1 SUBROUTINE ZGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
2 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
3 $ RCOND, RPVGRW, BERR, N_ERR_BNDS,
4 $ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
5 $ WORK, RWORK, INFO )
6 *
7 * -- LAPACK driver routine (version 3.2.2) --
8 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
9 * -- Jason Riedy of Univ. of California Berkeley. --
10 * -- June 2010 --
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, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
20 $ N_ERR_BNDS
21 DOUBLE PRECISION RCOND, RPVGRW
22 * ..
23 * .. Array Arguments ..
24 INTEGER IPIV( * )
25 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), 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 * ZGBSVXX 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. ZGBSVXX 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 * ZGBSVXX 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 * ZGBSVXX 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 ZGBSVXX 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 = 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 * KL (input) INTEGER
130 * The number of subdiagonals within the band of A. KL >= 0.
131 *
132 * KU (input) INTEGER
133 * The number of superdiagonals within the band of A. KU >= 0.
134 *
135 * NRHS (input) INTEGER
136 * The number of right hand sides, i.e., the number of columns
137 * of the matrices B and X. NRHS >= 0.
138 *
139 * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
140 * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
141 * The j-th column of A is stored in the j-th column of the
142 * array AB as follows:
143 * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
144 *
145 * If FACT = 'F' and EQUED is not 'N', then AB must have been
146 * equilibrated by the scaling factors in R and/or C. AB is not
147 * modified if FACT = 'F' or 'N', or if FACT = 'E' and
148 * EQUED = 'N' on exit.
149 *
150 * On exit, if EQUED .ne. 'N', A is scaled as follows:
151 * EQUED = 'R': A := diag(R) * A
152 * EQUED = 'C': A := A * diag(C)
153 * EQUED = 'B': A := diag(R) * A * diag(C).
154 *
155 * LDAB (input) INTEGER
156 * The leading dimension of the array AB. LDAB >= KL+KU+1.
157 *
158 * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
159 * If FACT = 'F', then AFB is an input argument and on entry
160 * contains details of the LU factorization of the band matrix
161 * A, as computed by ZGBTRF. U is stored as an upper triangular
162 * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
163 * and the multipliers used during the factorization are stored
164 * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
165 * the factored form of the equilibrated matrix A.
166 *
167 * If FACT = 'N', then AF is an output argument and on exit
168 * returns the factors L and U from the factorization A = P*L*U
169 * of the original matrix A.
170 *
171 * If FACT = 'E', then AF is an output argument and on exit
172 * returns the factors L and U from the factorization A = P*L*U
173 * of the equilibrated matrix A (see the description of A for
174 * the form of the equilibrated matrix).
175 *
176 * LDAFB (input) INTEGER
177 * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
178 *
179 * IPIV (input or output) INTEGER array, dimension (N)
180 * If FACT = 'F', then IPIV is an input argument and on entry
181 * contains the pivot indices from the factorization A = P*L*U
182 * as computed by DGETRF; row i of the matrix was interchanged
183 * with row IPIV(i).
184 *
185 * If FACT = 'N', then IPIV is an output argument and on exit
186 * contains the pivot indices from the factorization A = P*L*U
187 * of the original matrix A.
188 *
189 * If FACT = 'E', then IPIV is an output argument and on exit
190 * contains the pivot indices from the factorization A = P*L*U
191 * of the equilibrated matrix A.
192 *
193 * EQUED (input or output) CHARACTER*1
194 * Specifies the form of equilibration that was done.
195 * = 'N': No equilibration (always true if FACT = 'N').
196 * = 'R': Row equilibration, i.e., A has been premultiplied by
197 * diag(R).
198 * = 'C': Column equilibration, i.e., A has been postmultiplied
199 * by diag(C).
200 * = 'B': Both row and column equilibration, i.e., A has been
201 * replaced by diag(R) * A * diag(C).
202 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
203 * output argument.
204 *
205 * R (input or output) DOUBLE PRECISION array, dimension (N)
206 * The row scale factors for A. If EQUED = 'R' or 'B', A is
207 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
208 * is not accessed. R is an input argument if FACT = 'F';
209 * otherwise, R is an output argument. If FACT = 'F' and
210 * EQUED = 'R' or 'B', each element of R must be positive.
211 * If R is output, each element of R is a power of the radix.
212 * If R is input, each element of R should be a power of the radix
213 * to ensure a reliable solution and error estimates. Scaling by
214 * powers of the radix does not cause rounding errors unless the
215 * result underflows or overflows. Rounding errors during scaling
216 * lead to refining with a matrix that is not equivalent to the
217 * input matrix, producing error estimates that may not be
218 * reliable.
219 *
220 * C (input or output) DOUBLE PRECISION array, dimension (N)
221 * The column scale factors for A. If EQUED = 'C' or 'B', A is
222 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
223 * is not accessed. C is an input argument if FACT = 'F';
224 * otherwise, C is an output argument. If FACT = 'F' and
225 * EQUED = 'C' or 'B', each element of C must be positive.
226 * If C is output, each element of C is a power of the radix.
