LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
dgesvx.f
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1 *> \brief <b> DGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DGESVX + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvx.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
22 * EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
23 * WORK, IWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER EQUED, FACT, TRANS
27 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
28 * DOUBLE PRECISION RCOND
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IPIV( * ), IWORK( * )
32 * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
33 * $ BERR( * ), C( * ), FERR( * ), R( * ),
34 * $ WORK( * ), X( LDX, * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> DGESVX uses the LU factorization to compute the solution to a real
44 *> system of linear equations
45 *> A * X = B,
46 *> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
47 *>
48 *> Error bounds on the solution and a condition estimate are also
49 *> provided.
50 *> \endverbatim
51 *
52 *> \par Description:
53 * =================
54 *>
55 *> \verbatim
56 *>
57 *> The following steps are performed:
58 *>
59 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
60 *> the system:
61 *> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
62 *> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
63 *> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
64 *> Whether or not the system will be equilibrated depends on the
65 *> scaling of the matrix A, but if equilibration is used, A is
66 *> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
67 *> or diag(C)*B (if TRANS = 'T' or 'C').
68 *>
69 *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
70 *> matrix A (after equilibration if FACT = 'E') as
71 *> A = P * L * U,
72 *> where P is a permutation matrix, L is a unit lower triangular
73 *> matrix, and U is upper triangular.
74 *>
75 *> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
76 *> returns with INFO = i. Otherwise, the factored form of A is used
77 *> to estimate the condition number of the matrix A. If the
78 *> reciprocal of the condition number is less than machine precision,
79 *> INFO = N+1 is returned as a warning, but the routine still goes on
80 *> to solve for X and compute error bounds as described below.
81 *>
82 *> 4. The system of equations is solved for X using the factored form
83 *> of A.
84 *>
85 *> 5. Iterative refinement is applied to improve the computed solution
86 *> matrix and calculate error bounds and backward error estimates
87 *> for it.
88 *>
89 *> 6. If equilibration was used, the matrix X is premultiplied by
90 *> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
91 *> that it solves the original system before equilibration.
92 *> \endverbatim
93 *
94 * Arguments:
95 * ==========
96 *
97 *> \param[in] FACT
98 *> \verbatim
99 *> FACT is CHARACTER*1
100 *> Specifies whether or not the factored form of the matrix A is
101 *> supplied on entry, and if not, whether the matrix A should be
102 *> equilibrated before it is factored.
103 *> = 'F': On entry, AF and IPIV contain the factored form of A.
104 *> If EQUED is not 'N', the matrix A has been
105 *> equilibrated with scaling factors given by R and C.
106 *> A, AF, and IPIV are not modified.
107 *> = 'N': The matrix A will be copied to AF and factored.
108 *> = 'E': The matrix A will be equilibrated if necessary, then
109 *> copied to AF and factored.
110 *> \endverbatim
111 *>
112 *> \param[in] TRANS
113 *> \verbatim
114 *> TRANS is CHARACTER*1
115 *> Specifies the form of the system of equations:
116 *> = 'N': A * X = B (No transpose)
117 *> = 'T': A**T * X = B (Transpose)
118 *> = 'C': A**H * X = B (Transpose)
119 *> \endverbatim
120 *>
121 *> \param[in] N
122 *> \verbatim
123 *> N is INTEGER
124 *> The number of linear equations, i.e., the order of the
125 *> matrix A. N >= 0.
126 *> \endverbatim
127 *>
128 *> \param[in] NRHS
129 *> \verbatim
130 *> NRHS is INTEGER
131 *> The number of right hand sides, i.e., the number of columns
132 *> of the matrices B and X. NRHS >= 0.
133 *> \endverbatim
134 *>
135 *> \param[in,out] A
136 *> \verbatim
137 *> A is DOUBLE PRECISION array, dimension (LDA,N)
138 *> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
139 *> not 'N', then A must have been equilibrated by the scaling
140 *> factors in R and/or C. A is not modified if FACT = 'F' or
141 *> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
142 *>
143 *> On exit, if EQUED .ne. 'N', A is scaled as follows:
144 *> EQUED = 'R': A := diag(R) * A
145 *> EQUED = 'C': A := A * diag(C)
146 *> EQUED = 'B': A := diag(R) * A * diag(C).
