1 SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
2 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
3 $ RCOND, FERR, BERR, WORK, RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, TRANS
12 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
18 $ RWORK( * )
19 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
20 $ WORK( * ), X( LDX, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZGBSVX uses the LU factorization to compute the solution to a complex
27 * system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
28 * where A is a band matrix of order N with KL subdiagonals and KU
29 * superdiagonals, and X and B are N-by-NRHS matrices.
30 *
31 * Error bounds on the solution and a condition estimate are also
32 * provided.
33 *
34 * Description
35 * ===========
36 *
37 * The following steps are performed by this subroutine:
38 *
39 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
40 * the system:
41 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
42 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
43 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
44 * Whether or not the system will be equilibrated depends on the
45 * scaling of the matrix A, but if equilibration is used, A is
46 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
47 * or diag(C)*B (if TRANS = 'T' or 'C').
48 *
49 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
50 * matrix A (after equilibration if FACT = 'E') as
51 * A = L * U,
52 * where L is a product of permutation and unit lower triangular
53 * matrices with KL subdiagonals, and U is upper triangular with
54 * KL+KU superdiagonals.
55 *
56 * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
57 * returns with INFO = i. Otherwise, the factored form of A is used
58 * to estimate the condition number of the matrix A. If the
59 * reciprocal of the condition number is less than machine precision,
60 * INFO = N+1 is returned as a warning, but the routine still goes on
61 * to solve for X and compute error bounds as described below.
62 *
63 * 4. The system of equations is solved for X using the factored form
64 * of A.
65 *
66 * 5. Iterative refinement is applied to improve the computed solution
67 * matrix and calculate error bounds and backward error estimates
68 * for it.
69 *
70 * 6. If equilibration was used, the matrix X is premultiplied by
71 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
72 * that it solves the original system before equilibration.
73 *
74 * Arguments
75 * =========
76 *
77 * FACT (input) CHARACTER*1
78 * Specifies whether or not the factored form of the matrix A is
79 * supplied on entry, and if not, whether the matrix A should be
80 * equilibrated before it is factored.
81 * = 'F': On entry, AFB and IPIV contain the factored form of
82 * A. If EQUED is not 'N', the matrix A has been
83 * equilibrated with scaling factors given by R and C.
84 * AB, AFB, and IPIV are not modified.
85 * = 'N': The matrix A will be copied to AFB and factored.
86 * = 'E': The matrix A will be equilibrated if necessary, then
87 * copied to AFB and factored.
88 *
89 * TRANS (input) CHARACTER*1
90 * Specifies the form of the system of equations.
91 * = 'N': A * X = B (No transpose)
92 * = 'T': A**T * X = B (Transpose)
93 * = 'C': A**H * X = B (Conjugate transpose)
94 *
95 * N (input) INTEGER
96 * The number of linear equations, i.e., the order of the
97 * matrix A. N >= 0.
98 *
99 * KL (input) INTEGER
100 * The number of subdiagonals within the band of A. KL >= 0.
101 *
102 * KU (input) INTEGER
103 * The number of superdiagonals within the band of A. KU >= 0.
104 *
105 * NRHS (input) INTEGER
106 * The number of right hand sides, i.e., the number of columns
107 * of the matrices B and X. NRHS >= 0.
108 *
109 * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
110 * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
111 * The j-th column of A is stored in the j-th column of the
112 * array AB as follows:
113 * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
114 *
115 * If FACT = 'F' and EQUED is not 'N', then A must have been
116 * equilibrated by the scaling factors in R and/or C. AB is not
117 * modified if FACT = 'F' or 'N', or if FACT = 'E' and
118 * EQUED = 'N' on exit.
119 *
120 * On exit, if EQUED .ne. 'N', A is scaled as follows:
121 * EQUED = 'R': A := diag(R) * A
122 * EQUED = 'C': A := A * diag(C)
123 * EQUED = 'B': A := diag(R) * A * diag(C).
124 *
125 * LDAB (input) INTEGER
126 * The leading dimension of the array AB. LDAB >= KL+KU+1.
127 *
128 * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
129 * If FACT = 'F', then AFB is an input argument and on entry
130 * contains details of the LU factorization of the band matrix
131 * A, as computed by ZGBTRF. U is stored as an upper triangular
132 * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
133 * and the multipliers used during the factorization are stored
134 * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
135 * the factored form of the equilibrated matrix A.
