1 SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
2 $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
3 $ WORK, RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, TRANS
12 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
18 $ RWORK( * )
19 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
20 $ WORK( * ), X( LDX, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZGESVX uses the LU factorization to compute the solution to a complex
27 * system of linear equations
28 * A * X = B,
29 * where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
30 *
31 * Error bounds on the solution and a condition estimate are also
32 * provided.
33 *
34 * Description
35 * ===========
36 *
37 * The following steps are performed:
38 *
39 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
40 * the system:
41 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
42 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
43 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
44 * Whether or not the system will be equilibrated depends on the
45 * scaling of the matrix A, but if equilibration is used, A is
46 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
47 * or diag(C)*B (if TRANS = 'T' or 'C').
48 *
49 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
50 * matrix A (after equilibration if FACT = 'E') as
51 * A = P * L * U,
52 * where P is a permutation matrix, L is a unit lower triangular
53 * matrix, and U is upper triangular.
54 *
55 * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
56 * returns with INFO = i. Otherwise, the factored form of A is used
57 * to estimate the condition number of the matrix A. If the
58 * reciprocal of the condition number is less than machine precision,
59 * INFO = N+1 is returned as a warning, but the routine still goes on
60 * to solve for X and compute error bounds as described below.
61 *
62 * 4. The system of equations is solved for X using the factored form
63 * of A.
64 *
65 * 5. Iterative refinement is applied to improve the computed solution
66 * matrix and calculate error bounds and backward error estimates
67 * for it.
68 *
69 * 6. If equilibration was used, the matrix X is premultiplied by
70 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
71 * that it solves the original system before equilibration.
72 *
73 * Arguments
74 * =========
75 *
76 * FACT (input) CHARACTER*1
77 * Specifies whether or not the factored form of the matrix A is
78 * supplied on entry, and if not, whether the matrix A should be
79 * equilibrated before it is factored.
80 * = 'F': On entry, AF and IPIV contain the factored form of A.
81 * If EQUED is not 'N', the matrix A has been
82 * equilibrated with scaling factors given by R and C.
83 * A, AF, and IPIV are not modified.
84 * = 'N': The matrix A will be copied to AF and factored.
85 * = 'E': The matrix A will be equilibrated if necessary, then
86 * copied to AF and factored.
87 *
88 * TRANS (input) CHARACTER*1
89 * Specifies the form of the system of equations:
90 * = 'N': A * X = B (No transpose)
91 * = 'T': A**T * X = B (Transpose)
92 * = 'C': A**H * X = B (Conjugate transpose)
93 *
94 * N (input) INTEGER
95 * The number of linear equations, i.e., the order of the
96 * matrix A. N >= 0.
97 *
98 * NRHS (input) INTEGER
99 * The number of right hand sides, i.e., the number of columns
100 * of the matrices B and X. NRHS >= 0.
101 *
102 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
103 * On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
104 * not 'N', then A must have been equilibrated by the scaling
105 * factors in R and/or C. A is not modified if FACT = 'F' or
106 * 'N', or if FACT = 'E' and EQUED = 'N' on exit.
107 *
108 * On exit, if EQUED .ne. 'N', A is scaled as follows:
109 * EQUED = 'R': A := diag(R) * A
110 * EQUED = 'C': A := A * diag(C)
111 * EQUED = 'B': A := diag(R) * A * diag(C).
112 *
113 * LDA (input) INTEGER
114 * The leading dimension of the array A. LDA >= max(1,N).
115 *
116 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
117 * If FACT = 'F', then AF is an input argument and on entry
118 * contains the factors L and U from the factorization
119 * A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
120 * AF is the factored form of the equilibrated matrix A.
121 *
122 * If FACT = 'N', then AF is an output argument and on exit
123 * returns the factors L and U from the factorization A = P*L*U
124 * of the original matrix A.
125 *
126 * If FACT = 'E', then AF is an output argument and on exit
127 * returns the factors L and U from the factorization A = P*L*U
128 * of the equilibrated matrix A (see the description of A for
129 * the form of the equilibrated matrix).
130 *
131 * LDAF (input) INTEGER
132 * The leading dimension of the array AF. LDAF >= max(1,N).