147 *> \endverbatim
148 *>
149 *> \param[in] LDA
150 *> \verbatim
151 *> LDA is INTEGER
152 *> The leading dimension of the array A. LDA >= max(1,N).
153 *> \endverbatim
154 *>
155 *> \param[in,out] AF
156 *> \verbatim
157 *> AF is DOUBLE PRECISION array, dimension (LDAF,N)
158 *> If FACT = 'F', then AF is an input argument and on entry
159 *> contains the factors L and U from the factorization
160 *> A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
161 *> AF is the factored form of the equilibrated matrix A.
162 *>
163 *> If FACT = 'N', then AF is an output argument and on exit
164 *> returns the factors L and U from the factorization A = P*L*U
165 *> of the original matrix A.
166 *>
167 *> If FACT = 'E', then AF is an output argument and on exit
168 *> returns the factors L and U from the factorization A = P*L*U
169 *> of the equilibrated matrix A (see the description of A for
170 *> the form of the equilibrated matrix).
171 *> \endverbatim
172 *>
173 *> \param[in] LDAF
174 *> \verbatim
175 *> LDAF is INTEGER
176 *> The leading dimension of the array AF. LDAF >= max(1,N).
177 *> \endverbatim
178 *>
179 *> \param[in,out] IPIV
180 *> \verbatim
181 *> IPIV is INTEGER array, dimension (N)
182 *> If FACT = 'F', then IPIV is an input argument and on entry
183 *> contains the pivot indices from the factorization A = P*L*U
184 *> as computed by DGETRF; row i of the matrix was interchanged
185 *> with row IPIV(i).
186 *>
187 *> If FACT = 'N', then IPIV is an output argument and on exit
188 *> contains the pivot indices from the factorization A = P*L*U
189 *> of the original matrix A.
190 *>
191 *> If FACT = 'E', then IPIV is an output argument and on exit
192 *> contains the pivot indices from the factorization A = P*L*U
193 *> of the equilibrated matrix A.
194 *> \endverbatim
195 *>
196 *> \param[in,out] EQUED
197 *> \verbatim
198 *> EQUED is CHARACTER*1
199 *> Specifies the form of equilibration that was done.
200 *> = 'N': No equilibration (always true if FACT = 'N').
201 *> = 'R': Row equilibration, i.e., A has been premultiplied by
202 *> diag(R).
203 *> = 'C': Column equilibration, i.e., A has been postmultiplied
204 *> by diag(C).
205 *> = 'B': Both row and column equilibration, i.e., A has been
206 *> replaced by diag(R) * A * diag(C).
207 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
208 *> output argument.
209 *> \endverbatim
210 *>
211 *> \param[in,out] R
212 *> \verbatim
213 *> R is DOUBLE PRECISION array, dimension (N)
214 *> The row scale factors for A. If EQUED = 'R' or 'B', A is
215 *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
216 *> is not accessed. R is an input argument if FACT = 'F';
217 *> otherwise, R is an output argument. If FACT = 'F' and
218 *> EQUED = 'R' or 'B', each element of R must be positive.
219 *> \endverbatim
220 *>
221 *> \param[in,out] C
222 *> \verbatim
223 *> C is DOUBLE PRECISION array, dimension (N)
224 *> The column scale factors for A. If EQUED = 'C' or 'B', A is
225 *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
226 *> is not accessed. C is an input argument if FACT = 'F';
227 *> otherwise, C is an output argument. If FACT = 'F' and
228 *> EQUED = 'C' or 'B', each element of C must be positive.
229 *> \endverbatim
230 *>
231 *> \param[in,out] B
232 *> \verbatim
233 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
234 *> On entry, the N-by-NRHS right hand side matrix B.
235 *> On exit,
236 *> if EQUED = 'N', B is not modified;
237 *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
238 *> diag(R)*B;
239 *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
240 *> overwritten by diag(C)*B.
241 *> \endverbatim
242 *>
243 *> \param[in] LDB
244 *> \verbatim
245 *> LDB is INTEGER
246 *> The leading dimension of the array B. LDB >= max(1,N).