136 *
137 * If FACT = 'N', then AFB is an output argument and on exit
138 * returns details of the LU factorization of A.
139 *
140 * If FACT = 'E', then AFB is an output argument and on exit
141 * returns details of the LU factorization of the equilibrated
142 * matrix A (see the description of AB for the form of the
143 * equilibrated matrix).
144 *
145 * LDAFB (input) INTEGER
146 * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
147 *
148 * IPIV (input or output) INTEGER array, dimension (N)
149 * If FACT = 'F', then IPIV is an input argument and on entry
150 * contains the pivot indices from the factorization A = L*U
151 * as computed by ZGBTRF; row i of the matrix was interchanged
152 * with row IPIV(i).
153 *
154 * If FACT = 'N', then IPIV is an output argument and on exit
155 * contains the pivot indices from the factorization A = L*U
156 * of the original matrix A.
157 *
158 * If FACT = 'E', then IPIV is an output argument and on exit
159 * contains the pivot indices from the factorization A = L*U
160 * of the equilibrated matrix A.
161 *
162 * EQUED (input or output) CHARACTER*1
163 * Specifies the form of equilibration that was done.
164 * = 'N': No equilibration (always true if FACT = 'N').
165 * = 'R': Row equilibration, i.e., A has been premultiplied by
166 * diag(R).
167 * = 'C': Column equilibration, i.e., A has been postmultiplied
168 * by diag(C).
169 * = 'B': Both row and column equilibration, i.e., A has been
170 * replaced by diag(R) * A * diag(C).
171 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
172 * output argument.
173 *
174 * R (input or output) DOUBLE PRECISION array, dimension (N)
175 * The row scale factors for A. If EQUED = 'R' or 'B', A is
176 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
177 * is not accessed. R is an input argument if FACT = 'F';
178 * otherwise, R is an output argument. If FACT = 'F' and
179 * EQUED = 'R' or 'B', each element of R must be positive.
180 *
181 * C (input or output) DOUBLE PRECISION array, dimension (N)
182 * The column scale factors for A. If EQUED = 'C' or 'B', A is
183 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
184 * is not accessed. C is an input argument if FACT = 'F';
185 * otherwise, C is an output argument. If FACT = 'F' and
186 * EQUED = 'C' or 'B', each element of C must be positive.
187 *
188 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
189 * On entry, the right hand side matrix B.
190 * On exit,
191 * if EQUED = 'N', B is not modified;
192 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
193 * diag(R)*B;
194 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
195 * overwritten by diag(C)*B.
196 *
197 * LDB (input) INTEGER
198 * The leading dimension of the array B. LDB >= max(1,N).
199 *
200 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
201 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
202 * to the original system of equations. Note that A and B are
203 * modified on exit if EQUED .ne. 'N', and the solution to the
204 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
205 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
206 * and EQUED = 'R' or 'B'.
207 *
208 * LDX (input) INTEGER
209 * The leading dimension of the array X. LDX >= max(1,N).
210 *
211 * RCOND (output) DOUBLE PRECISION
212 * The estimate of the reciprocal condition number of the matrix
213 * A after equilibration (if done). If RCOND is less than the
214 * machine precision (in particular, if RCOND = 0), the matrix
215 * is singular to working precision. This condition is
216 * indicated by a return code of INFO > 0.
217 *
218 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
219 * The estimated forward error bound for each solution vector
220 * X(j) (the j-th column of the solution matrix X).
221 * If XTRUE is the true solution corresponding to X(j), FERR(j)
222 * is an estimated upper bound for the magnitude of the largest
223 * element in (X(j) - XTRUE) divided by the magnitude of the
224 * largest element in X(j). The estimate is as reliable as
225 * the estimate for RCOND, and is almost always a slight
226 * overestimate of the true error.
227 *
228 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
229 * The componentwise relative backward error of each solution
230 * vector X(j) (i.e., the smallest relative change in
231 * any element of A or B that makes X(j) an exact solution).