133 *
134 * IPIV (input or output) INTEGER array, dimension (N)
135 * If FACT = 'F', then IPIV is an input argument and on entry
136 * contains the pivot indices from the factorization A = P*L*U
137 * as computed by ZGETRF; row i of the matrix was interchanged
138 * with row IPIV(i).
139 *
140 * If FACT = 'N', then IPIV is an output argument and on exit
141 * contains the pivot indices from the factorization A = P*L*U
142 * of the original matrix A.
143 *
144 * If FACT = 'E', then IPIV is an output argument and on exit
145 * contains the pivot indices from the factorization A = P*L*U
146 * of the equilibrated matrix A.
147 *
148 * EQUED (input or output) CHARACTER*1
149 * Specifies the form of equilibration that was done.
150 * = 'N': No equilibration (always true if FACT = 'N').
151 * = 'R': Row equilibration, i.e., A has been premultiplied by
152 * diag(R).
153 * = 'C': Column equilibration, i.e., A has been postmultiplied
154 * by diag(C).
155 * = 'B': Both row and column equilibration, i.e., A has been
156 * replaced by diag(R) * A * diag(C).
157 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
158 * output argument.
159 *
160 * R (input or output) DOUBLE PRECISION array, dimension (N)
161 * The row scale factors for A. If EQUED = 'R' or 'B', A is
162 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
163 * is not accessed. R is an input argument if FACT = 'F';
164 * otherwise, R is an output argument. If FACT = 'F' and
165 * EQUED = 'R' or 'B', each element of R must be positive.
166 *
167 * C (input or output) DOUBLE PRECISION array, dimension (N)
168 * The column scale factors for A. If EQUED = 'C' or 'B', A is
169 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
170 * is not accessed. C is an input argument if FACT = 'F';
171 * otherwise, C is an output argument. If FACT = 'F' and
172 * EQUED = 'C' or 'B', each element of C must be positive.
173 *
174 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
175 * On entry, the N-by-NRHS right hand side matrix B.
176 * On exit,
177 * if EQUED = 'N', B is not modified;
178 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
179 * diag(R)*B;
180 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
181 * overwritten by diag(C)*B.
182 *
183 * LDB (input) INTEGER
184 * The leading dimension of the array B. LDB >= max(1,N).
185 *
186 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
187 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
188 * to the original system of equations. Note that A and B are
189 * modified on exit if EQUED .ne. 'N', and the solution to the
190 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
191 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
192 * and EQUED = 'R' or 'B'.
193 *
194 * LDX (input) INTEGER
195 * The leading dimension of the array X. LDX >= max(1,N).
196 *
197 * RCOND (output) DOUBLE PRECISION
198 * The estimate of the reciprocal condition number of the matrix
199 * A after equilibration (if done). If RCOND is less than the
200 * machine precision (in particular, if RCOND = 0), the matrix
201 * is singular to working precision. This condition is
202 * indicated by a return code of INFO > 0.
203 *
204 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
205 * The estimated forward error bound for each solution vector
206 * X(j) (the j-th column of the solution matrix X).
207 * If XTRUE is the true solution corresponding to X(j), FERR(j)
208 * is an estimated upper bound for the magnitude of the largest
209 * element in (X(j) - XTRUE) divided by the magnitude of the
210 * largest element in X(j). The estimate is as reliable as
211 * the estimate for RCOND, and is almost always a slight
212 * overestimate of the true error.
213 *
214 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
215 * The componentwise relative backward error of each solution
216 * vector X(j) (i.e., the smallest relative change in
217 * any element of A or B that makes X(j) an exact solution).
218 *
219 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
220 *
221 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (2*N)
222 * On exit, RWORK(1) contains the reciprocal pivot growth
223 * factor norm(A)/norm(U). The "max absolute element" norm is
224 * used. If RWORK(1) is much less than 1, then the stability
225 * of the LU factorization of the (equilibrated) matrix A
226 * could be poor. This also means that the solution X, condition
227 * estimator RCOND, and forward error bound FERR could be
228 * unreliable. If factorization fails with 0<INFO<=N, then
229 * RWORK(1) contains the reciprocal pivot growth factor for the
230 * leading INFO columns of A.
