1 SUBROUTINE ZGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
2 $ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
3 $ BERR, N_ERR_BNDS, ERR_BNDS_NORM,
4 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
5 $ INFO )
6 *
7 * -- LAPACK driver routine (version 3.2.1) --
8 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
9 * -- Jason Riedy of Univ. of California Berkeley. --
10 * -- April 2009 --
11 *
12 * -- LAPACK is a software package provided by Univ. of Tennessee, --
13 * -- Univ. of California Berkeley and NAG Ltd. --
14 *
15 IMPLICIT NONE
16 * ..
17 * .. Scalar Arguments ..
18 CHARACTER EQUED, FACT, TRANS
19 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
20 $ N_ERR_BNDS
21 DOUBLE PRECISION RCOND, RPVGRW
22 * ..
23 * .. Array Arguments ..
24 INTEGER IPIV( * )
25 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
26 $ X( LDX , * ),WORK( * )
27 DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
28 $ ERR_BNDS_NORM( NRHS, * ),
29 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
30 * ..
31 *
32 * Purpose
33 * =======
34 *
35 * ZGESVXX uses the LU factorization to compute the solution to a
36 * complex*16 system of linear equations A * X = B, where A is an
37 * N-by-N matrix and X and B are N-by-NRHS matrices.
38 *
39 * If requested, both normwise and maximum componentwise error bounds
40 * are returned. ZGESVXX will return a solution with a tiny
41 * guaranteed error (O(eps) where eps is the working machine
42 * precision) unless the matrix is very ill-conditioned, in which
43 * case a warning is returned. Relevant condition numbers also are
44 * calculated and returned.
45 *
46 * ZGESVXX accepts user-provided factorizations and equilibration
47 * factors; see the definitions of the FACT and EQUED options.
48 * Solving with refinement and using a factorization from a previous
49 * ZGESVXX call will also produce a solution with either O(eps)
50 * errors or warnings, but we cannot make that claim for general
51 * user-provided factorizations and equilibration factors if they
52 * differ from what ZGESVXX would itself produce.
53 *
54 * Description
55 * ===========
56 *
57 * The following steps are performed:
58 *
59 * 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
60 * the system:
61 *
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 *
66 * Whether or not the system will be equilibrated depends on the
67 * scaling of the matrix A, but if equilibration is used, A is
68 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
69 * or diag(C)*B (if TRANS = 'T' or 'C').
70 *
71 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
72 * the matrix A (after equilibration if FACT = 'E') as
73 *
74 * A = P * L * U,
75 *
76 * where P is a permutation matrix, L is a unit lower triangular
77 * matrix, and U is upper triangular.
78 *
79 * 3. If some U(i,i)=0, so that U is exactly singular, then the
80 * routine returns with INFO = i. Otherwise, the factored form of A
81 * is used to estimate the condition number of the matrix A (see
82 * argument RCOND). If the reciprocal of the condition number is less
83 * than machine precision, the routine still goes on to solve for X
84 * and compute error bounds as described below.
85 *
86 * 4. The system of equations is solved for X using the factored form
87 * of A.
88 *
89 * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
90 * the routine will use iterative refinement to try to get a small
91 * error and error bounds. Refinement calculates the residual to at
92 * least twice the working precision.
93 *
94 * 6. If equilibration was used, the matrix X is premultiplied by
95 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
96 * that it solves the original system before equilibration.
97 *
98 * Arguments
99 * =========
100 *
101 * Some optional parameters are bundled in the PARAMS array. These
102 * settings determine how refinement is performed, but often the
103 * defaults are acceptable. If the defaults are acceptable, users
104 * can pass NPARAMS = 0 which prevents the source code from accessing
105 * the PARAMS argument.
106 *
107 * FACT (input) CHARACTER*1
108 * Specifies whether or not the factored form of the matrix A is
109 * supplied on entry, and if not, whether the matrix A should be
110 * equilibrated before it is factored.
111 * = 'F': On entry, AF and IPIV contain the factored form of A.
112 * If EQUED is not 'N', the matrix A has been
113 * equilibrated with scaling factors given by R and C.
114 * A, AF, and IPIV are not modified.
115 * = 'N': The matrix A will be copied to AF and factored.
116 * = 'E': The matrix A will be equilibrated if necessary, then
117 * copied to AF and factored.
118 *
119 * TRANS (input) CHARACTER*1
120 * Specifies the form of the system of equations:
121 * = 'N': A * X = B (No transpose)
122 * = 'T': A**T * X = B (Transpose)
123 * = 'C': A**H * X = B (Conjugate Transpose)
124 *
125 * N (input) INTEGER
126 * The number of linear equations, i.e., the order of the
127 * matrix A. N >= 0.
128 *
129 * NRHS (input) INTEGER
130 * The number of right hand sides, i.e., the number of columns
131 * of the matrices B and X. NRHS >= 0.
132 *
133 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
134 * On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
135 * not 'N', then A must have been equilibrated by the scaling
136 * factors in R and/or C. A is not modified if FACT = 'F' or
137 * 'N', or if FACT = 'E' and EQUED = 'N' on exit.
138 *
139 * On exit, if EQUED .ne. 'N', A is scaled as follows:
140 * EQUED = 'R': A := diag(R) * A
141 * EQUED = 'C': A := A * diag(C)
142 * EQUED = 'B': A := diag(R) * A * diag(C).
143 *
144 * LDA (input) INTEGER
145 * The leading dimension of the array A. LDA >= max(1,N).
146 *
147 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
148 * If FACT = 'F', then AF is an input argument and on entry
149 * contains the factors L and U from the factorization
150 * A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
151 * AF is the factored form of the equilibrated matrix A.
152 *
153 * If FACT = 'N', then AF is an output argument and on exit
154 * returns the factors L and U from the factorization A = P*L*U
155 * of the original matrix A.
