of the radix.
267 *> If C is input, each element of C should be a power of the radix
268 *> to ensure a reliable solution and error estimates. Scaling by
269 *> powers of the radix does not cause rounding errors unless the
270 *> result underflows or overflows. Rounding errors during scaling
271 *> lead to refining with a matrix that is not equivalent to the
272 *> input matrix, producing error estimates that may not be
278 *> B is COMPLEX array, dimension (LDB,NRHS)
279 *> On entry, the N-by-NRHS right hand side matrix B.
281 *> if EQUED = 'N', B is not modified;
282 *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
284 *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
285 *> overwritten by diag(C)*B.
291 *> The leading dimension of the array B. LDB >= max(1,N).
296 *> X is COMPLEX array, dimension (LDX,NRHS)
297 *> If INFO = 0, the N-by-NRHS solution matrix X to the original
298 *> system of equations. Note that A and B are modified on exit
299 *> if EQUED .ne. 'N', and the solution to the equilibrated system is
300 *> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
301 *> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
307 *> The leading dimension of the array X. LDX >= max(1,N).
313 *> Reciprocal scaled condition number. This is an estimate of the
314 *> reciprocal Skeel condition number of the matrix A after
315 *> equilibration (if done). If this is less than the machine
316 *> precision (in particular, if it is zero), the matrix is singular
317 *> to working precision. Note that the error may still be small even
318 *> if this number is very small and the matrix appears ill-
322 *> \param[out] RPVGRW
325 *> Reciprocal pivot growth. On exit, this contains the reciprocal
326 *> pivot growth factor norm(A)/norm(U). The "max absolute element"
327 *> norm is used. If this is much less than 1, then the stability of
328 *> the LU factorization of the (equilibrated) matrix A could be poor.
329 *> This also means that the solution X, estimated condition numbers,
330 *> and error bounds could be unreliable. If factorization fails with
331 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
332 *> for the leading INFO columns of A. In CGESVX, this quantity is
333 *> returned in WORK(1).
338 *> BERR is REAL array, dimension (NRHS)
339 *> Componentwise relative backward error. This is the
340 *> componentwise relative backward error of each solution vector X(j)
341 *> (i.e., the smallest relative change in any element of A or B that
342 *> makes X(j) an exact solution).
345 *> \param[in] N_ERR_BNDS
347 *> N_ERR_BNDS is INTEGER
348 *> Number of error bounds to return for each right hand side
349 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
350 *> ERR_BNDS_COMP below.
353 *> \param[out] ERR_BNDS_NORM
355 *> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
356 *> For each right-hand side, this array contains information about
357 *> various error bounds and condition numbers corresponding to the
358 *> normwise relative error, which is defined as follows:
360 *> Normwise relative error in the ith solution vector:
361 *> max_j (abs(XTRUE(j,i) - X(j,i)))
362 *> ------------------------------
365 *> The array is indexed by the type of error information as described
366 *> below. There currently are up to three pieces of information
369 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
372 *> The second index in ERR_BNDS_NORM(:,err) contains the following
374 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
375 *> reciprocal condition number is less than the threshold
376 *> sqrt(n) * slamch('Epsilon').
378 *> err = 2 "Guaranteed" error bound: The estimated forward error,
379 *> almost certainly within a factor of 10 of the true error
380 *> so long as the next entry is greater than the threshold
381 *> sqrt(n) * slamch('Epsilon'). This error bound should only
382 *> be trusted if the previous boolean is true.
384 *> err = 3 Reciprocal condition number: Estimated normwise
385 *> reciprocal condition number. Compared with the threshold
386 *> sqrt(n) * slamch('Epsilon') to determine if the error
387 *> estimate is "guaranteed". These reciprocal condition
388 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
389 *> appropriately scaled matrix Z.
390 *> Let Z = S*A, where S scales each row by a power of the
391 *> radix so all absolute row sums of Z are approximately 1.
393 *> See Lapack Working Note 165 for further details and extra
397 *> \param[out] ERR_BNDS_COMP
399 *> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
400 *> For each right-hand side, this array contains information about
401 *> various error bounds and condition numbers corresponding to the
402 *> componentwise relative error, which is defined as follows:
404 *> Componentwise relative error in the ith solution vector:
405 *> abs(XTRUE(j,i) - X(j,i))
406 *> max_j ----------------------
409 *> The array is indexed by the right-hand side i (on which the
410 *> componentwise relative error depends), and the type of error
411 *> information as described below. There currently are up to three
412 *> pieces of information returned for each right-hand side. If
413 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
414 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
415 *> the first (:,N_ERR_BNDS) entries are returned.