232 *
233 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
234 *
235 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (N)
236 * On exit, RWORK(1) contains the reciprocal pivot growth
237 * factor norm(A)/norm(U). The "max absolute element" norm is
238 * used. If RWORK(1) is much less than 1, then the stability
239 * of the LU factorization of the (equilibrated) matrix A
240 * could be poor. This also means that the solution X, condition
241 * estimator RCOND, and forward error bound FERR could be
242 * unreliable. If factorization fails with 0<INFO<=N, then
243 * RWORK(1) contains the reciprocal pivot growth factor for the
244 * leading INFO columns of A.
245 *
246 * INFO (output) INTEGER
247 * = 0: successful exit
248 * < 0: if INFO = -i, the i-th argument had an illegal value
249 * > 0: if INFO = i, and i is
250 * <= N: U(i,i) is exactly zero. The factorization
251 * has been completed, but the factor U is exactly
252 * singular, so the solution and error bounds
253 * could not be computed. RCOND = 0 is returned.
254 * = N+1: U is nonsingular, but RCOND is less than machine
255 * precision, meaning that the matrix is singular
256 * to working precision. Nevertheless, the
257 * solution and error bounds are computed because
258 * there are a number of situations where the
259 * computed solution can be more accurate than the
260 * value of RCOND would suggest.
261 *
262 * =====================================================================
263 * Moved setting of INFO = N+1 so INFO does not subsequently get
264 * overwritten. Sven, 17 Mar 05.
265 * =====================================================================
266 *
267 * .. Parameters ..
268 DOUBLE PRECISION ZERO, ONE
269 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
270 * ..
271 * .. Local Scalars ..
272 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
273 CHARACTER NORM
274 INTEGER I, INFEQU, J, J1, J2
275 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
276 $ ROWCND, RPVGRW, SMLNUM
277 * ..
278 * .. External Functions ..
279 LOGICAL LSAME
280 DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB
281 EXTERNAL LSAME, DLAMCH, ZLANGB, ZLANTB
282 * ..
283 * .. External Subroutines ..
284 EXTERNAL XERBLA, ZCOPY, ZGBCON, ZGBEQU, ZGBRFS, ZGBTRF,
285 $ ZGBTRS, ZLACPY, ZLAQGB
286 * ..
287 * .. Intrinsic Functions ..
288 INTRINSIC ABS, MAX, MIN
289 * ..
290 * .. Executable Statements ..
291 *
292 INFO = 0
293 NOFACT = LSAME( FACT, 'N' )
294 EQUIL = LSAME( FACT, 'E' )
295 NOTRAN = LSAME( TRANS, 'N' )
296 IF( NOFACT .OR. EQUIL ) THEN
297 EQUED = 'N'
298 ROWEQU = .FALSE.
299 COLEQU = .FALSE.
300 ELSE
301 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
302 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
303 SMLNUM = DLAMCH( 'Safe minimum' )
304 BIGNUM = ONE / SMLNUM
305 END IF
306 *
307 * Test the input parameters.
308 *
309 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
310 $ THEN
311 INFO = -1
312 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
313 $ LSAME( TRANS, 'C' ) ) THEN
314 INFO = -2
315 ELSE IF( N.LT.0 ) THEN
316 INFO = -3
317 ELSE IF( KL.LT.0 ) THEN
318 INFO = -4
319 ELSE IF( KU.LT.0 ) THEN
320 INFO = -5
321 ELSE IF( NRHS.LT.0 ) THEN
322 INFO = -6
323 ELSE IF( LDAB.LT.KL+KU+1 ) THEN
324 INFO = -8
325 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
326 INFO = -10
327 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
328 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
329 INFO = -12
330 ELSE
331 IF( ROWEQU ) THEN
332 RCMIN = BIGNUM
333 RCMAX = ZERO
334 DO 10 J = 1, N
335 RCMIN = MIN( RCMIN, R( J ) )
336 RCMAX = MAX( RCMAX, R( J ) )
337 10 CONTINUE
338 IF( RCMIN.LE.ZERO ) THEN
339 INFO = -13
340 ELSE IF( N.GT.0 ) THEN
341 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
342 ELSE
343 ROWCND = ONE
344 END IF
345 END IF
346 IF( COLEQU .AND. INFO.EQ.0 ) THEN
347 RCMIN = BIGNUM
348 RCMAX = ZERO
349 DO 20 J = 1, N
350 RCMIN = MIN( RCMIN, C( J ) )
351 RCMAX = MAX( RCMAX, C( J ) )
352 20 CONTINUE
353 IF( RCMIN.LE.ZERO ) THEN
354 INFO = -14
355 ELSE IF( N.GT.0 ) THEN
356 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
357 ELSE
358 COLCND = ONE
359 END IF
360 END IF
361 IF( INFO.EQ.0 ) THEN
362 IF( LDB.LT.MAX( 1, N ) ) THEN
363 INFO = -16
364 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
365 INFO = -18
366 END IF
367 END IF
368 END IF
369 *
370 IF( INFO.NE.0 ) THEN
371 CALL XERBLA( 'ZGBSVX', -INFO )
372 RETURN
373 END IF
374 *
375 IF( EQUIL ) THEN
376 *
377 * Compute row and column scalings to equilibrate the matrix A.