227 * If C is input, each element of C should be a power of the radix
228 * to ensure a reliable solution and error estimates. Scaling by
229 * powers of the radix does not cause rounding errors unless the
230 * result underflows or overflows. Rounding errors during scaling
231 * lead to refining with a matrix that is not equivalent to the
232 * input matrix, producing error estimates that may not be
233 * reliable.
234 *
235 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
236 * On entry, the N-by-NRHS right hand side matrix B.
237 * On exit,
238 * if EQUED = 'N', B is not modified;
239 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
240 * diag(R)*B;
241 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
242 * overwritten by diag(C)*B.
243 *
244 * LDB (input) INTEGER
245 * The leading dimension of the array B. LDB >= max(1,N).
246 *
247 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
248 * If INFO = 0, the N-by-NRHS solution matrix X to the original
249 * system of equations. Note that A and B are modified on exit
250 * if EQUED .ne. 'N', and the solution to the equilibrated system is
251 * inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
252 * inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
253 *
254 * LDX (input) INTEGER
255 * The leading dimension of the array X. LDX >= max(1,N).
256 *
257 * RCOND (output) DOUBLE PRECISION
258 * Reciprocal scaled condition number. This is an estimate of the
259 * reciprocal Skeel condition number of the matrix A after
260 * equilibration (if done). If this is less than the machine
261 * precision (in particular, if it is zero), the matrix is singular
262 * to working precision. Note that the error may still be small even
263 * if this number is very small and the matrix appears ill-
264 * conditioned.
265 *
266 * RPVGRW (output) DOUBLE PRECISION
267 * Reciprocal pivot growth. On exit, this contains the reciprocal
268 * pivot growth factor norm(A)/norm(U). The "max absolute element"
269 * norm is used. If this is much less than 1, then the stability of
270 * the LU factorization of the (equilibrated) matrix A could be poor.
271 * This also means that the solution X, estimated condition numbers,
272 * and error bounds could be unreliable. If factorization fails with
273 * 0<INFO<=N, then this contains the reciprocal pivot growth factor
274 * for the leading INFO columns of A. In DGESVX, this quantity is
275 * returned in WORK(1).
276 *
277 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
278 * Componentwise relative backward error. This is the
279 * componentwise relative backward error of each solution vector X(j)
280 * (i.e., the smallest relative change in any element of A or B that
281 * makes X(j) an exact solution).
282 *
283 * N_ERR_BNDS (input) INTEGER
284 * Number of error bounds to return for each right hand side
285 * and each type (normwise or componentwise). See ERR_BNDS_NORM and
286 * ERR_BNDS_COMP below.
287 *
288 * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
289 * For each right-hand side, this array contains information about
290 * various error bounds and condition numbers corresponding to the
291 * normwise relative error, which is defined as follows:
292 *
293 * Normwise relative error in the ith solution vector:
294 * max_j (abs(XTRUE(j,i) - X(j,i)))
295 * ------------------------------
296 * max_j abs(X(j,i))
297 *
298 * The array is indexed by the type of error information as described
299 * below. There currently are up to three pieces of information
300 * returned.
301 *
302 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
303 * right-hand side.
304 *
305 * The second index in ERR_BNDS_NORM(:,err) contains the following
306 * three fields:
307 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
308 * reciprocal condition number is less than the threshold
309 * sqrt(n) * dlamch('Epsilon').
310 *
311 * err = 2 "Guaranteed" error bound: The estimated forward error,
312 * almost certainly within a factor of 10 of the true error
313 * so long as the next entry is greater than the threshold
314 * sqrt(n) * dlamch('Epsilon'). This error bound should only
315 * be trusted if the previous boolean is true.
316 *
317 * err = 3 Reciprocal condition number: Estimated normwise
318 * reciprocal condition number. Compared with the threshold
319 * sqrt(n) * dlamch('Epsilon') to determine if the error
320 * estimate is "guaranteed". These reciprocal condition
321 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
322 * appropriately scaled matrix Z.
323 * Let Z = S*A, where S scales each row by a power of the
324 * radix so all absolute row sums of Z are approximately 1.
325 *
326 * See Lapack Working Note 165 for further details and extra
327 * cautions.
328 *
329 * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
330 * For each right-hand side, this array contains information about
331 * various error bounds and condition numbers corresponding to the
332 * componentwise relative error, which is defined as follows:
333 *
334 * Componentwise relative error in the ith solution vector:
335 * abs(XTRUE(j,i) - X(j,i))
336 * max_j ----------------------
337 * abs(X(j,i))
338 *
339 * The array is indexed by the right-hand side i (on which the
340 * componentwise relative error depends), and the type of error
341 * information as described below. There currently are up to three
342 * pieces of information returned for each right-hand side. If
343 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
344 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
345 * the first (:,N_ERR_BNDS) entries are returned.
346 *
347 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
348 * right-hand side.
349 *
350 * The second index in ERR_BNDS_COMP(:,err) contains the following
351 * three fields:
352 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
353 * reciprocal condition number is less than the threshold
354 * sqrt(n) * dlamch('Epsilon').