247 *> \endverbatim
248 *>
249 *> \param[out] X
250 *> \verbatim
251 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
252 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
253 *> to the original system of equations. Note that A and B are
254 *> modified on exit if EQUED .ne. 'N', and the solution to the
255 *> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
256 *> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
257 *> and EQUED = 'R' or 'B'.
258 *> \endverbatim
259 *>
260 *> \param[in] LDX
261 *> \verbatim
262 *> LDX is INTEGER
263 *> The leading dimension of the array X. LDX >= max(1,N).
264 *> \endverbatim
265 *>
266 *> \param[out] RCOND
267 *> \verbatim
268 *> RCOND is DOUBLE PRECISION
269 *> The estimate of the reciprocal condition number of the matrix
270 *> A after equilibration (if done). If RCOND is less than the
271 *> machine precision (in particular, if RCOND = 0), the matrix
272 *> is singular to working precision. This condition is
273 *> indicated by a return code of INFO > 0.
274 *> \endverbatim
275 *>
276 *> \param[out] FERR
277 *> \verbatim
278 *> FERR is DOUBLE PRECISION array, dimension (NRHS)
279 *> The estimated forward error bound for each solution vector
280 *> X(j) (the j-th column of the solution matrix X).
281 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
282 *> is an estimated upper bound for the magnitude of the largest
283 *> element in (X(j) - XTRUE) divided by the magnitude of the
284 *> largest element in X(j). The estimate is as reliable as
285 *> the estimate for RCOND, and is almost always a slight
286 *> overestimate of the true error.
287 *> \endverbatim
288 *>
289 *> \param[out] BERR
290 *> \verbatim
291 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
292 *> The componentwise relative backward error of each solution
293 *> vector X(j) (i.e., the smallest relative change in
294 *> any element of A or B that makes X(j) an exact solution).
295 *> \endverbatim
296 *>
297 *> \param[out] WORK
298 *> \verbatim
299 *> WORK is DOUBLE PRECISION array, dimension (4*N)
300 *> On exit, WORK(1) contains the reciprocal pivot growth
301 *> factor norm(A)/norm(U). The "max absolute element" norm is
302 *> used. If WORK(1) is much less than 1, then the stability
303 *> of the LU factorization of the (equilibrated) matrix A
304 *> could be poor. This also means that the solution X, condition
305 *> estimator RCOND, and forward error bound FERR could be
306 *> unreliable. If factorization fails with 0<INFO<=N, then
307 *> WORK(1) contains the reciprocal pivot growth factor for the
308 *> leading INFO columns of A.
309 *> \endverbatim
310 *>
311 *> \param[out] IWORK
312 *> \verbatim
313 *> IWORK is INTEGER array, dimension (N)
314 *> \endverbatim
315 *>
316 *> \param[out] INFO
317 *> \verbatim
318 *> INFO is INTEGER
319 *> = 0: successful exit
320 *> < 0: if INFO = -i, the i-th argument had an illegal value
321 *> > 0: if INFO = i, and i is
322 *> <= N: U(i,i) is exactly zero. The factorization has
323 *> been completed, but the factor U is exactly
324 *> singular, so the solution and error bounds
325 *> could not be computed. RCOND = 0 is returned.
326 *> = N+1: U is nonsingular, but RCOND is less than machine
327 *> precision, meaning that the matrix is singular
328 *> to working precision. Nevertheless, the
329 *> solution and error bounds are computed because
330 *> there are a number of situations where the
331 *> computed solution can be more accurate than the
332 *> value of RCOND would suggest.
333 *> \endverbatim
334 *
335 * Authors:
336 * ========
337 *
338 *> \author Univ. of Tennessee
339 *> \author Univ. of California Berkeley
340 *> \author Univ. of Colorado Denver
341 *> \author NAG Ltd.
342 *
343 *> \date April 2012
344 *
345 *> \ingroup doubleGEsolve
346 *
347 * =====================================================================
348  SUBROUTINE dgesvx( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
349  $ equed, r, c, b, ldb, x, ldx, rcond, ferr, berr,
350  $ work, iwork, info )
351 *
352 * -- LAPACK driver routine (version 3.4.1) --
353 * -- LAPACK is a software package provided by Univ. of Tennessee, --
354 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
355 * April 2012
356 *
357 * .. Scalar Arguments ..