156 *
157 * If FACT = 'E', then AF is an output argument and on exit
158 * returns the factors L and U from the factorization A = P*L*U
159 * of the equilibrated matrix A (see the description of A for
160 * the form of the equilibrated matrix).
161 *
162 * LDAF (input) INTEGER
163 * The leading dimension of the array AF. LDAF >= max(1,N).
164 *
165 * IPIV (input or output) INTEGER array, dimension (N)
166 * If FACT = 'F', then IPIV is an input argument and on entry
167 * contains the pivot indices from the factorization A = P*L*U
168 * as computed by ZGETRF; row i of the matrix was interchanged
169 * with row IPIV(i).
170 *
171 * If FACT = 'N', then IPIV is an output argument and on exit
172 * contains the pivot indices from the factorization A = P*L*U
173 * of the original matrix A.
174 *
175 * If FACT = 'E', then IPIV is an output argument and on exit
176 * contains the pivot indices from the factorization A = P*L*U
177 * of the equilibrated matrix A.
178 *
179 * EQUED (input or output) CHARACTER*1
180 * Specifies the form of equilibration that was done.
181 * = 'N': No equilibration (always true if FACT = 'N').
182 * = 'R': Row equilibration, i.e., A has been premultiplied by
183 * diag(R).
184 * = 'C': Column equilibration, i.e., A has been postmultiplied
185 * by diag(C).
186 * = 'B': Both row and column equilibration, i.e., A has been
187 * replaced by diag(R) * A * diag(C).
188 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
189 * output argument.
190 *
191 * R (input or output) DOUBLE PRECISION array, dimension (N)
192 * The row scale factors for A. If EQUED = 'R' or 'B', A is
193 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
194 * is not accessed. R is an input argument if FACT = 'F';
195 * otherwise, R is an output argument. If FACT = 'F' and
196 * EQUED = 'R' or 'B', each element of R must be positive.
197 * If R is output, each element of R is a power of the radix.
198 * If R is input, each element of R should be a power of the radix
199 * to ensure a reliable solution and error estimates. Scaling by
200 * powers of the radix does not cause rounding errors unless the
201 * result underflows or overflows. Rounding errors during scaling
202 * lead to refining with a matrix that is not equivalent to the
203 * input matrix, producing error estimates that may not be
204 * reliable.
205 *
206 * C (input or output) DOUBLE PRECISION array, dimension (N)
207 * The column scale factors for A. If EQUED = 'C' or 'B', A is
208 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
209 * is not accessed. C is an input argument if FACT = 'F';
210 * otherwise, C is an output argument. If FACT = 'F' and
211 * EQUED = 'C' or 'B', each element of C must be positive.
212 * If C is output, each element of C is a power of the radix.
213 * If C is input, each element of C should be a power of the radix
214 * to ensure a reliable solution and error estimates. Scaling by
215 * powers of the radix does not cause rounding errors unless the
216 * result underflows or overflows. Rounding errors during scaling
217 * lead to refining with a matrix that is not equivalent to the
218 * input matrix, producing error estimates that may not be
219 * reliable.
220 *
221 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
222 * On entry, the N-by-NRHS right hand side matrix B.
223 * On exit,
224 * if EQUED = 'N', B is not modified;
225 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
226 * diag(R)*B;
227 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
228 * overwritten by diag(C)*B.
229 *
230 * LDB (input) INTEGER
231 * The leading dimension of the array B. LDB >= max(1,N).
232 *
233 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
234 * If INFO = 0, the N-by-NRHS solution matrix X to the original
235 * system of equations. Note that A and B are modified on exit
236 * if EQUED .ne. 'N', and the solution to the equilibrated system is
237 * inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
238 * inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
239 *
240 * LDX (input) INTEGER
241 * The leading dimension of the array X. LDX >= max(1,N).
242 *
243 * RCOND (output) DOUBLE PRECISION
244 * Reciprocal scaled condition number. This is an estimate of the
245 * reciprocal Skeel condition number of the matrix A after
246 * equilibration (if done). If this is less than the machine
247 * precision (in particular, if it is zero), the matrix is singular
248 * to working precision. Note that the error may still be small even
249 * if this number is very small and the matrix appears ill-
250 * conditioned.
251 *
252 * RPVGRW (output) DOUBLE PRECISION
253 * Reciprocal pivot growth. On exit, this contains the reciprocal
254 * pivot growth factor norm(A)/norm(U). The "max absolute element"
255 * norm is used. If this is much less than 1, then the stability of
256 * the LU factorization of the (equilibrated) matrix A could be poor.
257 * This also means that the solution X, estimated condition numbers,
258 * and error bounds could be unreliable. If factorization fails with
259 * 0<INFO<=N, then this contains the reciprocal pivot growth factor
260 * for the leading INFO columns of A. In ZGESVX, this quantity is
261 * returned in WORK(1).
262 *
263 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
264 * Componentwise relative backward error. This is the
265 * componentwise relative backward error of each solution vector X(j)
266 * (i.e., the smallest relative change in any element of A or B that
267 * makes X(j) an exact solution).
268 *
269 * N_ERR_BNDS (input) INTEGER
270 * Number of error bounds to return for each right hand side
271 * and each type (normwise or componentwise). See ERR_BNDS_NORM and
272 * ERR_BNDS_COMP below.
273 *
274 * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
275 * For each right-hand side, this array contains information about
276 * various error bounds and condition numbers corresponding to the
277 * normwise relative error, which is defined as follows:
278 *
279 * Normwise relative error in the ith solution vector:
280 * max_j (abs(XTRUE(j,i) - X(j,i)))
281 * ------------------------------
282 * max_j abs(X(j,i))
283 *
284 * The array is indexed by the type of error information as described
285 * below. There currently are up to three pieces of information
286 * returned.
287 *
288 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
289 * right-hand side.