355 *
356 * err = 2 "Guaranteed" error bound: The estimated forward error,
357 * almost certainly within a factor of 10 of the true error
358 * so long as the next entry is greater than the threshold
359 * sqrt(n) * dlamch('Epsilon'). This error bound should only
360 * be trusted if the previous boolean is true.
361 *
362 * err = 3 Reciprocal condition number: Estimated componentwise
363 * reciprocal condition number. Compared with the threshold
364 * sqrt(n) * dlamch('Epsilon') to determine if the error
365 * estimate is "guaranteed". These reciprocal condition
366 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
367 * appropriately scaled matrix Z.
368 * Let Z = S*(A*diag(x)), where x is the solution for the
369 * current right-hand side and S scales each row of
370 * A*diag(x) by a power of the radix so all absolute row
371 * sums of Z are approximately 1.
372 *
373 * See Lapack Working Note 165 for further details and extra
374 * cautions.
375 *
376 * NPARAMS (input) INTEGER
377 * Specifies the number of parameters set in PARAMS. If .LE. 0, the
378 * PARAMS array is never referenced and default values are used.
379 *
380 * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
381 * Specifies algorithm parameters. If an entry is .LT. 0.0, then
382 * that entry will be filled with default value used for that
383 * parameter. Only positions up to NPARAMS are accessed; defaults
384 * are used for higher-numbered parameters.
385 *
386 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
387 * refinement or not.
388 * Default: 1.0D+0
389 * = 0.0 : No refinement is performed, and no error bounds are
390 * computed.
391 * = 1.0 : Use the extra-precise refinement algorithm.
392 * (other values are reserved for future use)
393 *
394 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
395 * computations allowed for refinement.
396 * Default: 10
397 * Aggressive: Set to 100 to permit convergence using approximate
398 * factorizations or factorizations other than LU. If
399 * the factorization uses a technique other than
400 * Gaussian elimination, the guarantees in
401 * err_bnds_norm and err_bnds_comp may no longer be
402 * trustworthy.
403 *
404 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
405 * will attempt to find a solution with small componentwise
406 * relative error in the double-precision algorithm. Positive
407 * is true, 0.0 is false.
408 * Default: 1.0 (attempt componentwise convergence)
409 *
410 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
411 *
412 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
413 *
414 * INFO (output) INTEGER
415 * = 0: Successful exit. The solution to every right-hand side is
416 * guaranteed.
417 * < 0: If INFO = -i, the i-th argument had an illegal value
418 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
419 * has been completed, but the factor U is exactly singular, so
420 * the solution and error bounds could not be computed. RCOND = 0
421 * is returned.
422 * = N+J: The solution corresponding to the Jth right-hand side is
423 * not guaranteed. The solutions corresponding to other right-
424 * hand sides K with K > J may not be guaranteed as well, but
425 * only the first such right-hand side is reported. If a small
426 * componentwise error is not requested (PARAMS(3) = 0.0) then
427 * the Jth right-hand side is the first with a normwise error
428 * bound that is not guaranteed (the smallest J such
429 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
430 * the Jth right-hand side is the first with either a normwise or
431 * componentwise error bound that is not guaranteed (the smallest
432 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
433 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
434 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
435 * about all of the right-hand sides check ERR_BNDS_NORM or
436 * ERR_BNDS_COMP.
437 *
438 * ==================================================================
439 *
440 * .. Parameters ..
441 DOUBLE PRECISION ZERO, ONE
442 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
443 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
444 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
445 INTEGER CMP_ERR_I, PIV_GROWTH_I
446 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
447 $ BERR_I = 3 )
448 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
449 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
450 $ PIV_GROWTH_I = 9 )
451 * ..
452 * .. Local Scalars ..
453 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
454 INTEGER INFEQU, I, J, KL, KU
455 DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
456 $ ROWCND, SMLNUM
457 * ..
458 * .. External Functions ..
459 EXTERNAL LSAME, DLAMCH, ZLA_GBRPVGRW
460 LOGICAL LSAME
461 DOUBLE PRECISION DLAMCH, ZLA_GBRPVGRW
462 * ..
463 * .. External Subroutines ..
464 EXTERNAL ZGBEQUB, ZGBTRF, ZGBTRS, ZLACPY, ZLAQGB,
465 $ XERBLA, ZLASCL2, ZGBRFSX
466 * ..
467 * .. Intrinsic Functions ..
468 INTRINSIC MAX, MIN
469 * ..
470 * .. Executable Statements ..
471 *
472 INFO = 0
473 NOFACT = LSAME( FACT, 'N' )
474 EQUIL = LSAME( FACT, 'E' )
475 NOTRAN = LSAME( TRANS, 'N' )
476 SMLNUM = DLAMCH( 'Safe minimum' )
477 BIGNUM = ONE / SMLNUM
478 IF( NOFACT .OR. EQUIL ) THEN
479 EQUED = 'N'
480 ROWEQU = .FALSE.
481 COLEQU = .FALSE.
482 ELSE
483 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
484 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
485 END IF
486 *
487 * Default is failure. If an input parameter is wrong or
488 * factorization fails, make everything look horrible. Only the
489 * pivot growth is set here, the rest is initialized in ZGBRFSX.
