Annotation of rpl/lapack/lapack/zgesvx.f, revision 1.17
1.8 bertrand 1: *> \brief <b> ZGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.8 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 9: *> Download ZGESVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgesvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgesvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgesvx.f">
1.8 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.8 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
22: * EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
23: * WORK, RWORK, INFO )
1.15 bertrand 24: *
1.8 bertrand 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( * )
32: * DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
33: * $ RWORK( * )
34: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35: * $ WORK( * ), X( LDX, * )
36: * ..
1.15 bertrand 37: *
1.8 bertrand 38: *
39: *> \par Purpose:
40: * =============
41: *>
42: *> \verbatim
43: *>
44: *> ZGESVX uses the LU factorization to compute the solution to a complex
45: *> system of linear equations
46: *> A * X = B,
47: *> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
48: *>
49: *> Error bounds on the solution and a condition estimate are also
50: *> provided.
51: *> \endverbatim
52: *
53: *> \par Description:
54: * =================
55: *>
56: *> \verbatim
57: *>
58: *> The following steps are performed:
59: *>
60: *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
61: *> the system:
62: *> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
63: *> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
64: *> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
65: *> Whether or not the system will be equilibrated depends on the
66: *> scaling of the matrix A, but if equilibration is used, A is
67: *> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
68: *> or diag(C)*B (if TRANS = 'T' or 'C').
69: *>
70: *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
71: *> matrix A (after equilibration if FACT = 'E') as
72: *> A = P * L * U,
73: *> where P is a permutation matrix, L is a unit lower triangular
74: *> matrix, and U is upper triangular.
75: *>
76: *> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
77: *> returns with INFO = i. Otherwise, the factored form of A is used
78: *> to estimate the condition number of the matrix A. If the
79: *> reciprocal of the condition number is less than machine precision,
80: *> INFO = N+1 is returned as a warning, but the routine still goes on
81: *> to solve for X and compute error bounds as described below.
82: *>
83: *> 4. The system of equations is solved for X using the factored form
84: *> of A.
85: *>
86: *> 5. Iterative refinement is applied to improve the computed solution
87: *> matrix and calculate error bounds and backward error estimates
88: *> for it.
89: *>
90: *> 6. If equilibration was used, the matrix X is premultiplied by
91: *> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
92: *> that it solves the original system before equilibration.
93: *> \endverbatim
94: *
95: * Arguments:
96: * ==========
97: *
98: *> \param[in] FACT
99: *> \verbatim
100: *> FACT is CHARACTER*1
101: *> Specifies whether or not the factored form of the matrix A is
102: *> supplied on entry, and if not, whether the matrix A should be
103: *> equilibrated before it is factored.
104: *> = 'F': On entry, AF and IPIV contain the factored form of A.
105: *> If EQUED is not 'N', the matrix A has been
106: *> equilibrated with scaling factors given by R and C.
107: *> A, AF, and IPIV are not modified.
108: *> = 'N': The matrix A will be copied to AF and factored.
109: *> = 'E': The matrix A will be equilibrated if necessary, then
110: *> copied to AF and factored.
111: *> \endverbatim
112: *>
113: *> \param[in] TRANS
114: *> \verbatim
115: *> TRANS is CHARACTER*1
116: *> Specifies the form of the system of equations:
117: *> = 'N': A * X = B (No transpose)
118: *> = 'T': A**T * X = B (Transpose)
119: *> = 'C': A**H * X = B (Conjugate transpose)
120: *> \endverbatim
121: *>
122: *> \param[in] N
123: *> \verbatim
124: *> N is INTEGER
125: *> The number of linear equations, i.e., the order of the
126: *> matrix A. N >= 0.
127: *> \endverbatim
128: *>
129: *> \param[in] NRHS
130: *> \verbatim
131: *> NRHS is INTEGER
132: *> The number of right hand sides, i.e., the number of columns
133: *> of the matrices B and X. NRHS >= 0.
134: *> \endverbatim
135: *>
136: *> \param[in,out] A
137: *> \verbatim
138: *> A is COMPLEX*16 array, dimension (LDA,N)
139: *> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
140: *> not 'N', then A must have been equilibrated by the scaling
141: *> factors in R and/or C. A is not modified if FACT = 'F' or
142: *> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
143: *>
144: *> On exit, if EQUED .ne. 'N', A is scaled as follows:
145: *> EQUED = 'R': A := diag(R) * A
146: *> EQUED = 'C': A := A * diag(C)
147: *> EQUED = 'B': A := diag(R) * A * diag(C).
148: *> \endverbatim
149: *>
150: *> \param[in] LDA
151: *> \verbatim
152: *> LDA is INTEGER
153: *> The leading dimension of the array A. LDA >= max(1,N).
154: *> \endverbatim
155: *>
156: *> \param[in,out] AF
157: *> \verbatim
1.10 bertrand 158: *> AF is COMPLEX*16 array, dimension (LDAF,N)
1.8 bertrand 159: *> If FACT = 'F', then AF is an input argument and on entry
160: *> contains the factors L and U from the factorization
161: *> A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
162: *> AF is the factored form of the equilibrated matrix A.