231 *
232 * INFO (output) INTEGER
233 * = 0: successful exit
234 * < 0: if INFO = -i, the i-th argument had an illegal value
235 * > 0: if INFO = i, and i is
236 * <= N: U(i,i) is exactly zero. The factorization has
237 * been completed, but the factor U is exactly
238 * singular, so the solution and error bounds
239 * could not be computed. RCOND = 0 is returned.
240 * = N+1: U is nonsingular, but RCOND is less than machine
241 * precision, meaning that the matrix is singular
242 * to working precision. Nevertheless, the
243 * solution and error bounds are computed because
244 * there are a number of situations where the
245 * computed solution can be more accurate than the
246 * value of RCOND would suggest.
247 *
248 * =====================================================================
249 *
250 * .. Parameters ..
251 DOUBLE PRECISION ZERO, ONE
252 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
253 * ..
254 * .. Local Scalars ..
255 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
256 CHARACTER NORM
257 INTEGER I, INFEQU, J
258 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
259 $ ROWCND, RPVGRW, SMLNUM
260 * ..
261 * .. External Functions ..
262 LOGICAL LSAME
263 DOUBLE PRECISION DLAMCH, ZLANGE, ZLANTR
264 EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANTR
265 * ..
266 * .. External Subroutines ..
267 EXTERNAL XERBLA, ZGECON, ZGEEQU, ZGERFS, ZGETRF, ZGETRS,
268 $ ZLACPY, ZLAQGE
269 * ..
270 * .. Intrinsic Functions ..
271 INTRINSIC MAX, MIN
272 * ..
273 * .. Executable Statements ..
274 *
275 INFO = 0
276 NOFACT = LSAME( FACT, 'N' )
277 EQUIL = LSAME( FACT, 'E' )
278 NOTRAN = LSAME( TRANS, 'N' )
279 IF( NOFACT .OR. EQUIL ) THEN
280 EQUED = 'N'
281 ROWEQU = .FALSE.
282 COLEQU = .FALSE.
283 ELSE
284 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
285 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
286 SMLNUM = DLAMCH( 'Safe minimum' )
287 BIGNUM = ONE / SMLNUM
288 END IF
289 *
290 * Test the input parameters.
291 *
292 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
293 $ THEN
294 INFO = -1
295 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
296 $ LSAME( TRANS, 'C' ) ) THEN
297 INFO = -2
298 ELSE IF( N.LT.0 ) THEN
299 INFO = -3
300 ELSE IF( NRHS.LT.0 ) THEN
301 INFO = -4
302 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
303 INFO = -6
304 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
305 INFO = -8
306 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
307 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
308 INFO = -10
309 ELSE
310 IF( ROWEQU ) THEN
311 RCMIN = BIGNUM
312 RCMAX = ZERO
313 DO 10 J = 1, N
314 RCMIN = MIN( RCMIN, R( J ) )
315 RCMAX = MAX( RCMAX, R( J ) )
316 10 CONTINUE
317 IF( RCMIN.LE.ZERO ) THEN
318 INFO = -11
319 ELSE IF( N.GT.0 ) THEN
320 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
321 ELSE
322 ROWCND = ONE
323 END IF
324 END IF
325 IF( COLEQU .AND. INFO.EQ.0 ) THEN
326 RCMIN = BIGNUM
327 RCMAX = ZERO
328 DO 20 J = 1, N
329 RCMIN = MIN( RCMIN, C( J ) )
330 RCMAX = MAX( RCMAX, C( J ) )
331 20 CONTINUE
332 IF( RCMIN.LE.ZERO ) THEN
333 INFO = -12
334 ELSE IF( N.GT.0 ) THEN
335 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
336 ELSE
337 COLCND = ONE
338 END IF
339 END IF
340 IF( INFO.EQ.0 ) THEN
341 IF( LDB.LT.MAX( 1, N ) ) THEN
342 INFO = -14
343 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
344 INFO = -16
345 END IF
346 END IF
347 END IF
348 *
349 IF( INFO.NE.0 ) THEN
350 CALL XERBLA( 'ZGESVX', -INFO )
351 RETURN
352 END IF
353 *
354 IF( EQUIL ) THEN
355 *
356 * Compute row and column scalings to equilibrate the matrix A.