290 *
291 * The second index in ERR_BNDS_NORM(:,err) contains the following
292 * three fields:
293 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
294 * reciprocal condition number is less than the threshold
295 * sqrt(n) * dlamch('Epsilon').
296 *
297 * err = 2 "Guaranteed" error bound: The estimated forward error,
298 * almost certainly within a factor of 10 of the true error
299 * so long as the next entry is greater than the threshold
300 * sqrt(n) * dlamch('Epsilon'). This error bound should only
301 * be trusted if the previous boolean is true.
302 *
303 * err = 3 Reciprocal condition number: Estimated normwise
304 * reciprocal condition number. Compared with the threshold
305 * sqrt(n) * dlamch('Epsilon') to determine if the error
306 * estimate is "guaranteed". These reciprocal condition
307 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
308 * appropriately scaled matrix Z.
309 * Let Z = S*A, where S scales each row by a power of the
310 * radix so all absolute row sums of Z are approximately 1.
311 *
312 * See Lapack Working Note 165 for further details and extra
313 * cautions.
314 *
315 * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
316 * For each right-hand side, this array contains information about
317 * various error bounds and condition numbers corresponding to the
318 * componentwise relative error, which is defined as follows:
319 *
320 * Componentwise relative error in the ith solution vector:
321 * abs(XTRUE(j,i) - X(j,i))
322 * max_j ----------------------
323 * abs(X(j,i))
324 *
325 * The array is indexed by the right-hand side i (on which the
326 * componentwise relative error depends), and the type of error
327 * information as described below. There currently are up to three
328 * pieces of information returned for each right-hand side. If
329 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
330 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
331 * the first (:,N_ERR_BNDS) entries are returned.
332 *
333 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
334 * right-hand side.
335 *
336 * The second index in ERR_BNDS_COMP(:,err) contains the following
337 * three fields:
338 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
339 * reciprocal condition number is less than the threshold
340 * sqrt(n) * dlamch('Epsilon').
341 *
342 * err = 2 "Guaranteed" error bound: The estimated forward error,
343 * almost certainly within a factor of 10 of the true error
344 * so long as the next entry is greater than the threshold
345 * sqrt(n) * dlamch('Epsilon'). This error bound should only
346 * be trusted if the previous boolean is true.
347 *
348 * err = 3 Reciprocal condition number: Estimated componentwise
349 * reciprocal condition number. Compared with the threshold
350 * sqrt(n) * dlamch('Epsilon') to determine if the error
351 * estimate is "guaranteed". These reciprocal condition
352 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
353 * appropriately scaled matrix Z.
354 * Let Z = S*(A*diag(x)), where x is the solution for the
355 * current right-hand side and S scales each row of
356 * A*diag(x) by a power of the radix so all absolute row
357 * sums of Z are approximately 1.
358 *
359 * See Lapack Working Note 165 for further details and extra
360 * cautions.
361 *
362 * NPARAMS (input) INTEGER
363 * Specifies the number of parameters set in PARAMS. If .LE. 0, the
364 * PARAMS array is never referenced and default values are used.
365 *
366 * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
367 * Specifies algorithm parameters. If an entry is .LT. 0.0, then
368 * that entry will be filled with default value used for that
369 * parameter. Only positions up to NPARAMS are accessed; defaults
370 * are used for higher-numbered parameters.
371 *
372 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
373 * refinement or not.
374 * Default: 1.0D+0
375 * = 0.0 : No refinement is performed, and no error bounds are
376 * computed.
377 * = 1.0 : Use the extra-precise refinement algorithm.
378 * (other values are reserved for future use)
379 *
380 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
381 * computations allowed for refinement.
382 * Default: 10
383 * Aggressive: Set to 100 to permit convergence using approximate
384 * factorizations or factorizations other than LU. If
385 * the factorization uses a technique other than
386 * Gaussian elimination, the guarantees in
387 * err_bnds_norm and err_bnds_comp may no longer be
388 * trustworthy.
389 *
390 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
391 * will attempt to find a solution with small componentwise
392 * relative error in the double-precision algorithm. Positive
393 * is true, 0.0 is false.
394 * Default: 1.0 (attempt componentwise convergence)
395 *
396 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
397 *
398 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
399 *
400 * INFO (output) INTEGER
401 * = 0: Successful exit. The solution to every right-hand side is
402 * guaranteed.
403 * < 0: If INFO = -i, the i-th argument had an illegal value
404 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
405 * has been completed, but the factor U is exactly singular, so
406 * the solution and error bounds could not be computed. RCOND = 0
407 * is returned.
408 * = N+J: The solution corresponding to the Jth right-hand side is
409 * not guaranteed. The solutions corresponding to other right-
410 * hand sides K with K > J may not be guaranteed as well, but
411 * only the first such right-hand side is reported. If a small
412 * componentwise error is not requested (PARAMS(3) = 0.0) then
413 * the Jth right-hand side is the first with a normwise error
414 * bound that is not guaranteed (the smallest J such
415 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
416 * the Jth right-hand side is the first with either a normwise or
417 * componentwise error bound that is not guaranteed (the smallest
418 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
419 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
420 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
421 * about all of the right-hand sides check ERR_BNDS_NORM or
422 * ERR_BNDS_COMP.
423 *
424 * ==================================================================
425 *
426 * .. Parameters ..
427 DOUBLE PRECISION ZERO, ONE
428 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
429 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
430 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
431 INTEGER CMP_ERR_I, PIV_GROWTH_I
432 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
433 $ BERR_I = 3 )
434 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
435 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
436 $ PIV_GROWTH_I = 9 )
437 * ..
438 * .. Local Scalars ..
439 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
440 INTEGER INFEQU, J
441 DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
442 $ ROWCND, SMLNUM
443 * ..
444 * .. External Functions ..
445 EXTERNAL LSAME, DLAMCH, ZLA_RPVGRW
446 LOGICAL LSAME
447 DOUBLE PRECISION DLAMCH, ZLA_RPVGRW
448 * ..