163: *>
164: *> If FACT = 'N', then AF is an output argument and on exit
165: *> returns the factors L and U from the factorization A = P*L*U
166: *> of the original matrix A.
167: *>
168: *> If FACT = 'E', then AF is an output argument and on exit
169: *> returns the factors L and U from the factorization A = P*L*U
170: *> of the equilibrated matrix A (see the description of A for
171: *> the form of the equilibrated matrix).
172: *> \endverbatim
173: *>
174: *> \param[in] LDAF
175: *> \verbatim
176: *> LDAF is INTEGER
177: *> The leading dimension of the array AF. LDAF >= max(1,N).
178: *> \endverbatim
179: *>
180: *> \param[in,out] IPIV
181: *> \verbatim
1.10 bertrand 182: *> IPIV is INTEGER array, dimension (N)
1.8 bertrand 183: *> If FACT = 'F', then IPIV is an input argument and on entry
184: *> contains the pivot indices from the factorization A = P*L*U
185: *> as computed by ZGETRF; row i of the matrix was interchanged
186: *> with row IPIV(i).
187: *>
188: *> If FACT = 'N', then IPIV is an output argument and on exit
189: *> contains the pivot indices from the factorization A = P*L*U
190: *> of the original matrix A.
191: *>
192: *> If FACT = 'E', then IPIV is an output argument and on exit
193: *> contains the pivot indices from the factorization A = P*L*U
194: *> of the equilibrated matrix A.
195: *> \endverbatim
196: *>
197: *> \param[in,out] EQUED
198: *> \verbatim
1.10 bertrand 199: *> EQUED is CHARACTER*1
1.8 bertrand 200: *> Specifies the form of equilibration that was done.
201: *> = 'N': No equilibration (always true if FACT = 'N').
202: *> = 'R': Row equilibration, i.e., A has been premultiplied by
203: *> diag(R).
204: *> = 'C': Column equilibration, i.e., A has been postmultiplied
205: *> by diag(C).
206: *> = 'B': Both row and column equilibration, i.e., A has been
207: *> replaced by diag(R) * A * diag(C).
208: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
209: *> output argument.
210: *> \endverbatim
211: *>
212: *> \param[in,out] R
213: *> \verbatim
1.10 bertrand 214: *> R is DOUBLE PRECISION array, dimension (N)
1.8 bertrand 215: *> The row scale factors for A. If EQUED = 'R' or 'B', A is
216: *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
217: *> is not accessed. R is an input argument if FACT = 'F';
218: *> otherwise, R is an output argument. If FACT = 'F' and
219: *> EQUED = 'R' or 'B', each element of R must be positive.
220: *> \endverbatim
221: *>
222: *> \param[in,out] C
223: *> \verbatim
1.10 bertrand 224: *> C is DOUBLE PRECISION array, dimension (N)
1.8 bertrand 225: *> The column scale factors for A. If EQUED = 'C' or 'B', A is
226: *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
227: *> is not accessed. C is an input argument if FACT = 'F';
228: *> otherwise, C is an output argument. If FACT = 'F' and
229: *> EQUED = 'C' or 'B', each element of C must be positive.
230: *> \endverbatim
231: *>
232: *> \param[in,out] B
233: *> \verbatim
234: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
235: *> On entry, the N-by-NRHS right hand side matrix B.
236: *> On exit,
237: *> if EQUED = 'N', B is not modified;
238: *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
239: *> diag(R)*B;
240: *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
241: *> overwritten by diag(C)*B.
242: *> \endverbatim
243: *>
244: *> \param[in] LDB
245: *> \verbatim
246: *> LDB is INTEGER
247: *> The leading dimension of the array B. LDB >= max(1,N).
248: *> \endverbatim
249: *>
250: *> \param[out] X
251: *> \verbatim
252: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
253: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
254: *> to the original system of equations. Note that A and B are
255: *> modified on exit if EQUED .ne. 'N', and the solution to the
256: *> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
257: *> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
258: *> and EQUED = 'R' or 'B'.
259: *> \endverbatim
260: *>
261: *> \param[in] LDX
262: *> \verbatim
263: *> LDX is INTEGER
264: *> The leading dimension of the array X. LDX >= max(1,N).
265: *> \endverbatim
266: *>
267: *> \param[out] RCOND
268: *> \verbatim
269: *> RCOND is DOUBLE PRECISION
270: *> The estimate of the reciprocal condition number of the matrix
271: *> A after equilibration (if done). If RCOND is less than the
272: *> machine precision (in particular, if RCOND = 0), the matrix
273: *> is singular to working precision. This condition is
274: *> indicated by a return code of INFO > 0.
275: *> \endverbatim
276: *>
277: *> \param[out] FERR
278: *> \verbatim
279: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
280: *> The estimated forward error bound for each solution vector
281: *> X(j) (the j-th column of the solution matrix X).
282: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
283: *> is an estimated upper bound for the magnitude of the largest
284: *> element in (X(j) - XTRUE) divided by the magnitude of the
285: *> largest element in X(j). The estimate is as reliable as
286: *> the estimate for RCOND, and is almost always a slight
287: *> overestimate of the true error.