490 *
491 RPVGRW = ZERO
492 *
493 * Test the input parameters. PARAMS is not tested until DGERFSX.
494 *
495 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
496 $ LSAME( FACT, 'F' ) ) THEN
497 INFO = -1
498 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
499 $ LSAME( TRANS, 'C' ) ) THEN
500 INFO = -2
501 ELSE IF( N.LT.0 ) THEN
502 INFO = -3
503 ELSE IF( KL.LT.0 ) THEN
504 INFO = -4
505 ELSE IF( KU.LT.0 ) THEN
506 INFO = -5
507 ELSE IF( NRHS.LT.0 ) THEN
508 INFO = -6
509 ELSE IF( LDAB.LT.KL+KU+1 ) THEN
510 INFO = -8
511 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
512 INFO = -10
513 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
514 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
515 INFO = -12
516 ELSE
517 IF( ROWEQU ) THEN
518 RCMIN = BIGNUM
519 RCMAX = ZERO
520 DO 10 J = 1, N
521 RCMIN = MIN( RCMIN, R( J ) )
522 RCMAX = MAX( RCMAX, R( J ) )
523 10 CONTINUE
524 IF( RCMIN.LE.ZERO ) THEN
525 INFO = -13
526 ELSE IF( N.GT.0 ) THEN
527 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
528 ELSE
529 ROWCND = ONE
530 END IF
531 END IF
532 IF( COLEQU .AND. INFO.EQ.0 ) THEN
533 RCMIN = BIGNUM
534 RCMAX = ZERO
535 DO 20 J = 1, N
536 RCMIN = MIN( RCMIN, C( J ) )
537 RCMAX = MAX( RCMAX, C( J ) )
538 20 CONTINUE
539 IF( RCMIN.LE.ZERO ) THEN
540 INFO = -14
541 ELSE IF( N.GT.0 ) THEN
542 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
543 ELSE
544 COLCND = ONE
545 END IF
546 END IF
547 IF( INFO.EQ.0 ) THEN
548 IF( LDB.LT.MAX( 1, N ) ) THEN
549 INFO = -15
550 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
551 INFO = -16
552 END IF
553 END IF
554 END IF
555 *
556 IF( INFO.NE.0 ) THEN
557 CALL XERBLA( 'ZGBSVXX', -INFO )
558 RETURN
559 END IF
560 *
561 IF( EQUIL ) THEN
562 *
563 * Compute row and column scalings to equilibrate the matrix A.
564 *
565 CALL ZGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
566 $ AMAX, INFEQU )
567 IF( INFEQU.EQ.0 ) THEN
568 *
569 * Equilibrate the matrix.
570 *
571 CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
572 $ AMAX, EQUED )
573 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
574 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
575 END IF
576 *
577 * If the scaling factors are not applied, set them to 1.0.
578 *
579 IF ( .NOT.ROWEQU ) THEN
580 DO J = 1, N
581 R( J ) = 1.0D+0
582 END DO
583 END IF
584 IF ( .NOT.COLEQU ) THEN
585 DO J = 1, N
586 C( J ) = 1.0D+0
587 END DO
588 END IF
589 END IF
590 *
591 * Scale the right-hand side.
592 *
593 IF( NOTRAN ) THEN
594 IF( ROWEQU ) CALL ZLASCL2( N, NRHS, R, B, LDB )
595 ELSE
596 IF( COLEQU ) CALL ZLASCL2( N, NRHS, C, B, LDB )
597 END IF
598 *
599 IF( NOFACT .OR. EQUIL ) THEN
600 *
601 * Compute the LU factorization of A.
602 *
603 DO 40, J = 1, N
604 DO 30, I = KL+1, 2*KL+KU+1
605 AFB( I, J ) = AB( I-KL, J )
606 30 CONTINUE
607 40 CONTINUE
608 CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
609 *
610 * Return if INFO is non-zero.
611 *
612 IF( INFO.GT.0 ) THEN
613 *
614 * Pivot in column INFO is exactly 0
615 * Compute the reciprocal pivot growth factor of the
616 * leading rank-deficient INFO columns of A.
617 *
618 RPVGRW = ZLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB,
619 $ LDAFB )
620 RETURN
621 END IF
622 END IF
623 *
624 * Compute the reciprocal pivot growth factor RPVGRW.
625 *
626 RPVGRW = ZLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB )
627 *
628 * Compute the solution matrix X.
629 *
630 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
631 CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
632 $ INFO )
633 *
634 * Use iterative refinement to improve the computed solution and
635 * compute error bounds and backward error estimates for it.
636 *
637 CALL ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
638 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
639 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
640 $ WORK, RWORK, INFO )
641
642 *
643 * Scale solutions.