358  CHARACTER EQUED, FACT, TRANS
359  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
360  DOUBLE PRECISION RCOND
361 * ..
362 * .. Array Arguments ..
363  INTEGER IPIV( * ), IWORK( * )
364  DOUBLE PRECISION A( lda, * ), AF( ldaf, * ), B( ldb, * ),
365  $ berr( * ), c( * ), ferr( * ), r( * ),
366  $ work( * ), x( ldx, * )
367 * ..
368 *
369 * =====================================================================
370 *
371 * .. Parameters ..
372  DOUBLE PRECISION ZERO, ONE
373  parameter( zero = 0.0d+0, one = 1.0d+0 )
374 * ..
375 * .. Local Scalars ..
376  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
377  CHARACTER NORM
378  INTEGER I, INFEQU, J
379  DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
380  $ rowcnd, rpvgrw, smlnum
381 * ..
382 * .. External Functions ..
383  LOGICAL LSAME
384  DOUBLE PRECISION DLAMCH, DLANGE, DLANTR
385  EXTERNAL lsame, dlamch, dlange, dlantr
386 * ..
387 * .. External Subroutines ..
388  EXTERNAL dgecon, dgeequ, dgerfs, dgetrf, dgetrs, dlacpy,
389  $ dlaqge, xerbla
390 * ..
391 * .. Intrinsic Functions ..
392  INTRINSIC max, min
393 * ..
394 * .. Executable Statements ..
395 *
396  info = 0
397  nofact = lsame( fact, 'N' )
398  equil = lsame( fact, 'E' )
399  notran = lsame( trans, 'N' )
400  IF( nofact .OR. equil ) THEN
401  equed = 'N'
402  rowequ = .false.
403  colequ = .false.
404  ELSE
405  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
406  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
407  smlnum = dlamch( 'Safe minimum' )
408  bignum = one / smlnum
409  END IF
410 *
411 * Test the input parameters.
412 *
413  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
414  $ THEN
415  info = -1
416  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
417  $ lsame( trans, 'C' ) ) THEN
418  info = -2
419  ELSE IF( n.LT.0 ) THEN
420  info = -3
421  ELSE IF( nrhs.LT.0 ) THEN
422  info = -4
423  ELSE IF( lda.LT.max( 1, n ) ) THEN
424  info = -6
425  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
426  info = -8
427  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
428  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
429  info = -10
430  ELSE
431  IF( rowequ ) THEN
432  rcmin = bignum
433  rcmax = zero
434  DO 10 j = 1, n
435  rcmin = min( rcmin, r( j ) )
436  rcmax = max( rcmax, r( j ) )
437  10 CONTINUE
438  IF( rcmin.LE.zero ) THEN
439  info = -11
440  ELSE IF( n.GT.0 ) THEN
441  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
442  ELSE
443  rowcnd = one
444  END IF
445  END IF
446  IF( colequ .AND. info.EQ.0 ) THEN
447  rcmin = bignum
448  rcmax = zero
449  DO 20 j = 1, n
450  rcmin = min( rcmin, c( j ) )
451  rcmax = max( rcmax, c( j ) )
452  20 CONTINUE
453  IF( rcmin.LE.zero ) THEN
454  info = -12
455  ELSE IF( n.GT.0 ) THEN
456  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
457  ELSE
458  colcnd = one
459  END IF
460  END IF
461  IF( info.EQ.0 ) THEN
462  IF( ldb.LT.max( 1, n ) ) THEN
463  info = -14
464  ELSE IF( ldx.LT.max( 1, n ) ) THEN
465  info = -16
466  END IF
467  END IF
468  END IF
469 *
470  IF( info.NE.0 ) THEN
471  CALL xerbla( 'DGESVX', -info )
472  RETURN
473  END IF
474 *
475  IF( equil ) THEN
476 *
477 * Compute row and column scalings to equilibrate the matrix A.
478 *
479  CALL dgeequ( n, n, a, lda, r, c, rowcnd, colcnd, amax, infequ )
480  IF( infequ.EQ.0 ) THEN
481 *
482 * Equilibrate the matrix.
483 *
484  CALL dlaqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
485  $ equed )
486  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
487  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
488  END IF
489  END IF
490 *
491 * Scale the right hand side.