357 *
358 CALL ZGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
359 IF( INFEQU.EQ.0 ) THEN
360 *
361 * Equilibrate the matrix.
362 *
363 CALL ZLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
364 $ EQUED )
365 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
366 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
367 END IF
368 END IF
369 *
370 * Scale the right hand side.
371 *
372 IF( NOTRAN ) THEN
373 IF( ROWEQU ) THEN
374 DO 40 J = 1, NRHS
375 DO 30 I = 1, N
376 B( I, J ) = R( I )*B( I, J )
377 30 CONTINUE
378 40 CONTINUE
379 END IF
380 ELSE IF( COLEQU ) THEN
381 DO 60 J = 1, NRHS
382 DO 50 I = 1, N
383 B( I, J ) = C( I )*B( I, J )
384 50 CONTINUE
385 60 CONTINUE
386 END IF
387 *
388 IF( NOFACT .OR. EQUIL ) THEN
389 *
390 * Compute the LU factorization of A.
391 *
392 CALL ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
393 CALL ZGETRF( N, N, AF, LDAF, IPIV, INFO )
394 *
395 * Return if INFO is non-zero.
396 *
397 IF( INFO.GT.0 ) THEN
398 *
399 * Compute the reciprocal pivot growth factor of the
400 * leading rank-deficient INFO columns of A.
401 *
402 RPVGRW = ZLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
403 $ RWORK )
404 IF( RPVGRW.EQ.ZERO ) THEN
405 RPVGRW = ONE
406 ELSE
407 RPVGRW = ZLANGE( 'M', N, INFO, A, LDA, RWORK ) /
408 $ RPVGRW
409 END IF
410 RWORK( 1 ) = RPVGRW
411 RCOND = ZERO
412 RETURN
413 END IF
414 END IF
415 *
416 * Compute the norm of the matrix A and the
417 * reciprocal pivot growth factor RPVGRW.
418 *
419 IF( NOTRAN ) THEN
420 NORM = '1'
421 ELSE
422 NORM = 'I'
423 END IF
424 ANORM = ZLANGE( NORM, N, N, A, LDA, RWORK )
425 RPVGRW = ZLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK )
426 IF( RPVGRW.EQ.ZERO ) THEN
427 RPVGRW = ONE
428 ELSE
429 RPVGRW = ZLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW
430 END IF
431 *
432 * Compute the reciprocal of the condition number of A.
433 *
434 CALL ZGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
435 *
436 * Compute the solution matrix X.
437 *
438 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
439 CALL ZGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
440 *
441 * Use iterative refinement to improve the computed solution and
442 * compute error bounds and backward error estimates for it.
443 *
444 CALL ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
445 $ LDX, FERR, BERR, WORK, RWORK, INFO )
446 *
447 * Transform the solution matrix X to a solution of the original
448 * system.
449 *
450 IF( NOTRAN ) THEN
451 IF( COLEQU ) THEN
452 DO 80 J = 1, NRHS
453 DO 70 I = 1, N
454 X( I, J ) = C( I )*X( I, J )
455 70 CONTINUE
456 80 CONTINUE
457 DO 90 J = 1, NRHS
458 FERR( J ) = FERR( J ) / COLCND
459 90 CONTINUE
460 END IF
461 ELSE IF( ROWEQU ) THEN
462 DO 110 J = 1, NRHS
463 DO 100 I = 1, N
464 X( I, J ) = R( I )*X( I, J )
465 100 CONTINUE
466 110 CONTINUE
467 DO 120 J = 1, NRHS
468 FERR( J ) = FERR( J ) / ROWCND
469 120 CONTINUE
470 END IF
471 *
472 * Set INFO = N+1 if the matrix is singular to working precision.