449 * .. External Subroutines ..
450 EXTERNAL ZGEEQUB, ZGETRF, ZGETRS, ZLACPY, ZLAQGE,
451 $ XERBLA, ZLASCL2, ZGERFSX
452 * ..
453 * .. Intrinsic Functions ..
454 INTRINSIC MAX, MIN
455 * ..
456 * .. Executable Statements ..
457 *
458 INFO = 0
459 NOFACT = LSAME( FACT, 'N' )
460 EQUIL = LSAME( FACT, 'E' )
461 NOTRAN = LSAME( TRANS, 'N' )
462 SMLNUM = DLAMCH( 'Safe minimum' )
463 BIGNUM = ONE / SMLNUM
464 IF( NOFACT .OR. EQUIL ) THEN
465 EQUED = 'N'
466 ROWEQU = .FALSE.
467 COLEQU = .FALSE.
468 ELSE
469 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
470 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
471 END IF
472 *
473 * Default is failure. If an input parameter is wrong or
474 * factorization fails, make everything look horrible. Only the
475 * pivot growth is set here, the rest is initialized in ZGERFSX.
476 *
477 RPVGRW = ZERO
478 *
479 * Test the input parameters. PARAMS is not tested until ZGERFSX.
480 *
481 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
482 $ LSAME( FACT, 'F' ) ) THEN
483 INFO = -1
484 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
485 $ LSAME( TRANS, 'C' ) ) THEN
486 INFO = -2
487 ELSE IF( N.LT.0 ) THEN
488 INFO = -3
489 ELSE IF( NRHS.LT.0 ) THEN
490 INFO = -4
491 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
492 INFO = -6
493 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
494 INFO = -8
495 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
496 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
497 INFO = -10
498 ELSE
499 IF( ROWEQU ) THEN
500 RCMIN = BIGNUM
501 RCMAX = ZERO
502 DO 10 J = 1, N
503 RCMIN = MIN( RCMIN, R( J ) )
504 RCMAX = MAX( RCMAX, R( J ) )
505 10 CONTINUE
506 IF( RCMIN.LE.ZERO ) THEN
507 INFO = -11
508 ELSE IF( N.GT.0 ) THEN
509 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
510 ELSE
511 ROWCND = ONE
512 END IF
513 END IF
514 IF( COLEQU .AND. INFO.EQ.0 ) THEN
515 RCMIN = BIGNUM
516 RCMAX = ZERO
517 DO 20 J = 1, N
518 RCMIN = MIN( RCMIN, C( J ) )
519 RCMAX = MAX( RCMAX, C( J ) )
520 20 CONTINUE
521 IF( RCMIN.LE.ZERO ) THEN
522 INFO = -12
523 ELSE IF( N.GT.0 ) THEN
524 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
525 ELSE
526 COLCND = ONE
527 END IF
528 END IF
529 IF( INFO.EQ.0 ) THEN
530 IF( LDB.LT.MAX( 1, N ) ) THEN
531 INFO = -14
532 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
533 INFO = -16
534 END IF
535 END IF
536 END IF
537 *
538 IF( INFO.NE.0 ) THEN
539 CALL XERBLA( 'ZGESVXX', -INFO )
540 RETURN
541 END IF
542 *
543 IF( EQUIL ) THEN
544 *
545 * Compute row and column scalings to equilibrate the matrix A.
546 *
547 CALL ZGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
548 $ INFEQU )
549 IF( INFEQU.EQ.0 ) THEN
550 *
551 * Equilibrate the matrix.
552 *
553 CALL ZLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
554 $ EQUED )
555 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
556 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
557 END IF
558 *
559 * If the scaling factors are not applied, set them to 1.0.
560 *
561 IF ( .NOT.ROWEQU ) THEN
562 DO J = 1, N
563 R( J ) = 1.0D+0
564 END DO
565 END IF
566 IF ( .NOT.COLEQU ) THEN
567 DO J = 1, N
568 C( J ) = 1.0D+0
569 END DO
570 END IF
571 END IF
572 *
573 * Scale the right-hand side.
574 *
575 IF( NOTRAN ) THEN
576 IF( ROWEQU ) CALL ZLASCL2( N, NRHS, R, B, LDB )
577 ELSE
578 IF( COLEQU ) CALL ZLASCL2( N, NRHS, C, B, LDB )
579 END IF
580 *
581 IF( NOFACT .OR. EQUIL ) THEN
582 *
583 * Compute the LU factorization of A.
584 *
585 CALL ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
586 CALL ZGETRF( N, N, AF, LDAF, IPIV, INFO )
587 *
588 * Return if INFO is non-zero.
589 *
590 IF( INFO.GT.0 ) THEN
591 *
592 * Pivot in column INFO is exactly 0
593 * Compute the reciprocal pivot growth factor of the
594 * leading rank-deficient INFO columns of A.
595 *
596 RPVGRW = ZLA_RPVGRW( N, INFO, A, LDA, AF, LDAF )
597 RETURN
598 END IF
599 END IF
600 *
601 * Compute the reciprocal pivot growth factor RPVGRW.
602 *
603 RPVGRW = ZLA_RPVGRW( N, N, A, LDA, AF, LDAF )
604 *
605 * Compute the solution matrix X.
606 *
607 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
608 CALL ZGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
609 *
610 * Use iterative refinement to improve the computed solution and
611 * compute error bounds and backward error estimates for it.
612 *
613 CALL ZGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF,
614 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
615 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
616 $ WORK, RWORK, INFO )
617 *
618 * Scale solutions.