417 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
420 *> The second index in ERR_BNDS_COMP(:,err) contains the following
422 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
423 *> reciprocal condition number is less than the threshold
424 *> sqrt(n) * slamch('Epsilon').
426 *> err = 2 "Guaranteed" error bound: The estimated forward error,
427 *> almost certainly within a factor of 10 of the true error
428 *> so long as the next entry is greater than the threshold
429 *> sqrt(n) * slamch('Epsilon'). This error bound should only
430 *> be trusted if the previous boolean is true.
432 *> err = 3 Reciprocal condition number: Estimated componentwise
433 *> reciprocal condition number. Compared with the threshold
434 *> sqrt(n) * slamch('Epsilon') to determine if the error
435 *> estimate is "guaranteed". These reciprocal condition
436 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
437 *> appropriately scaled matrix Z.
438 *> Let Z = S*(A*diag(x)), where x is the solution for the
439 *> current right-hand side and S scales each row of
440 *> A*diag(x) by a power of the radix so all absolute row
441 *> sums of Z are approximately 1.
443 *> See Lapack Working Note 165 for further details and extra
447 *> \param[in] NPARAMS
449 *> NPARAMS is INTEGER
450 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
451 *> PARAMS array is never referenced and default values are used.
454 *> \param[in,out] PARAMS
456 *> PARAMS is REAL array, dimension NPARAMS
457 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
458 *> that entry will be filled with default value used for that
459 *> parameter. Only positions up to NPARAMS are accessed; defaults
460 *> are used for higher-numbered parameters.
462 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
463 *> refinement or not.
465 *> = 0.0 : No refinement is performed, and no error bounds are
467 *> = 1.0 : Use the double-precision refinement algorithm,
468 *> possibly with doubled-single computations if the
469 *> compilation environment does not support DOUBLE
471 *> (other values are reserved for future use)
473 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
474 *> computations allowed for refinement.
476 *> Aggressive: Set to 100 to permit convergence using approximate
477 *> factorizations or factorizations other than LU. If
478 *> the factorization uses a technique other than
479 *> Gaussian elimination, the guarantees in
480 *> err_bnds_norm and err_bnds_comp may no longer be
483 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
484 *> will attempt to find a solution with small componentwise
485 *> relative error in the double-precision algorithm. Positive
486 *> is true, 0.0 is false.
487 *> Default: 1.0 (attempt componentwise convergence)
492 *> WORK is COMPLEX array, dimension (2*N)
497 *> RWORK is REAL array, dimension (2*N)
503 *> = 0: Successful exit. The solution to every right-hand side is
505 *> < 0: If INFO = -i, the i-th argument had an illegal value
506 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
507 *> has been completed, but the factor U is exactly singular, so
508 *> the solution and error bounds could not be computed. RCOND = 0
510 *> = N+J: The solution corresponding to the Jth right-hand side is
511 *> not guaranteed. The solutions corresponding to other right-
512 *> hand sides K with K > J may not be guaranteed as well, but
513 *> only the first such right-hand side is reported. If a small
514 *> componentwise error is not requested (PARAMS(3) = 0.0) then
515 *> the Jth right-hand side is the first with a normwise error
516 *> bound that is not guaranteed (the smallest J such
517 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
518 *> the Jth right-hand side is the first with either a normwise or
519 *> componentwise error bound that is not guaranteed (the smallest
520 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
521 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
522 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
523 *> about all of the right-hand sides check ERR_BNDS_NORM or
530 *> \author Univ. of Tennessee
531 *> \author Univ. of California Berkeley
532 *> \author Univ. of Colorado Denver
537 *> \ingroup complexGEsolve
539 * =====================================================================
540 SUBROUTINE CGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
541 $ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
542 $ BERR, N_ERR_BNDS, ERR_BNDS_NORM,
543 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
546 * -- LAPACK driver routine (version 3.4.1) --
547 * -- LAPACK is a software package provided by Univ. of Tennessee, --
548 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
551 * .. Scalar Arguments ..
552 CHARACTER EQUED, FACT, TRANS
553 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
557 * .. Array Arguments ..