378 *
379 CALL ZGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
380 $ AMAX, INFEQU )
381 IF( INFEQU.EQ.0 ) THEN
382 *
383 * Equilibrate the matrix.
384 *
385 CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
386 $ AMAX, EQUED )
387 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
388 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
389 END IF
390 END IF
391 *
392 * Scale the right hand side.
393 *
394 IF( NOTRAN ) THEN
395 IF( ROWEQU ) THEN
396 DO 40 J = 1, NRHS
397 DO 30 I = 1, N
398 B( I, J ) = R( I )*B( I, J )
399 30 CONTINUE
400 40 CONTINUE
401 END IF
402 ELSE IF( COLEQU ) THEN
403 DO 60 J = 1, NRHS
404 DO 50 I = 1, N
405 B( I, J ) = C( I )*B( I, J )
406 50 CONTINUE
407 60 CONTINUE
408 END IF
409 *
410 IF( NOFACT .OR. EQUIL ) THEN
411 *
412 * Compute the LU factorization of the band matrix A.
413 *
414 DO 70 J = 1, N
415 J1 = MAX( J-KU, 1 )
416 J2 = MIN( J+KL, N )
417 CALL ZCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
418 $ AFB( KL+KU+1-J+J1, J ), 1 )
419 70 CONTINUE
420 *
421 CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
422 *
423 * Return if INFO is non-zero.
424 *
425 IF( INFO.GT.0 ) THEN
426 *
427 * Compute the reciprocal pivot growth factor of the
428 * leading rank-deficient INFO columns of A.
429 *
430 ANORM = ZERO
431 DO 90 J = 1, INFO
432 DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
433 ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
434 80 CONTINUE
435 90 CONTINUE
436 RPVGRW = ZLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
437 $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
438 $ RWORK )
439 IF( RPVGRW.EQ.ZERO ) THEN
440 RPVGRW = ONE
441 ELSE
442 RPVGRW = ANORM / RPVGRW
443 END IF
444 RWORK( 1 ) = RPVGRW
445 RCOND = ZERO
446 RETURN
447 END IF
448 END IF
449 *
450 * Compute the norm of the matrix A and the
451 * reciprocal pivot growth factor RPVGRW.
452 *
453 IF( NOTRAN ) THEN
454 NORM = '1'
455 ELSE
456 NORM = 'I'
457 END IF
458 ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
459 RPVGRW = ZLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
460 IF( RPVGRW.EQ.ZERO ) THEN
461 RPVGRW = ONE
462 ELSE
463 RPVGRW = ZLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW
464 END IF
465 *
466 * Compute the reciprocal of the condition number of A.
467 *
468 CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
469 $ WORK, RWORK, INFO )
470 *
471 * Compute the solution matrix X.
472 *
473 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
474 CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
475 $ INFO )
476 *
477 * Use iterative refinement to improve the computed solution and
478 * compute error bounds and backward error estimates for it.
479 *
480 CALL ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
481 $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
482 *
483 * Transform the solution matrix X to a solution of the original
484 * system.
