Annotation of rpl/lapack/lapack/dgesvx.f, revision 1.10
1.8 bertrand 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
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvx.f">
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
1.10 ! bertrand 157: *> AF is DOUBLE PRECISION array, dimension (LDAF,N)
1.8 bertrand 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
1.10 ! bertrand 181: *> IPIV is INTEGER array, dimension (N)
1.8 bertrand 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
1.10 ! bertrand 198: *> EQUED is CHARACTER*1
1.8 bertrand 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
1.10 ! bertrand 213: *> R is DOUBLE PRECISION array, dimension (N)
1.8 bertrand 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
1.10 ! bertrand 223: *> C is DOUBLE PRECISION array, dimension (N)
1.8 bertrand 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: *
1.10 ! bertrand 343: *> \date April 2012
1.8 bertrand 344: *
345: *> \ingroup doubleGEsolve
346: *
347: * =====================================================================
1.1 bertrand 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: *
1.10 ! bertrand 352: * -- LAPACK driver routine (version 3.4.1) --
1.1 bertrand 353: * -- LAPACK is a software package provided by Univ. of Tennessee, --
354: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.10 ! bertrand 355: * April 2012
1.1 bertrand 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
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