559 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
560 $ X( LDX , * ),WORK( * )
561 REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
562 $ ERR_BNDS_NORM( NRHS, * ),
563 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
566 * ==================================================================
570 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
571 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
572 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
573 INTEGER CMP_ERR_I, PIV_GROWTH_I
574 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
576 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
577 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
580 * .. Local Scalars ..
581 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
583 REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
586 * .. External Functions ..
587 EXTERNAL LSAME, SLAMCH, CLA_GERPVGRW
589 REAL SLAMCH, CLA_GERPVGRW
591 * .. External Subroutines ..
592 EXTERNAL CGEEQUB, CGETRF, CGETRS, CLACPY, CLAQGE,
593 $ XERBLA, CLASCL2, CGERFSX
595 * .. Intrinsic Functions ..
598 * .. Executable Statements ..
601 NOFACT = LSAME( FACT, 'N' )
602 EQUIL = LSAME( FACT, 'E' )
603 NOTRAN = LSAME( TRANS, 'N' )
604 SMLNUM = SLAMCH( 'Safe minimum' )
605 BIGNUM = ONE / SMLNUM
606 IF( NOFACT .OR. EQUIL ) THEN
611 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
612 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
615 * Default is failure. If an input parameter is wrong or
616 * factorization fails, make everything look horrible. Only the
617 * pivot growth is set here, the rest is initialized in CGERFSX.
621 * Test the input parameters. PARAMS is not tested until CGERFSX.
623 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
624 $ LSAME( FACT, 'F' ) ) THEN
626 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
627 $ LSAME( TRANS, 'C' ) ) THEN
629 ELSE IF( N.LT.0 ) THEN
631 ELSE IF( NRHS.LT.0 ) THEN
633 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
635 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
637 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
638 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
645 RCMIN = MIN( RCMIN, R( J ) )
646 RCMAX = MAX( RCMAX, R( J ) )
648 IF( RCMIN.LE.ZERO ) THEN
650 ELSE IF( N.GT.0 ) THEN
651 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
656 IF( COLEQU .AND. INFO.EQ.0 ) THEN
660 RCMIN = MIN( RCMIN, C( J ) )
661 RCMAX = MAX( RCMAX, C( J ) )
663 IF( RCMIN.LE.ZERO ) THEN
665 ELSE IF( N.GT.0 ) THEN
666 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
672 IF( LDB.LT.MAX( 1, N ) ) THEN
674 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
681 CALL XERBLA( 'CGESVXX', -INFO )
687 * Compute row and column scalings to equilibrate the matrix A.
689 CALL CGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
691 IF( INFEQU.EQ.0 ) THEN
693 * Equilibrate the matrix.
695 CALL CLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
697 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
698 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
701 * If the scaling factors are not applied, set them to 1.0.
703 IF ( .NOT.ROWEQU ) THEN
708 IF ( .NOT.COLEQU ) THEN
715 * Scale the right-hand side.
718 IF( ROWEQU ) CALL CLASCL2( N, NRHS, R, B, LDB )
720 IF( COLEQU ) CALL CLASCL2( N, NRHS, C, B, LDB )
723 IF( NOFACT .OR. EQUIL ) THEN
725 * Compute the LU factorization of A.
727 CALL CLACPY( 'Full', N, N, A, LDA, AF, LDAF )
728 CALL CGETRF( N, N, AF, LDAF, IPIV, INFO )
730 * Return if INFO is non-zero.
734 * Pivot in column INFO is exactly 0
735 * Compute the reciprocal pivot growth factor of the
736 * leading rank-deficient INFO columns of A.
738 RPVGRW = CLA_GERPVGRW( N, INFO, A, LDA, AF, LDAF )
743 * Compute the reciprocal pivot growth factor RPVGRW.
745 RPVGRW = CLA_GERPVGRW( N, N, A, LDA, AF, LDAF )
747 * Compute the solution matrix X.
749 CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
750 CALL CGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
752 * Use iterative refinement to improve the computed solution and
753 * compute error bounds and backward error estimates for it.
755 CALL CGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF,
756 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
757 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
758 $ WORK, RWORK, INFO )
762 IF ( COLEQU .AND. NOTRAN ) THEN
763 CALL CLASCL2 ( N, NRHS, C, X, LDX )
764 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
765 CALL CLASCL2 ( N, NRHS, R, X, LDX )