485 *
486 IF( NOTRAN ) THEN
487 IF( COLEQU ) THEN
488 DO 110 J = 1, NRHS
489 DO 100 I = 1, N
490 X( I, J ) = C( I )*X( I, J )
491 100 CONTINUE
492 110 CONTINUE
493 DO 120 J = 1, NRHS
494 FERR( J ) = FERR( J ) / COLCND
495 120 CONTINUE
496 END IF
497 ELSE IF( ROWEQU ) THEN
498 DO 140 J = 1, NRHS
499 DO 130 I = 1, N
500 X( I, J ) = R( I )*X( I, J )
501 130 CONTINUE
502 140 CONTINUE
503 DO 150 J = 1, NRHS
504 FERR( J ) = FERR( J ) / ROWCND
505 150 CONTINUE
506 END IF
507 *
508 * Set INFO = N+1 if the matrix is singular to working precision.
509 *
510 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
511 $ INFO = N + 1
512 *
513 RWORK( 1 ) = RPVGRW
514 RETURN
515 *
516 * End of ZGBSVX
517 *
518 END
2 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
3 $ RCOND, FERR, BERR, WORK, RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, TRANS
12 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
18 $ RWORK( * )
19 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
20 $ WORK( * ), X( LDX, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZGBSVX uses the LU factorization to compute the solution to a complex
27 * system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
28 * where A is a band matrix of order N with KL subdiagonals and KU
29 * superdiagonals, and X and B are N-by-NRHS matrices.
30 *
31 * Error bounds on the solution and a condition estimate are also
32 * provided.
33 *
34 * Description
35 * ===========
36 *
37 * The following steps are performed by this subroutine:
38 *
39 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
40 * the system:
41 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
42 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
43 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
44 * Whether or not the system will be equilibrated depends on the
45 * scaling of the matrix A, but if equilibration is used, A is
46 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
47 * or diag(C)*B (if TRANS = 'T' or 'C').
48 *
49 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
50 * matrix A (after equilibration if FACT = 'E') as
51 * A = L * U,
52 * where L is a product of permutation and unit lower triangular
53 * matrices with KL subdiagonals, and U is upper triangular with
54 * KL+KU superdiagonals.
55 *
56 * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
57 * returns with INFO = i. Otherwise, the factored form of A is used
58 * to estimate the condition number of the matrix A. If the
59 * reciprocal of the condition number is less than machine precision,
60 * INFO = N+1 is returned as a warning, but the routine still goes on
61 * to solve for X and compute error bounds as described below.
62 *
63 * 4. The system of equations is solved for X using the factored form
64 * of A.
65 *
66 * 5. Iterative refinement is applied to improve the computed solution
67 * matrix and calculate error bounds and backward error estimates
68 * for it.
69 *
70 * 6. If equilibration was used, the matrix X is premultiplied by
71 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
72 * that it solves the original system before equilibration.
73 *
74 * Arguments
75 * =========
76 *
77 * FACT (input) CHARACTER*1
78 * Specifies whether or not the factored form of the matrix A is
79 * supplied on entry, and if not, whether the matrix A should be
80 * equilibrated before it is factored.
81 * = 'F': On entry, AFB and IPIV contain the factored form of
82 * A. If EQUED is not 'N', the matrix A has been
83 * equilibrated with scaling factors given by R and C.
84 * AB, AFB, and IPIV are not modified.
85 * = 'N': The matrix A will be copied to AFB and factored.
86 * = 'E': The matrix A will be equilibrated if necessary, then
87 * copied to AFB and factored.
88 *
89 * TRANS (input) CHARACTER*1
90 * Specifies the form of the system of equations.
91 * = 'N': A * X = B (No transpose)
92 * = 'T': A**T * X = B (Transpose)
93 * = 'C': A**H * X = B (Conjugate transpose)
94 *
95 * N (input) INTEGER
96 * The number of linear equations, i.e., the order of the
97 * matrix A. N >= 0.
98 *
99 * KL (input) INTEGER
100 * The number of subdiagonals within the band of A. KL >= 0.
101 *
102 * KU (input) INTEGER
103 * The number of superdiagonals within the band of A. KU >= 0.
104 *
105 * NRHS (input) INTEGER
106 * The number of right hand sides, i.e., the number of columns
107 * of the matrices B and X. NRHS >= 0.
108 *
109 * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
110 * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
111 * The j-th column of A is stored in the j-th column of the
112 * array AB as follows:
113 * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
114 *
115 * If FACT = 'F' and EQUED is not 'N', then A must have been
116 * equilibrated by the scaling factors in R and/or C. AB is not
117 * modified if FACT = 'F' or 'N', or if FACT = 'E' and
118 * EQUED = 'N' on exit.