492 *
493  IF( notran ) THEN
494  IF( rowequ ) THEN
495  DO 40 j = 1, nrhs
496  DO 30 i = 1, n
497  b( i, j ) = r( i )*b( i, j )
498  30 CONTINUE
499  40 CONTINUE
500  END IF
501  ELSE IF( colequ ) THEN
502  DO 60 j = 1, nrhs
503  DO 50 i = 1, n
504  b( i, j ) = c( i )*b( i, j )
505  50 CONTINUE
506  60 CONTINUE
507  END IF
508 *
509  IF( nofact .OR. equil ) THEN
510 *
511 * Compute the LU factorization of A.
512 *
513  CALL dlacpy( 'Full', n, n, a, lda, af, ldaf )
514  CALL dgetrf( n, n, af, ldaf, ipiv, info )
515 *
516 * Return if INFO is non-zero.
517 *
518  IF( info.GT.0 ) THEN
519 *
520 * Compute the reciprocal pivot growth factor of the
521 * leading rank-deficient INFO columns of A.
522 *
523  rpvgrw = dlantr( 'M', 'U', 'N', info, info, af, ldaf,
524  $ work )
525  IF( rpvgrw.EQ.zero ) THEN
526  rpvgrw = one
527  ELSE
528  rpvgrw = dlange( 'M', n, info, a, lda, work ) / rpvgrw
529  END IF
530  work( 1 ) = rpvgrw
531  rcond = zero
532  RETURN
533  END IF
534  END IF
535 *
536 * Compute the norm of the matrix A and the
537 * reciprocal pivot growth factor RPVGRW.
538 *
539  IF( notran ) THEN
540  norm = '1'
541  ELSE
542  norm = 'I'
543  END IF
544  anorm = dlange( norm, n, n, a, lda, work )
545  rpvgrw = dlantr( 'M', 'U', 'N', n, n, af, ldaf, work )
546  IF( rpvgrw.EQ.zero ) THEN
547  rpvgrw = one
548  ELSE
549  rpvgrw = dlange( 'M', n, n, a, lda, work ) / rpvgrw
550  END IF
551 *
552 * Compute the reciprocal of the condition number of A.
553 *
554  CALL dgecon( norm, n, af, ldaf, anorm, rcond, work, iwork, info )
555 *
556 * Compute the solution matrix X.
557 *
558  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
559  CALL dgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
560 *
561 * Use iterative refinement to improve the computed solution and
562 * compute error bounds and backward error estimates for it.
563 *
564  CALL dgerfs( trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
565  $ ldx, ferr, berr, work, iwork, info )
566 *
567 * Transform the solution matrix X to a solution of the original
568 * system.
569 *
570  IF( notran ) THEN
571  IF( colequ ) THEN
572  DO 80 j = 1, nrhs
573  DO 70 i = 1, n
574  x( i, j ) = c( i )*x( i, j )
575  70 CONTINUE
576  80 CONTINUE
577  DO 90 j = 1, nrhs
578  ferr( j ) = ferr( j ) / colcnd
579  90 CONTINUE
580  END IF
581  ELSE IF( rowequ ) THEN
582  DO 110 j = 1, nrhs
583  DO 100 i = 1, n
584  x( i, j ) = r( i )*x( i, j )
585  100 CONTINUE
586  110 CONTINUE
587  DO 120 j = 1, nrhs
588  ferr( j ) = ferr( j ) / rowcnd
589  120 CONTINUE
590  END IF
591 *
592  work( 1 ) = rpvgrw
593 *
594 * Set INFO = N+1 if the matrix is singular to working precision.
595 *
596  IF( rcond.LT.dlamch( 'Epsilon' ) )
597  $ info = n + 1
598  RETURN
599 *
600 * End of DGESVX
601 *
602  END
subroutine dgesvx(FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
DGESVX computes the solution to system of linear equations A * X = B for GE matrices ...
Definition: dgesvx.f:351
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
subroutine dgerfs(TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DGERFS
Definition: dgerfs.f:187
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:123
subroutine dgetrf(M, N, A, LDA, IPIV, INFO)
DGETRF
Definition: dgetrf.f:110
subroutine dgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
DGECON
Definition: dgecon.f:126
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:144
subroutine dgeequ(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
DGEEQU
Definition: dgeequ.f:141