619 *
620 IF ( COLEQU .AND. NOTRAN ) THEN
621 CALL ZLASCL2 ( N, NRHS, C, X, LDX )
622 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
623 CALL ZLASCL2 ( N, NRHS, R, X, LDX )
624 END IF
625 *
626 RETURN
627 *
628 * End of ZGESVXX
629 *
630 END
2 $ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
3 $ BERR, N_ERR_BNDS, ERR_BNDS_NORM,
4 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
5 $ INFO )
6 *
7 * -- LAPACK driver routine (version 3.2.1) --
8 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
9 * -- Jason Riedy of Univ. of California Berkeley. --
10 * -- April 2009 --
11 *
12 * -- LAPACK is a software package provided by Univ. of Tennessee, --
13 * -- Univ. of California Berkeley and NAG Ltd. --
14 *
15 IMPLICIT NONE
16 * ..
17 * .. Scalar Arguments ..
18 CHARACTER EQUED, FACT, TRANS
19 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
20 $ N_ERR_BNDS
21 DOUBLE PRECISION RCOND, RPVGRW
22 * ..
23 * .. Array Arguments ..
24 INTEGER IPIV( * )
25 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
26 $ X( LDX , * ),WORK( * )
27 DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
28 $ ERR_BNDS_NORM( NRHS, * ),
29 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
30 * ..
31 *
32 * Purpose
33 * =======
34 *
35 * ZGESVXX uses the LU factorization to compute the solution to a
36 * complex*16 system of linear equations A * X = B, where A is an
37 * N-by-N matrix and X and B are N-by-NRHS matrices.
38 *
39 * If requested, both normwise and maximum componentwise error bounds
40 * are returned. ZGESVXX will return a solution with a tiny
41 * guaranteed error (O(eps) where eps is the working machine
42 * precision) unless the matrix is very ill-conditioned, in which
43 * case a warning is returned. Relevant condition numbers also are
44 * calculated and returned.
45 *
46 * ZGESVXX accepts user-provided factorizations and equilibration
47 * factors; see the definitions of the FACT and EQUED options.
48 * Solving with refinement and using a factorization from a previous
49 * ZGESVXX call will also produce a solution with either O(eps)
50 * errors or warnings, but we cannot make that claim for general
51 * user-provided factorizations and equilibration factors if they
52 * differ from what ZGESVXX would itself produce.
53 *
54 * Description
55 * ===========
56 *
57 * The following steps are performed:
58 *
59 * 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
60 * the system:
61 *
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 *
66 * Whether or not the system will be equilibrated depends on the
67 * scaling of the matrix A, but if equilibration is used, A is
68 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
69 * or diag(C)*B (if TRANS = 'T' or 'C').
70 *
71 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
72 * the matrix A (after equilibration if FACT = 'E') as
73 *
74 * A = P * L * U,
75 *
76 * where P is a permutation matrix, L is a unit lower triangular
77 * matrix, and U is upper triangular.
78 *
79 * 3. If some U(i,i)=0, so that U is exactly singular, then the
80 * routine returns with INFO = i. Otherwise, the factored form of A
81 * is used to estimate the condition number of the matrix A (see
82 * argument RCOND). If the reciprocal of the condition number is less
83 * than machine precision, the routine still goes on to solve for X
84 * and compute error bounds as described below.
85 *
86 * 4. The system of equations is solved for X using the factored form
87 * of A.
88 *
89 * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
90 * the routine will use iterative refinement to try to get a small
91 * error and error bounds. Refinement calculates the residual to at
92 * least twice the working precision.
93 *
94 * 6. If equilibration was used, the matrix X is premultiplied by
95 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
96 * that it solves the original system before equilibration.
97 *
98 * Arguments
99 * =========
100 *
101 * Some optional parameters are bundled in the PARAMS array. These
102 * settings determine how refinement is performed, but often the
103 * defaults are acceptable. If the defaults are acceptable, users
104 * can pass NPARAMS = 0 which prevents the source code from accessing
105 * the PARAMS argument.
106 *
107 * FACT (input) CHARACTER*1
108 * Specifies whether or not the factored form of the matrix A is
109 * supplied on entry, and if not, whether the matrix A should be
110 * equilibrated before it is factored.
111 * = 'F': On entry, AF and IPIV contain the factored form of A.
112 * If EQUED is not 'N', the matrix A has been
113 * equilibrated with scaling factors given by R and C.
114 * A, AF, and IPIV are not modified.
115 * = 'N': The matrix A will be copied to AF and factored.
116 * = 'E': The matrix A will be equilibrated if necessary, then
117 * copied to AF and factored.
118 *
119 * TRANS (input) CHARACTER*1
120 * Specifies the form of the system of equations:
121 * = 'N': A * X = B (No transpose)
122 * = 'T': A**T * X = B (Transpose)
123 * = 'C': A**H * X = B (Conjugate Transpose)
124 *
125 * N (input) INTEGER
126 * The number of linear equations, i.e., the order of the
127 * matrix A. N >= 0.
128 *
129 * NRHS (input) INTEGER
130 * The number of right hand sides, i.e., the number of columns
131 * of the matrices B and X. NRHS >= 0.
132 *
133 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
134 * On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
135 * not 'N', then A must have been equilibrated by the scaling
136 * factors in R and/or C. A is not modified if FACT = 'F' or
137 * 'N', or if FACT = 'E' and EQUED = 'N' on exit.
138 *
139 * On exit, if EQUED .ne. 'N', A is scaled as follows:
140 * EQUED = 'R': A := diag(R) * A
141 * EQUED = 'C': A := A * diag(C)
142 * EQUED = 'B': A := diag(R) * A * diag(C).
143 *
144 * LDA (input) INTEGER
145 * The leading dimension of the array A. LDA >= max(1,N).
146 *
147 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
148 * If FACT = 'F', then AF is an input argument and on entry
149 * contains the factors L and U from the factorization
150 * A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
151 * AF is the factored form of the equilibrated matrix A.
152 *
153 * If FACT = 'N', then AF is an output argument and on exit
154 * returns the factors L and U from the factorization A = P*L*U
155 * of the original matrix A.