644 *
645 IF ( COLEQU .AND. NOTRAN ) THEN
646 CALL ZLASCL2( N, NRHS, C, X, LDX )
647 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
648 CALL ZLASCL2( N, NRHS, R, X, LDX )
649 END IF
650 *
651 RETURN
652 *
653 * End of ZGBSVXX
654 *
655 END
2 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
3 $ RCOND, RPVGRW, BERR, N_ERR_BNDS,
4 $ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
5 $ WORK, RWORK, INFO )
6 *
7 * -- LAPACK driver routine (version 3.2.2) --
8 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
9 * -- Jason Riedy of Univ. of California Berkeley. --
10 * -- June 2010 --
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, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
20 $ N_ERR_BNDS
21 DOUBLE PRECISION RCOND, RPVGRW
22 * ..
23 * .. Array Arguments ..
24 INTEGER IPIV( * )
25 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), 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 * ZGBSVXX 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. ZGBSVXX 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 * ZGBSVXX 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 * ZGBSVXX 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 ZGBSVXX 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 = 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 * KL (input) INTEGER
130 * The number of subdiagonals within the band of A. KL >= 0.
131 *
132 * KU (input) INTEGER
133 * The number of superdiagonals within the band of A. KU >= 0.
134 *
135 * NRHS (input) INTEGER
136 * The number of right hand sides, i.e., the number of columns
137 * of the matrices B and X. NRHS >= 0.
138 *
139 * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
140 * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
141 * The j-th column of A is stored in the j-th column of the
142 * array AB as follows:
143 * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
144 *
145 * If FACT = 'F' and EQUED is not 'N', then AB must have been
146 * equilibrated by the scaling factors in R and/or C. AB is not
147 * modified if FACT = 'F' or 'N', or if FACT = 'E' and
148 * EQUED = 'N' on exit.
149 *
150 * On exit, if EQUED .ne. 'N', A is scaled as follows:
151 * EQUED = 'R': A := diag(R) * A
152 * EQUED = 'C': A := A * diag(C)
153 * EQUED = 'B': A := diag(R) * A * diag(C).
154 *
155 * LDAB (input) INTEGER
156 * The leading dimension of the array AB. LDAB >= KL+KU+1.
157 *
158 * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
159 * If FACT = 'F', then AFB is an input argument and on entry
160 * contains details of the LU factorization of the band matrix
161 * A, as computed by ZGBTRF. U is stored as an upper triangular
162 * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
163 * and the multipliers used during the factorization are stored
164 * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
165 * the factored form of the equilibrated matrix A.
166 *
167 * If FACT = 'N', then AF is an output argument and on exit
168 * returns the factors L and U from the factorization A = P*L*U
169 * of the original matrix A.
170 *
171 * If FACT = 'E', then AF is an output argument and on exit
172 * returns the factors L and U from the factorization A = P*L*U
173 * of the equilibrated matrix A (see the description of A for
174 * the form of the equilibrated matrix).
175 *
176 * LDAFB (input) INTEGER
177 * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
178 *
179 * IPIV (input or output) INTEGER array, dimension (N)
180 * If FACT = 'F', then IPIV is an input argument and on entry
181 * contains the pivot indices from the factorization A = P*L*U
182 * as computed by DGETRF; row i of the matrix was interchanged
183 * with row IPIV(i).
184 *
185 * If FACT = 'N', then IPIV is an output argument and on exit
186 * contains the pivot indices from the factorization A = P*L*U
187 * of the original matrix A.
188 *
189 * If FACT = 'E', then IPIV is an output argument and on exit
190 * contains the pivot indices from the factorization A = P*L*U
191 * of the equilibrated matrix A.
192 *
193 * EQUED (input or output) CHARACTER*1
194 * Specifies the form of equilibration that was done.
195 * = 'N': No equilibration (always true if FACT = 'N').
196 * = 'R': Row equilibration, i.e., A has been premultiplied by
197 * diag(R).
198 * = 'C': Column equilibration, i.e., A has been postmultiplied
199 * by diag(C).
200 * = 'B': Both row and column equilibration, i.e., A has been
201 * replaced by diag(R) * A * diag(C).
202 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
203 * output argument.
204 *
205 * R (input or output) DOUBLE PRECISION array, dimension (N)
206 * The row scale factors for A. If EQUED = 'R' or 'B', A is
207 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
208 * is not accessed. R is an input argument if FACT = 'F';
209 * otherwise, R is an output argument. If FACT = 'F' and
210 * EQUED = 'R' or 'B', each element of R must be positive.
211 * If R is output, each element of R is a power of the radix.
212 * If R is input, each element of R should be a power of the radix
213 * to ensure a reliable solution and error estimates. Scaling by
214 * powers of the radix does not cause rounding errors unless the
215 * result underflows or overflows. Rounding errors during scaling
216 * lead to refining with a matrix that is not equivalent to the
217 * input matrix, producing error estimates that may not be
218 * reliable.
219 *
220 * C (input or output) DOUBLE PRECISION array, dimension (N)
221 * The column scale factors for A. If EQUED = 'C' or 'B', A is
222 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
223 * is not accessed. C is an input argument if FACT = 'F';
224 * otherwise, C is an output argument. If FACT = 'F' and
225 * EQUED = 'C' or 'B', each element of C must be positive.
226 * If C is output, each element of C is a power of the radix.