473 *
474 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
475 $ INFO = N + 1
476 *
477 RWORK( 1 ) = RPVGRW
478 RETURN
479 *
480 * End of ZGESVX
481 *
482 END
2 $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
3 $ WORK, RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, TRANS
12 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
18 $ RWORK( * )
19 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
20 $ WORK( * ), X( LDX, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZGESVX uses the LU factorization to compute the solution to a complex
27 * system of linear equations
28 * A * X = B,
29 * where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
30 *
31 * Error bounds on the solution and a condition estimate are also
32 * provided.
33 *
34 * Description
35 * ===========
36 *
37 * The following steps are performed:
38 *
39 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
40 * the system:
41 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
42 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
43 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
44 * Whether or not the system will be equilibrated depends on the
45 * scaling of the matrix A, but if equilibration is used, A is
46 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
47 * or diag(C)*B (if TRANS = 'T' or 'C').
48 *
49 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
50 * matrix A (after equilibration if FACT = 'E') as
51 * A = P * L * U,
52 * where P is a permutation matrix, L is a unit lower triangular
53 * matrix, and U is upper triangular.
54 *
55 * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
56 * returns with INFO = i. Otherwise, the factored form of A is used
57 * to estimate the condition number of the matrix A. If the
58 * reciprocal of the condition number is less than machine precision,
59 * INFO = N+1 is returned as a warning, but the routine still goes on
60 * to solve for X and compute error bounds as described below.
61 *
62 * 4. The system of equations is solved for X using the factored form
63 * of A.
64 *
65 * 5. Iterative refinement is applied to improve the computed solution
66 * matrix and calculate error bounds and backward error estimates
67 * for it.
68 *
69 * 6. If equilibration was used, the matrix X is premultiplied by
70 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
71 * that it solves the original system before equilibration.
72 *
73 * Arguments
74 * =========
75 *
76 * FACT (input) CHARACTER*1
77 * Specifies whether or not the factored form of the matrix A is
78 * supplied on entry, and if not, whether the matrix A should be
79 * equilibrated before it is factored.
80 * = 'F': On entry, AF and IPIV contain the factored form of A.
81 * If EQUED is not 'N', the matrix A has been
82 * equilibrated with scaling factors given by R and C.
83 * A, AF, and IPIV are not modified.
84 * = 'N': The matrix A will be copied to AF and factored.
85 * = 'E': The matrix A will be equilibrated if necessary, then
86 * copied to AF and factored.
87 *
88 * TRANS (input) CHARACTER*1
89 * Specifies the form of the system of equations:
90 * = 'N': A * X = B (No transpose)
91 * = 'T': A**T * X = B (Transpose)
92 * = 'C': A**H * X = B (Conjugate transpose)
93 *
94 * N (input) INTEGER
95 * The number of linear equations, i.e., the order of the
96 * matrix A. N >= 0.
97 *
98 * NRHS (input) INTEGER
99 * The number of right hand sides, i.e., the number of columns
100 * of the matrices B and X. NRHS >= 0.
101 *
102 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
103 * On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
104 * not 'N', then A must have been equilibrated by the scaling
105 * factors in R and/or C. A is not modified if FACT = 'F' or
106 * 'N', or if FACT = 'E' and EQUED = 'N' on exit.
107 *
108 * On exit, if EQUED .ne. 'N', A is scaled as follows:
109 * EQUED = 'R': A := diag(R) * A
110 * EQUED = 'C': A := A * diag(C)
111 * EQUED = 'B': A := diag(R) * A * diag(C).
112 *
113 * LDA (input) INTEGER
114 * The leading dimension of the array A. LDA >= max(1,N).
115 *
116 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
117 * If FACT = 'F', then AF is an input argument and on entry
118 * contains the factors L and U from the factorization
119 * A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
120 * AF is the factored form of the equilibrated matrix A.
121 *
122 * If FACT = 'N', then AF is an output argument and on exit
123 * returns the factors L and U from the factorization A = P*L*U
124 * of the original matrix A.
125 *
126 * If FACT = 'E', then AF is an output argument and on exit
127 * returns the factors L and U from the factorization A = P*L*U
128 * of the equilibrated matrix A (see the description of A for
129 * the form of the equilibrated matrix).
130 *
131 * LDAF (input) INTEGER
132 * The leading dimension of the array AF. LDAF >= max(1,N).