156 *
157 * If FACT = 'E', then AF is an output argument and on exit
158 * returns the factors L and U from the factorization A = P*L*U
159 * of the equilibrated matrix A (see the description of A for
160 * the form of the equilibrated matrix).
161 *
162 * LDAF (input) INTEGER
163 * The leading dimension of the array AF. LDAF >= max(1,N).
164 *
165 * IPIV (input or output) INTEGER array, dimension (N)
166 * If FACT = 'F', then IPIV is an input argument and on entry
167 * contains the pivot indices from the factorization A = P*L*U
168 * as computed by ZGETRF; row i of the matrix was interchanged
169 * with row IPIV(i).
170 *
171 * If FACT = 'N', then IPIV is an output argument and on exit
172 * contains the pivot indices from the factorization A = P*L*U
173 * of the original matrix A.
174 *
175 * If FACT = 'E', then IPIV is an output argument and on exit
176 * contains the pivot indices from the factorization A = P*L*U
177 * of the equilibrated matrix A.
178 *
179 * EQUED (input or output) CHARACTER*1
180 * Specifies the form of equilibration that was done.
181 * = 'N': No equilibration (always true if FACT = 'N').
182 * = 'R': Row equilibration, i.e., A has been premultiplied by
183 * diag(R).
184 * = 'C': Column equilibration, i.e., A has been postmultiplied
185 * by diag(C).
186 * = 'B': Both row and column equilibration, i.e., A has been
187 * replaced by diag(R) * A * diag(C).
188 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
189 * output argument.
190 *
191 * R (input or output) DOUBLE PRECISION array, dimension (N)
192 * The row scale factors for A. If EQUED = 'R' or 'B', A is
193 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
194 * is not accessed. R is an input argument if FACT = 'F';
195 * otherwise, R is an output argument. If FACT = 'F' and
196 * EQUED = 'R' or 'B', each element of R must be positive.
197 * If R is output, each element of R is a power of the radix.
198 * If R is input, each element of R should be a power of the radix
199 * to ensure a reliable solution and error estimates. Scaling by
200 * powers of the radix does not cause rounding errors unless the
201 * result underflows or overflows. Rounding errors during scaling
202 * lead to refining with a matrix that is not equivalent to the
203 * input matrix, producing error estimates that may not be
204 * reliable.
205 *
206 * C (input or output) DOUBLE PRECISION array, dimension (N)
207 * The column scale factors for A. If EQUED = 'C' or 'B', A is
208 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
209 * is not accessed. C is an input argument if FACT = 'F';
210 * otherwise, C is an output argument. If FACT = 'F' and
211 * EQUED = 'C' or 'B', each element of C must be positive.
212 * If C is output, each element of C is a power of the radix.
213 * If C is input, each element of C should be a power of the radix
214 * to ensure a reliable solution and error estimates. Scaling by
215 * powers of the radix does not cause rounding errors unless the
216 * result underflows or overflows. Rounding errors during scaling
217 * lead to refining with a matrix that is not equivalent to the
218 * input matrix, producing error estimates that may not be
219 * reliable.
220 *
221 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
222 * On entry, the N-by-NRHS right hand side matrix B.
223 * On exit,
224 * if EQUED = 'N', B is not modified;
225 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
226 * diag(R)*B;
227 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
228 * overwritten by diag(C)*B.
229 *
230 * LDB (input) INTEGER
231 * The leading dimension of the array B. LDB >= max(1,N).
232 *
233 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
234 * If INFO = 0, the N-by-NRHS solution matrix X to the original
235 * system of equations. Note that A and B are modified on exit
236 * if EQUED .ne. 'N', and the solution to the equilibrated system is
237 * inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
238 * inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
239 *
240 * LDX (input) INTEGER
241 * The leading dimension of the array X. LDX >= max(1,N).
242 *
243 * RCOND (output) DOUBLE PRECISION
244 * Reciprocal scaled condition number. This is an estimate of the
245 * reciprocal Skeel condition number of the matrix A after
246 * equilibration (if done). If this is less than the machine
247 * precision (in particular, if it is zero), the matrix is singular
248 * to working precision. Note that the error may still be small even
249 * if this number is very small and the matrix appears ill-
250 * conditioned.
251 *
252 * RPVGRW (output) DOUBLE PRECISION
253 * Reciprocal pivot growth. On exit, this contains the reciprocal
254 * pivot growth factor norm(A)/norm(U). The "max absolute element"
255 * norm is used. If this is much less than 1, then the stability of
256 * the LU factorization of the (equilibrated) matrix A could be poor.
257 * This also means that the solution X, estimated condition numbers,
258 * and error bounds could be unreliable. If factorization fails with
259 * 0<INFO<=N, then this contains the reciprocal pivot growth factor
260 * for the leading INFO columns of A. In ZGESVX, this quantity is
261 * returned in WORK(1).
262 *
263 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
264 * Componentwise relative backward error. This is the
265 * componentwise relative backward error of each solution vector X(j)
266 * (i.e., the smallest relative change in any element of A or B that
267 * makes X(j) an exact solution).
268 *
269 * N_ERR_BNDS (input) INTEGER
270 * Number of error bounds to return for each right hand side
271 * and each type (normwise or componentwise). See ERR_BNDS_NORM and
272 * ERR_BNDS_COMP below.
273 *
274 * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
275 * For each right-hand side, this array contains information about
276 * various error bounds and condition numbers corresponding to the
277 * normwise relative error, which is defined as follows:
278 *
279 * Normwise relative error in the ith solution vector:
280 * max_j (abs(XTRUE(j,i) - X(j,i)))
281 * ------------------------------
282 * max_j abs(X(j,i))
283 *
284 * The array is indexed by the type of error information as described
285 * below. There currently are up to three pieces of information
286 * returned.
287 *
288 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
289 * right-hand side.
290 *
291 * The second index in ERR_BNDS_NORM(:,err) contains the following
292 * three fields:
293 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
294 * reciprocal condition number is less than the threshold
295 * sqrt(n) * dlamch('Epsilon').