227 * If C is input, each element of C should be a power of the radix
228 * to ensure a reliable solution and error estimates. Scaling by
229 * powers of the radix does not cause rounding errors unless the
230 * result underflows or overflows. Rounding errors during scaling
231 * lead to refining with a matrix that is not equivalent to the
232 * input matrix, producing error estimates that may not be
233 * reliable.
234 *
235 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
236 * On entry, the N-by-NRHS right hand side matrix B.
237 * On exit,
238 * if EQUED = 'N', B is not modified;
239 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
240 * diag(R)*B;
241 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
242 * overwritten by diag(C)*B.
243 *
244 * LDB (input) INTEGER
245 * The leading dimension of the array B. LDB >= max(1,N).
246 *
247 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
248 * If INFO = 0, the N-by-NRHS solution matrix X to the original
249 * system of equations. Note that A and B are modified on exit
250 * if EQUED .ne. 'N', and the solution to the equilibrated system is
251 * inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
252 * inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
253 *
254 * LDX (input) INTEGER
255 * The leading dimension of the array X. LDX >= max(1,N).
256 *
257 * RCOND (output) DOUBLE PRECISION
258 * Reciprocal scaled condition number. This is an estimate of the
259 * reciprocal Skeel condition number of the matrix A after
260 * equilibration (if done). If this is less than the machine
261 * precision (in particular, if it is zero), the matrix is singular
262 * to working precision. Note that the error may still be small even
263 * if this number is very small and the matrix appears ill-
264 * conditioned.
265 *
266 * RPVGRW (output) DOUBLE PRECISION
267 * Reciprocal pivot growth. On exit, this contains the reciprocal
268 * pivot growth factor norm(A)/norm(U). The "max absolute element"
269 * norm is used. If this is much less than 1, then the stability of
270 * the LU factorization of the (equilibrated) matrix A could be poor.
271 * This also means that the solution X, estimated condition numbers,
272 * and error bounds could be unreliable. If factorization fails with
273 * 0<INFO<=N, then this contains the reciprocal pivot growth factor
274 * for the leading INFO columns of A. In DGESVX, this quantity is
275 * returned in WORK(1).
276 *
277 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
278 * Componentwise relative backward error. This is the
279 * componentwise relative backward error of each solution vector X(j)
280 * (i.e., the smallest relative change in any element of A or B that
281 * makes X(j) an exact solution).
282 *
283 * N_ERR_BNDS (input) INTEGER
284 * Number of error bounds to return for each right hand side
285 * and each type (normwise or componentwise). See ERR_BNDS_NORM and
286 * ERR_BNDS_COMP below.
287 *
288 * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
289 * For each right-hand side, this array contains information about
290 * various error bounds and condition numbers corresponding to the
291 * normwise relative error, which is defined as follows:
292 *
293 * Normwise relative error in the ith solution vector:
294 * max_j (abs(XTRUE(j,i) - X(j,i)))
295 * ------------------------------
296 * max_j abs(X(j,i))
297 *
298 * The array is indexed by the type of error information as described
299 * below. There currently are up to three pieces of information
300 * returned.
301 *
302 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
303 * right-hand side.
304 *
305 * The second index in ERR_BNDS_NORM(:,err) contains the following
306 * three fields:
307 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
308 * reciprocal condition number is less than the threshold
309 * sqrt(n) * dlamch('Epsilon').
310 *
311 * err = 2 "Guaranteed" error bound: The estimated forward error,
312 * almost certainly within a factor of 10 of the true error
313 * so long as the next entry is greater than the threshold
314 * sqrt(n) * dlamch('Epsilon'). This error bound should only
315 * be trusted if the previous boolean is true.
316 *
317 * err = 3 Reciprocal condition number: Estimated normwise
318 * reciprocal condition number. Compared with the threshold
319 * sqrt(n) * dlamch('Epsilon') to determine if the error
320 * estimate is "guaranteed". These reciprocal condition
321 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
322 * appropriately scaled matrix Z.
323 * Let Z = S*A, where S scales each row by a power of the
324 * radix so all absolute row sums of Z are approximately 1.
325 *
326 * See Lapack Working Note 165 for further details and extra
327 * cautions.
328 *
329 * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
330 * For each right-hand side, this array contains information about
331 * various error bounds and condition numbers corresponding to the
332 * componentwise relative error, which is defined as follows:
333 *
334 * Componentwise relative error in the ith solution vector:
335 * abs(XTRUE(j,i) - X(j,i))
336 * max_j ----------------------
337 * abs(X(j,i))
338 *
339 * The array is indexed by the right-hand side i (on which the
340 * componentwise relative error depends), and the type of error
341 * information as described below. There currently are up to three
342 * pieces of information returned for each right-hand side. If
343 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
344 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
345 * the first (:,N_ERR_BNDS) entries are returned.
346 *
347 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
348 * right-hand side.
349 *
350 * The second index in ERR_BNDS_COMP(:,err) contains the following
351 * three fields:
352 * err = 1 "Trust/don't trust" boolean. Trust the answer if the
353 * reciprocal condition number is less than the threshold
354 * sqrt(n) * dlamch('Epsilon').