133 *
134 * IPIV (input or output) INTEGER array, dimension (N)
135 * If FACT = 'F', then IPIV is an input argument and on entry
136 * contains the pivot indices from the factorization A = P*L*U
137 * as computed by ZGETRF; row i of the matrix was interchanged
138 * with row IPIV(i).
139 *
140 * If FACT = 'N', then IPIV is an output argument and on exit
141 * contains the pivot indices from the factorization A = P*L*U
142 * of the original matrix A.
143 *
144 * If FACT = 'E', then IPIV is an output argument and on exit
145 * contains the pivot indices from the factorization A = P*L*U
146 * of the equilibrated matrix A.
147 *
148 * EQUED (input or output) CHARACTER*1
149 * Specifies the form of equilibration that was done.
150 * = 'N': No equilibration (always true if FACT = 'N').
151 * = 'R': Row equilibration, i.e., A has been premultiplied by
152 * diag(R).
153 * = 'C': Column equilibration, i.e., A has been postmultiplied
154 * by diag(C).
155 * = 'B': Both row and column equilibration, i.e., A has been
156 * replaced by diag(R) * A * diag(C).
157 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
158 * output argument.
159 *
160 * R (input or output) DOUBLE PRECISION array, dimension (N)
161 * The row scale factors for A. If EQUED = 'R' or 'B', A is
162 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
163 * is not accessed. R is an input argument if FACT = 'F';
164 * otherwise, R is an output argument. If FACT = 'F' and
165 * EQUED = 'R' or 'B', each element of R must be positive.
166 *
167 * C (input or output) DOUBLE PRECISION array, dimension (N)
168 * The column scale factors for A. If EQUED = 'C' or 'B', A is
169 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
170 * is not accessed. C is an input argument if FACT = 'F';
171 * otherwise, C is an output argument. If FACT = 'F' and
172 * EQUED = 'C' or 'B', each element of C must be positive.
173 *
174 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
175 * On entry, the N-by-NRHS right hand side matrix B.
176 * On exit,
177 * if EQUED = 'N', B is not modified;
178 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
179 * diag(R)*B;
180 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
181 * overwritten by diag(C)*B.
182 *
183 * LDB (input) INTEGER
184 * The leading dimension of the array B. LDB >= max(1,N).
185 *
186 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
187 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
188 * to the original system of equations. Note that A and B are
189 * modified on exit if EQUED .ne. 'N', and the solution to the
190 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
191 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
192 * and EQUED = 'R' or 'B'.
193 *
194 * LDX (input) INTEGER
195 * The leading dimension of the array X. LDX >= max(1,N).
196 *
197 * RCOND (output) DOUBLE PRECISION
198 * The estimate of the reciprocal condition number of the matrix
199 * A after equilibration (if done). If RCOND is less than the
200 * machine precision (in particular, if RCOND = 0), the matrix
201 * is singular to working precision. This condition is
202 * indicated by a return code of INFO > 0.
203 *
204 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
205 * The estimated forward error bound for each solution vector
206 * X(j) (the j-th column of the solution matrix X).
207 * If XTRUE is the true solution corresponding to X(j), FERR(j)
208 * is an estimated upper bound for the magnitude of the largest
209 * element in (X(j) - XTRUE) divided by the magnitude of the
210 * largest element in X(j). The estimate is as reliable as
211 * the estimate for RCOND, and is almost always a slight
212 * overestimate of the true error.
213 *
214 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
215 * The componentwise relative backward error of each solution
216 * vector X(j) (i.e., the smallest relative change in
217 * any element of A or B that makes X(j) an exact solution).
218 *
219 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
220 *
221 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (2*N)
222 * On exit, RWORK(1) contains the reciprocal pivot growth
223 * factor norm(A)/norm(U). The "max absolute element" norm is
224 * used. If RWORK(1) is much less than 1, then the stability
225 * of the LU factorization of the (equilibrated) matrix A
226 * could be poor. This also means that the solution X, condition
227 * estimator RCOND, and forward error bound FERR could be
228 * unreliable. If factorization fails with 0<INFO<=N, then
229 * RWORK(1) contains the reciprocal pivot growth factor for the
230 * leading INFO columns of A.