296 *
297 * err = 2 "Guaranteed" error bound: The estimated forward error,
298 * almost certainly within a factor of 10 of the true error
299 * so long as the next entry is greater than the threshold
300 * sqrt(n) * dlamch('Epsilon'). This error bound should only
301 * be trusted if the previous boolean is true.
302 *
303 * err = 3 Reciprocal condition number: Estimated normwise
304 * reciprocal condition number. Compared with the threshold
305 * sqrt(n) * dlamch('Epsilon') to determine if the error
306 * estimate is "guaranteed". These reciprocal condition
307 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
308 * appropriately scaled matrix Z.
309 * Let Z = S*A, where S scales each row by a power of the
310 * radix so all absolute row sums of Z are approximately 1.
311 *
312 * See Lapack Working Note 165 for further details and extra
313 * cautions.
314 *
315 * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
316 * For each right-hand side, this array contains information about
317 * various error bounds and condition numbers corresponding to the
318 * componentwise relative error, which is defined as follows:
319 *
320 * Componentwise relative error in the ith solution vector:
321 * abs(XTRUE(j,i) - X(j,i))
322 * max_j ----------------------
323 * abs(X(j,i))
324 *
325 * The array is indexed by the right-hand side i (on which the
326 * componentwise relative error depends), and the type of error
327 * information as described below. There currently are up to three
328 * pieces of information returned for each right-hand side. If
329 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
330 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
331 * the first (:,N_ERR_BNDS) entries are returned.
332 *
333 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
334 * right-hand side.
335 *
336 * The second index in ERR_BNDS_COMP(:,err) contains the following
337 * three fields:
338 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
339 * reciprocal condition number is less than the threshold
340 * sqrt(n) * dlamch('Epsilon').
341 *
342 * err = 2 "Guaranteed" error bound: The estimated forward error,
343 * almost certainly within a factor of 10 of the true error
344 * so long as the next entry is greater than the threshold
345 * sqrt(n) * dlamch('Epsilon'). This error bound should only
346 * be trusted if the previous boolean is true.
347 *
348 * err = 3 Reciprocal condition number: Estimated componentwise
349 * reciprocal condition number. Compared with the threshold
350 * sqrt(n) * dlamch('Epsilon') to determine if the error
351 * estimate is "guaranteed". These reciprocal condition
352 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
353 * appropriately scaled matrix Z.
354 * Let Z = S*(A*diag(x)), where x is the solution for the
355 * current right-hand side and S scales each row of
356 * A*diag(x) by a power of the radix so all absolute row
357 * sums of Z are approximately 1.
358 *
359 * See Lapack Working Note 165 for further details and extra
360 * cautions.
361 *
362 * NPARAMS (input) INTEGER
363 * Specifies the number of parameters set in PARAMS. If .LE. 0, the
364 * PARAMS array is never referenced and default values are used.
365 *
366 * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
367 * Specifies algorithm parameters. If an entry is .LT. 0.0, then
368 * that entry will be filled with default value used for that
369 * parameter. Only positions up to NPARAMS are accessed; defaults
370 * are used for higher-numbered parameters.
371 *
372 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
373 * refinement or not.
374 * Default: 1.0D+0
375 * = 0.0 : No refinement is performed, and no error bounds are
376 * computed.
377 * = 1.0 : Use the extra-precise refinement algorithm.
378 * (other values are reserved for future use)
379 *
380 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
381 * computations allowed for refinement.
382 * Default: 10
383 * Aggressive: Set to 100 to permit convergence using approximate
384 * factorizations or factorizations other than LU. If
385 * the factorization uses a technique other than
386 * Gaussian elimination, the guarantees in
387 * err_bnds_norm and err_bnds_comp may no longer be
388 * trustworthy.
389 *
390 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
391 * will attempt to find a solution with small componentwise
392 * relative error in the double-precision algorithm. Positive
393 * is true, 0.0 is false.
394 * Default: 1.0 (attempt componentwise convergence)
395 *
396 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
397 *
398 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
399 *
400 * INFO (output) INTEGER
401 * = 0: Successful exit. The solution to every right-hand side is
402 * guaranteed.
403 * < 0: If INFO = -i, the i-th argument had an illegal value
404 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
405 * has been completed, but the factor U is exactly singular, so
406 * the solution and error bounds could not be computed. RCOND = 0
407 * is returned.
408 * = N+J: The solution corresponding to the Jth right-hand side is
409 * not guaranteed. The solutions corresponding to other right-
410 * hand sides K with K > J may not be guaranteed as well, but
411 * only the first such right-hand side is reported. If a small
412 * componentwise error is not requested (PARAMS(3) = 0.0) then
413 * the Jth right-hand side is the first with a normwise error
414 * bound that is not guaranteed (the smallest J such
415 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
416 * the Jth right-hand side is the first with either a normwise or
417 * componentwise error bound that is not guaranteed (the smallest
418 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
419 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
420 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
421 * about all of the right-hand sides check ERR_BNDS_NORM or
422 * ERR_BNDS_COMP.
423 *
424 * ==================================================================
425 *
426 * .. Parameters ..
427 DOUBLE PRECISION ZERO, ONE
428 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
429 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
430 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
431 INTEGER CMP_ERR_I, PIV_GROWTH_I
432 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
433 $ BERR_I = 3 )
434 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
435 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
436 $ PIV_GROWTH_I = 9 )
437 * ..
438 * .. Local Scalars ..
439 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
440 INTEGER INFEQU, J
441 DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
442 $ ROWCND, SMLNUM
443 * ..
444 * .. External Functions ..
445 EXTERNAL LSAME, DLAMCH, ZLA_RPVGRW
446 LOGICAL LSAME
447 DOUBLE PRECISION DLAMCH, ZLA_RPVGRW
448 * ..