288: *> \endverbatim
289: *>
290: *> \param[out] BERR
291: *> \verbatim
292: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
293: *> The componentwise relative backward error of each solution
294: *> vector X(j) (i.e., the smallest relative change in
295: *> any element of A or B that makes X(j) an exact solution).
296: *> \endverbatim
297: *>
298: *> \param[out] WORK
299: *> \verbatim
300: *> WORK is COMPLEX*16 array, dimension (2*N)
301: *> \endverbatim
302: *>
303: *> \param[out] RWORK
304: *> \verbatim
305: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
306: *> On exit, RWORK(1) contains the reciprocal pivot growth
307: *> factor norm(A)/norm(U). The "max absolute element" norm is
308: *> used. If RWORK(1) is much less than 1, then the stability
309: *> of the LU factorization of the (equilibrated) matrix A
310: *> could be poor. This also means that the solution X, condition
311: *> estimator RCOND, and forward error bound FERR could be
312: *> unreliable. If factorization fails with 0<INFO<=N, then
313: *> RWORK(1) contains the reciprocal pivot growth factor for the
314: *> leading INFO columns of A.
315: *> \endverbatim
316: *>
317: *> \param[out] INFO
318: *> \verbatim
319: *> INFO is INTEGER
320: *> = 0: successful exit
321: *> < 0: if INFO = -i, the i-th argument had an illegal value
322: *> > 0: if INFO = i, and i is
323: *> <= N: U(i,i) is exactly zero. The factorization has
324: *> been completed, but the factor U is exactly
325: *> singular, so the solution and error bounds
326: *> could not be computed. RCOND = 0 is returned.
327: *> = N+1: U is nonsingular, but RCOND is less than machine
328: *> precision, meaning that the matrix is singular
329: *> to working precision. Nevertheless, the
330: *> solution and error bounds are computed because
331: *> there are a number of situations where the
332: *> computed solution can be more accurate than the
333: *> value of RCOND would suggest.
334: *> \endverbatim
335: *
336: * Authors:
337: * ========
338: *
1.15 bertrand 339: *> \author Univ. of Tennessee
340: *> \author Univ. of California Berkeley
341: *> \author Univ. of Colorado Denver
342: *> \author NAG Ltd.
1.8 bertrand 343: *
1.10 bertrand 344: *> \date April 2012
1.8 bertrand 345: *
346: *> \ingroup complex16GEsolve
347: *
348: * =====================================================================
1.1 bertrand 349: SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
350: $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
351: $ WORK, RWORK, INFO )
352: *
1.15 bertrand 353: * -- LAPACK driver routine (version 3.7.0) --
1.1 bertrand 354: * -- LAPACK is a software package provided by Univ. of Tennessee, --
355: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.10 bertrand 356: * April 2012
1.1 bertrand 357: *
358: * .. Scalar Arguments ..
359: CHARACTER EQUED, FACT, TRANS
360: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
361: DOUBLE PRECISION RCOND
362: * ..
363: * .. Array Arguments ..
364: INTEGER IPIV( * )
365: DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
366: $ RWORK( * )
367: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
368: $ WORK( * ), X( LDX, * )
369: * ..
370: *
371: * =====================================================================
372: *
373: * .. Parameters ..
374: DOUBLE PRECISION ZERO, ONE
375: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
376: * ..
377: * .. Local Scalars ..
378: LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
379: CHARACTER NORM
380: INTEGER I, INFEQU, J
381: DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
382: $ ROWCND, RPVGRW, SMLNUM
383: * ..
384: * .. External Functions ..
385: LOGICAL LSAME
386: DOUBLE PRECISION DLAMCH, ZLANGE, ZLANTR
387: EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANTR
388: * ..
389: * .. External Subroutines ..
390: EXTERNAL XERBLA, ZGECON, ZGEEQU, ZGERFS, ZGETRF, ZGETRS,
391: $ ZLACPY, ZLAQGE
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 = DLAMCH( '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( 'ZGESVX', -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 ZGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
482: IF( INFEQU.EQ.0 ) THEN
483: *
484: * Equilibrate the matrix.
485: *
486: CALL ZLAQGE( 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 ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
516: CALL ZGETRF( 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 = ZLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
526: $ RWORK )
527: IF( RPVGRW.EQ.ZERO ) THEN
528: RPVGRW = ONE
529: ELSE
530: RPVGRW = ZLANGE( '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 = ZLANGE( NORM, N, N, A, LDA, RWORK )
548: RPVGRW = ZLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK )
549: IF( RPVGRW.EQ.ZERO ) THEN
550: RPVGRW = ONE
551: ELSE
552: RPVGRW = ZLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW
553: END IF
554: *
555: * Compute the reciprocal of the condition number of A.
556: *
557: CALL ZGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
558: *
559: * Compute the solution matrix X.
560: *
561: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
562: CALL ZGETRS( 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 ZGERFS( 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.DLAMCH( 'Epsilon' ) )
598: $ INFO = N + 1
599: *
600: RWORK( 1 ) = RPVGRW
601: RETURN
602: *
603: * End of ZGESVX
604: *
605: END
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