231 *
232 * INFO (output) INTEGER
233 * = 0: successful exit
234 * < 0: if INFO = -i, the i-th argument had an illegal value
235 * > 0: if INFO = i, and i is
236 * <= N: U(i,i) is exactly zero. The factorization has
237 * been completed, but the factor U is exactly
238 * singular, so the solution and error bounds
239 * could not be computed. RCOND = 0 is returned.
240 * = N+1: U is nonsingular, but RCOND is less than machine
241 * precision, meaning that the matrix is singular
242 * to working precision. Nevertheless, the
243 * solution and error bounds are computed because
244 * there are a number of situations where the
245 * computed solution can be more accurate than the
246 * value of RCOND would suggest.
247 *
248 * =====================================================================
249 *
250 * .. Parameters ..
251 DOUBLE PRECISION ZERO, ONE
252 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
253 * ..
254 * .. Local Scalars ..
255 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
256 CHARACTER NORM
257 INTEGER I, INFEQU, J
258 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
259 $ ROWCND, RPVGRW, SMLNUM
260 * ..
261 * .. External Functions ..
262 LOGICAL LSAME
263 DOUBLE PRECISION DLAMCH, ZLANGE, ZLANTR
264 EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANTR
265 * ..
266 * .. External Subroutines ..
267 EXTERNAL XERBLA, ZGECON, ZGEEQU, ZGERFS, ZGETRF, ZGETRS,
268 $ ZLACPY, ZLAQGE
269 * ..
270 * .. Intrinsic Functions ..
271 INTRINSIC MAX, MIN
272 * ..
273 * .. Executable Statements ..
274 *
275 INFO = 0
276 NOFACT = LSAME( FACT, 'N' )
277 EQUIL = LSAME( FACT, 'E' )
278 NOTRAN = LSAME( TRANS, 'N' )
279 IF( NOFACT .OR. EQUIL ) THEN
280 EQUED = 'N'
281 ROWEQU = .FALSE.
282 COLEQU = .FALSE.
283 ELSE
284 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
285 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
286 SMLNUM = DLAMCH( 'Safe minimum' )
287 BIGNUM = ONE / SMLNUM
288 END IF
289 *
290 * Test the input parameters.
291 *
292 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
293 $ THEN
294 INFO = -1
295 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
296 $ LSAME( TRANS, 'C' ) ) THEN
297 INFO = -2
298 ELSE IF( N.LT.0 ) THEN
299 INFO = -3
300 ELSE IF( NRHS.LT.0 ) THEN
301 INFO = -4
302 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
303 INFO = -6
304 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
305 INFO = -8
306 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
307 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
308 INFO = -10
309 ELSE
310 IF( ROWEQU ) THEN
311 RCMIN = BIGNUM
312 RCMAX = ZERO
313 DO 10 J = 1, N
314 RCMIN = MIN( RCMIN, R( J ) )
315 RCMAX = MAX( RCMAX, R( J ) )
316 10 CONTINUE
317 IF( RCMIN.LE.ZERO ) THEN
318 INFO = -11
319 ELSE IF( N.GT.0 ) THEN
320 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
321 ELSE
322 ROWCND = ONE
323 END IF
324 END IF
325 IF( COLEQU .AND. INFO.EQ.0 ) THEN
326 RCMIN = BIGNUM
327 RCMAX = ZERO
328 DO 20 J = 1, N
329 RCMIN = MIN( RCMIN, C( J ) )
330 RCMAX = MAX( RCMAX, C( J ) )
331 20 CONTINUE
332 IF( RCMIN.LE.ZERO ) THEN
333 INFO = -12
334 ELSE IF( N.GT.0 ) THEN
335 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
336 ELSE
337 COLCND = ONE
338 END IF
339 END IF
340 IF( INFO.EQ.0 ) THEN
341 IF( LDB.LT.MAX( 1, N ) ) THEN
342 INFO = -14
343 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
344 INFO = -16
345 END IF
346 END IF
347 END IF
348 *
349 IF( INFO.NE.0 ) THEN
350 CALL XERBLA( 'ZGESVX', -INFO )
351 RETURN
352 END IF
353 *
354 IF( EQUIL ) THEN
355 *
356 * Compute row and column scalings to equilibrate the matrix A.