449 * .. External Subroutines ..
450 EXTERNAL ZGEEQUB, ZGETRF, ZGETRS, ZLACPY, ZLAQGE,
451 $ XERBLA, ZLASCL2, ZGERFSX
452 * ..
453 * .. Intrinsic Functions ..
454 INTRINSIC MAX, MIN
455 * ..
456 * .. Executable Statements ..
457 *
458 INFO = 0
459 NOFACT = LSAME( FACT, 'N' )
460 EQUIL = LSAME( FACT, 'E' )
461 NOTRAN = LSAME( TRANS, 'N' )
462 SMLNUM = DLAMCH( 'Safe minimum' )
463 BIGNUM = ONE / SMLNUM
464 IF( NOFACT .OR. EQUIL ) THEN
465 EQUED = 'N'
466 ROWEQU = .FALSE.
467 COLEQU = .FALSE.
468 ELSE
469 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
470 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
471 END IF
472 *
473 * Default is failure. If an input parameter is wrong or
474 * factorization fails, make everything look horrible. Only the
475 * pivot growth is set here, the rest is initialized in ZGERFSX.
476 *
477 RPVGRW = ZERO
478 *
479 * Test the input parameters. PARAMS is not tested until ZGERFSX.
480 *
481 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
482 $ LSAME( FACT, 'F' ) ) THEN
483 INFO = -1
484 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
485 $ LSAME( TRANS, 'C' ) ) THEN
486 INFO = -2
487 ELSE IF( N.LT.0 ) THEN
488 INFO = -3
489 ELSE IF( NRHS.LT.0 ) THEN
490 INFO = -4
491 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
492 INFO = -6
493 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
494 INFO = -8
495 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
496 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
497 INFO = -10
498 ELSE
499 IF( ROWEQU ) THEN
500 RCMIN = BIGNUM
501 RCMAX = ZERO
502 DO 10 J = 1, N
503 RCMIN = MIN( RCMIN, R( J ) )
504 RCMAX = MAX( RCMAX, R( J ) )
505 10 CONTINUE
506 IF( RCMIN.LE.ZERO ) THEN
507 INFO = -11
508 ELSE IF( N.GT.0 ) THEN
509 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
510 ELSE
511 ROWCND = ONE
512 END IF
513 END IF
514 IF( COLEQU .AND. INFO.EQ.0 ) THEN
515 RCMIN = BIGNUM
516 RCMAX = ZERO
517 DO 20 J = 1, N
518 RCMIN = MIN( RCMIN, C( J ) )
519 RCMAX = MAX( RCMAX, C( J ) )
520 20 CONTINUE
521 IF( RCMIN.LE.ZERO ) THEN
522 INFO = -12
523 ELSE IF( N.GT.0 ) THEN
524 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
525 ELSE
526 COLCND = ONE
527 END IF
528 END IF
529 IF( INFO.EQ.0 ) THEN
530 IF( LDB.LT.MAX( 1, N ) ) THEN
531 INFO = -14
532 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
533 INFO = -16
534 END IF
535 END IF
536 END IF
537 *
538 IF( INFO.NE.0 ) THEN
539 CALL XERBLA( 'ZGESVXX', -INFO )
540 RETURN
541 END IF
542 *
543 IF( EQUIL ) THEN
544 *
545 * Compute row and column scalings to equilibrate the matrix A.
546 *
547 CALL ZGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
548 $ INFEQU )
549 IF( INFEQU.EQ.0 ) THEN
550 *
551 * Equilibrate the matrix.
552 *
553 CALL ZLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
554 $ EQUED )
555 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
556 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
557 END IF
558 *
559 * If the scaling factors are not applied, set them to 1.0.
560 *
561 IF ( .NOT.ROWEQU ) THEN
562 DO J = 1, N
563 R( J ) = 1.0D+0
564 END DO
565 END IF
566 IF ( .NOT.COLEQU ) THEN
567 DO J = 1, N
568 C( J ) = 1.0D+0
569 END DO
570 END IF
571 END IF
572 *
573 * Scale the right-hand side.
574 *
575 IF( NOTRAN ) THEN
576 IF( ROWEQU ) CALL ZLASCL2( N, NRHS, R, B, LDB )
577 ELSE
578 IF( COLEQU ) CALL ZLASCL2( N, NRHS, C, B, LDB )
579 END IF
580 *
581 IF( NOFACT .OR. EQUIL ) THEN
582 *
583 * Compute the LU factorization of A.
584 *
585 CALL ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
586 CALL ZGETRF( N, N, AF, LDAF, IPIV, INFO )
587 *
588 * Return if INFO is non-zero.
589 *
590 IF( INFO.GT.0 ) THEN
591 *
592 * Pivot in column INFO is exactly 0
593 * Compute the reciprocal pivot growth factor of the
594 * leading rank-deficient INFO columns of A.
595 *
596 RPVGRW = ZLA_RPVGRW( N, INFO, A, LDA, AF, LDAF )
597 RETURN
598 END IF
599 END IF
600 *
601 * Compute the reciprocal pivot growth factor RPVGRW.
602 *
603 RPVGRW = ZLA_RPVGRW( N, N, A, LDA, AF, LDAF )
604 *
605 * Compute the solution matrix X.
606 *
607 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
608 CALL ZGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
609 *
610 * Use iterative refinement to improve the computed solution and
611 * compute error bounds and backward error estimates for it.
612 *
613 CALL ZGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF,
614 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
615 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
616 $ WORK, RWORK, INFO )
617 *
618 * Scale solutions.
619 *
620 IF ( COLEQU .AND. NOTRAN ) THEN
621 CALL ZLASCL2 ( N, NRHS, C, X, LDX )
622 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
623 CALL ZLASCL2 ( N, NRHS, R, X, LDX )
624 END IF
625 *
626 RETURN
627 *
628 * End of ZGESVXX
629 *
630 END