355 *
356 * err = 2 "Guaranteed" error bound: The estimated forward error,
357 * almost certainly within a factor of 10 of the true error
358 * so long as the next entry is greater than the threshold
359 * sqrt(n) * dlamch('Epsilon'). This error bound should only
360 * be trusted if the previous boolean is true.
361 *
362 * err = 3 Reciprocal condition number: Estimated componentwise
363 * reciprocal condition number. Compared with the threshold
364 * sqrt(n) * dlamch('Epsilon') to determine if the error
365 * estimate is "guaranteed". These reciprocal condition
366 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
367 * appropriately scaled matrix Z.
368 * Let Z = S*(A*diag(x)), where x is the solution for the
369 * current right-hand side and S scales each row of
370 * A*diag(x) by a power of the radix so all absolute row
371 * sums of Z are approximately 1.
372 *
373 * See Lapack Working Note 165 for further details and extra
374 * cautions.
375 *
376 * NPARAMS (input) INTEGER
377 * Specifies the number of parameters set in PARAMS. If .LE. 0, the
378 * PARAMS array is never referenced and default values are used.
379 *
380 * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
381 * Specifies algorithm parameters. If an entry is .LT. 0.0, then
382 * that entry will be filled with default value used for that
383 * parameter. Only positions up to NPARAMS are accessed; defaults
384 * are used for higher-numbered parameters.
385 *
386 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
387 * refinement or not.
388 * Default: 1.0D+0
389 * = 0.0 : No refinement is performed, and no error bounds are
390 * computed.
391 * = 1.0 : Use the extra-precise refinement algorithm.
392 * (other values are reserved for future use)
393 *
394 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
395 * computations allowed for refinement.
396 * Default: 10
397 * Aggressive: Set to 100 to permit convergence using approximate
398 * factorizations or factorizations other than LU. If
399 * the factorization uses a technique other than
400 * Gaussian elimination, the guarantees in
401 * err_bnds_norm and err_bnds_comp may no longer be
402 * trustworthy.
403 *
404 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
405 * will attempt to find a solution with small componentwise
406 * relative error in the double-precision algorithm. Positive
407 * is true, 0.0 is false.
408 * Default: 1.0 (attempt componentwise convergence)
409 *
410 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
411 *
412 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
413 *
414 * INFO (output) INTEGER
415 * = 0: Successful exit. The solution to every right-hand side is
416 * guaranteed.
417 * < 0: If INFO = -i, the i-th argument had an illegal value
418 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
419 * has been completed, but the factor U is exactly singular, so
420 * the solution and error bounds could not be computed. RCOND = 0
421 * is returned.
422 * = N+J: The solution corresponding to the Jth right-hand side is
423 * not guaranteed. The solutions corresponding to other right-
424 * hand sides K with K > J may not be guaranteed as well, but
425 * only the first such right-hand side is reported. If a small
426 * componentwise error is not requested (PARAMS(3) = 0.0) then
427 * the Jth right-hand side is the first with a normwise error
428 * bound that is not guaranteed (the smallest J such
429 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
430 * the Jth right-hand side is the first with either a normwise or
431 * componentwise error bound that is not guaranteed (the smallest
432 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
433 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
434 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
435 * about all of the right-hand sides check ERR_BNDS_NORM or
436 * ERR_BNDS_COMP.
437 *
438 * ==================================================================
439 *
440 * .. Parameters ..
441 DOUBLE PRECISION ZERO, ONE
442 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
443 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
444 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
445 INTEGER CMP_ERR_I, PIV_GROWTH_I
446 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
447 $ BERR_I = 3 )
448 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
449 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
450 $ PIV_GROWTH_I = 9 )
451 * ..
452 * .. Local Scalars ..
453 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
454 INTEGER INFEQU, I, J, KL, KU
455 DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
456 $ ROWCND, SMLNUM
457 * ..
458 * .. External Functions ..
459 EXTERNAL LSAME, DLAMCH, ZLA_GBRPVGRW
460 LOGICAL LSAME
461 DOUBLE PRECISION DLAMCH, ZLA_GBRPVGRW
462 * ..
463 * .. External Subroutines ..
464 EXTERNAL ZGBEQUB, ZGBTRF, ZGBTRS, ZLACPY, ZLAQGB,
465 $ XERBLA, ZLASCL2, ZGBRFSX
466 * ..
467 * .. Intrinsic Functions ..
468 INTRINSIC MAX, MIN
469 * ..
470 * .. Executable Statements ..
471 *
472 INFO = 0
473 NOFACT = LSAME( FACT, 'N' )
474 EQUIL = LSAME( FACT, 'E' )
475 NOTRAN = LSAME( TRANS, 'N' )
476 SMLNUM = DLAMCH( 'Safe minimum' )
477 BIGNUM = ONE / SMLNUM
478 IF( NOFACT .OR. EQUIL ) THEN
479 EQUED = 'N'
480 ROWEQU = .FALSE.
481 COLEQU = .FALSE.
482 ELSE
483 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
484 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
485 END IF
486 *
487 * Default is failure. If an input parameter is wrong or
488 * factorization fails, make everything look horrible. Only the
489 * pivot growth is set here, the rest is initialized in ZGBRFSX.