357 *
358 CALL ZGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
359 IF( INFEQU.EQ.0 ) THEN
360 *
361 * Equilibrate the matrix.
362 *
363 CALL ZLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
364 $ EQUED )
365 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
366 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
367 END IF
368 END IF
369 *
370 * Scale the right hand side.
371 *
372 IF( NOTRAN ) THEN
373 IF( ROWEQU ) THEN
374 DO 40 J = 1, NRHS
375 DO 30 I = 1, N
376 B( I, J ) = R( I )*B( I, J )
377 30 CONTINUE
378 40 CONTINUE
379 END IF
380 ELSE IF( COLEQU ) THEN
381 DO 60 J = 1, NRHS
382 DO 50 I = 1, N
383 B( I, J ) = C( I )*B( I, J )
384 50 CONTINUE
385 60 CONTINUE
386 END IF
387 *
388 IF( NOFACT .OR. EQUIL ) THEN
389 *
390 * Compute the LU factorization of A.
391 *
392 CALL ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
393 CALL ZGETRF( N, N, AF, LDAF, IPIV, INFO )
394 *
395 * Return if INFO is non-zero.
396 *
397 IF( INFO.GT.0 ) THEN
398 *
399 * Compute the reciprocal pivot growth factor of the
400 * leading rank-deficient INFO columns of A.
401 *
402 RPVGRW = ZLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
403 $ RWORK )
404 IF( RPVGRW.EQ.ZERO ) THEN
405 RPVGRW = ONE
406 ELSE
407 RPVGRW = ZLANGE( 'M', N, INFO, A, LDA, RWORK ) /
408 $ RPVGRW
409 END IF
410 RWORK( 1 ) = RPVGRW
411 RCOND = ZERO
412 RETURN
413 END IF
414 END IF
415 *
416 * Compute the norm of the matrix A and the
417 * reciprocal pivot growth factor RPVGRW.
418 *
419 IF( NOTRAN ) THEN
420 NORM = '1'
421 ELSE
422 NORM = 'I'
423 END IF
424 ANORM = ZLANGE( NORM, N, N, A, LDA, RWORK )
425 RPVGRW = ZLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK )
426 IF( RPVGRW.EQ.ZERO ) THEN
427 RPVGRW = ONE
428 ELSE
429 RPVGRW = ZLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW
430 END IF
431 *
432 * Compute the reciprocal of the condition number of A.
433 *
434 CALL ZGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
435 *
436 * Compute the solution matrix X.
437 *
438 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
439 CALL ZGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
440 *
441 * Use iterative refinement to improve the computed solution and
442 * compute error bounds and backward error estimates for it.
443 *
444 CALL ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
445 $ LDX, FERR, BERR, WORK, RWORK, INFO )
446 *
447 * Transform the solution matrix X to a solution of the original
448 * system.
449 *
450 IF( NOTRAN ) THEN
451 IF( COLEQU ) THEN
452 DO 80 J = 1, NRHS
453 DO 70 I = 1, N
454 X( I, J ) = C( I )*X( I, J )
455 70 CONTINUE
456 80 CONTINUE
457 DO 90 J = 1, NRHS
458 FERR( J ) = FERR( J ) / COLCND
459 90 CONTINUE
460 END IF
461 ELSE IF( ROWEQU ) THEN
462 DO 110 J = 1, NRHS
463 DO 100 I = 1, N
464 X( I, J ) = R( I )*X( I, J )
465 100 CONTINUE
466 110 CONTINUE
467 DO 120 J = 1, NRHS
468 FERR( J ) = FERR( J ) / ROWCND
469 120 CONTINUE
470 END IF
471 *
472 * Set INFO = N+1 if the matrix is singular to working precision.
473 *
474 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
475 $ INFO = N + 1
476 *
477 RWORK( 1 ) = RPVGRW
478 RETURN
479 *
480 * End of ZGESVX
481 *
482 END