490 *
491 RPVGRW = ZERO
492 *
493 * Test the input parameters. PARAMS is not tested until DGERFSX.
494 *
495 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
496 $ LSAME( FACT, 'F' ) ) THEN
497 INFO = -1
498 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
499 $ LSAME( TRANS, 'C' ) ) THEN
500 INFO = -2
501 ELSE IF( N.LT.0 ) THEN
502 INFO = -3
503 ELSE IF( KL.LT.0 ) THEN
504 INFO = -4
505 ELSE IF( KU.LT.0 ) THEN
506 INFO = -5
507 ELSE IF( NRHS.LT.0 ) THEN
508 INFO = -6
509 ELSE IF( LDAB.LT.KL+KU+1 ) THEN
510 INFO = -8
511 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
512 INFO = -10
513 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
514 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
515 INFO = -12
516 ELSE
517 IF( ROWEQU ) THEN
518 RCMIN = BIGNUM
519 RCMAX = ZERO
520 DO 10 J = 1, N
521 RCMIN = MIN( RCMIN, R( J ) )
522 RCMAX = MAX( RCMAX, R( J ) )
523 10 CONTINUE
524 IF( RCMIN.LE.ZERO ) THEN
525 INFO = -13
526 ELSE IF( N.GT.0 ) THEN
527 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
528 ELSE
529 ROWCND = ONE
530 END IF
531 END IF
532 IF( COLEQU .AND. INFO.EQ.0 ) THEN
533 RCMIN = BIGNUM
534 RCMAX = ZERO
535 DO 20 J = 1, N
536 RCMIN = MIN( RCMIN, C( J ) )
537 RCMAX = MAX( RCMAX, C( J ) )
538 20 CONTINUE
539 IF( RCMIN.LE.ZERO ) THEN
540 INFO = -14
541 ELSE IF( N.GT.0 ) THEN
542 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
543 ELSE
544 COLCND = ONE
545 END IF
546 END IF
547 IF( INFO.EQ.0 ) THEN
548 IF( LDB.LT.MAX( 1, N ) ) THEN
549 INFO = -15
550 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
551 INFO = -16
552 END IF
553 END IF
554 END IF
555 *
556 IF( INFO.NE.0 ) THEN
557 CALL XERBLA( 'ZGBSVXX', -INFO )
558 RETURN
559 END IF
560 *
561 IF( EQUIL ) THEN
562 *
563 * Compute row and column scalings to equilibrate the matrix A.
564 *
565 CALL ZGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
566 $ AMAX, INFEQU )
567 IF( INFEQU.EQ.0 ) THEN
568 *
569 * Equilibrate the matrix.
570 *
571 CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
572 $ AMAX, EQUED )
573 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
574 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
575 END IF
576 *
577 * If the scaling factors are not applied, set them to 1.0.
578 *
579 IF ( .NOT.ROWEQU ) THEN
580 DO J = 1, N
581 R( J ) = 1.0D+0
582 END DO
583 END IF
584 IF ( .NOT.COLEQU ) THEN
585 DO J = 1, N
586 C( J ) = 1.0D+0
587 END DO
588 END IF
589 END IF
590 *
591 * Scale the right-hand side.
592 *
593 IF( NOTRAN ) THEN
594 IF( ROWEQU ) CALL ZLASCL2( N, NRHS, R, B, LDB )
595 ELSE
596 IF( COLEQU ) CALL ZLASCL2( N, NRHS, C, B, LDB )
597 END IF
598 *
599 IF( NOFACT .OR. EQUIL ) THEN
600 *
601 * Compute the LU factorization of A.
602 *
603 DO 40, J = 1, N
604 DO 30, I = KL+1, 2*KL+KU+1
605 AFB( I, J ) = AB( I-KL, J )
606 30 CONTINUE
607 40 CONTINUE
608 CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
609 *
610 * Return if INFO is non-zero.
611 *
612 IF( INFO.GT.0 ) THEN
613 *
614 * Pivot in column INFO is exactly 0
615 * Compute the reciprocal pivot growth factor of the
616 * leading rank-deficient INFO columns of A.
617 *
618 RPVGRW = ZLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB,
619 $ LDAFB )
620 RETURN
621 END IF
622 END IF
623 *
624 * Compute the reciprocal pivot growth factor RPVGRW.
625 *
626 RPVGRW = ZLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB )
627 *
628 * Compute the solution matrix X.
629 *
630 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
631 CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
632 $ INFO )
633 *
634 * Use iterative refinement to improve the computed solution and
635 * compute error bounds and backward error estimates for it.
636 *
637 CALL ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
638 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
639 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
640 $ WORK, RWORK, INFO )
641
642 *
643 * Scale solutions.
644 *
645 IF ( COLEQU .AND. NOTRAN ) THEN
646 CALL ZLASCL2( N, NRHS, C, X, LDX )
647 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
648 CALL ZLASCL2( N, NRHS, R, X, LDX )
649 END IF
650 *
651 RETURN
652 *
653 * End of ZGBSVXX
654 *
655 END