119 *
120 * On exit, if EQUED .ne. 'N', A is scaled as follows:
121 * EQUED = 'R': A := diag(R) * A
122 * EQUED = 'C': A := A * diag(C)
123 * EQUED = 'B': A := diag(R) * A * diag(C).
124 *
125 * LDAB (input) INTEGER
126 * The leading dimension of the array AB. LDAB >= KL+KU+1.
127 *
128 * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
129 * If FACT = 'F', then AFB is an input argument and on entry
130 * contains details of the LU factorization of the band matrix
131 * A, as computed by ZGBTRF. U is stored as an upper triangular
132 * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
133 * and the multipliers used during the factorization are stored
134 * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
135 * the factored form of the equilibrated matrix A.
136 *
137 * If FACT = 'N', then AFB is an output argument and on exit
138 * returns details of the LU factorization of A.
139 *
140 * If FACT = 'E', then AFB is an output argument and on exit
141 * returns details of the LU factorization of the equilibrated
142 * matrix A (see the description of AB for the form of the
143 * equilibrated matrix).
144 *
145 * LDAFB (input) INTEGER
146 * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
147 *
148 * IPIV (input or output) INTEGER array, dimension (N)
149 * If FACT = 'F', then IPIV is an input argument and on entry
150 * contains the pivot indices from the factorization A = L*U
151 * as computed by ZGBTRF; row i of the matrix was interchanged
152 * with row IPIV(i).
153 *
154 * If FACT = 'N', then IPIV is an output argument and on exit
155 * contains the pivot indices from the factorization A = L*U
156 * of the original matrix A.
157 *
158 * If FACT = 'E', then IPIV is an output argument and on exit
159 * contains the pivot indices from the factorization A = L*U
160 * of the equilibrated matrix A.
161 *
162 * EQUED (input or output) CHARACTER*1
163 * Specifies the form of equilibration that was done.
164 * = 'N': No equilibration (always true if FACT = 'N').
165 * = 'R': Row equilibration, i.e., A has been premultiplied by
166 * diag(R).
167 * = 'C': Column equilibration, i.e., A has been postmultiplied
168 * by diag(C).
169 * = 'B': Both row and column equilibration, i.e., A has been
170 * replaced by diag(R) * A * diag(C).
171 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
172 * output argument.
173 *
174 * R (input or output) DOUBLE PRECISION array, dimension (N)
175 * The row scale factors for A. If EQUED = 'R' or 'B', A is
176 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
177 * is not accessed. R is an input argument if FACT = 'F';
178 * otherwise, R is an output argument. If FACT = 'F' and
179 * EQUED = 'R' or 'B', each element of R must be positive.
180 *
181 * C (input or output) DOUBLE PRECISION array, dimension (N)
182 * The column scale factors for A. If EQUED = 'C' or 'B', A is
183 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
184 * is not accessed. C is an input argument if FACT = 'F';
185 * otherwise, C is an output argument. If FACT = 'F' and
186 * EQUED = 'C' or 'B', each element of C must be positive.
187 *
188 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
189 * On entry, the right hand side matrix B.
190 * On exit,
191 * if EQUED = 'N', B is not modified;
192 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
193 * diag(R)*B;
194 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
195 * overwritten by diag(C)*B.
196 *
197 * LDB (input) INTEGER
198 * The leading dimension of the array B. LDB >= max(1,N).
199 *
200 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
201 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
202 * to the original system of equations. Note that A and B are
203 * modified on exit if EQUED .ne. 'N', and the solution to the
204 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
205 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
206 * and EQUED = 'R' or 'B'.
207 *
208 * LDX (input) INTEGER
209 * The leading dimension of the array X. LDX >= max(1,N).
210 *
211 * RCOND (output) DOUBLE PRECISION
212 * The estimate of the reciprocal condition number of the matrix
213 * A after equilibration (if done). If RCOND is less than the
214 * machine precision (in particular, if RCOND = 0), the matrix
215 * is singular to working precision. This condition is
216 * indicated by a return code of INFO > 0.
217 *
218 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
219 * The estimated forward error bound for each solution vector
220 * X(j) (the j-th column of the solution matrix X).
221 * If XTRUE is the true solution corresponding to X(j), FERR(j)
222 * is an estimated upper bound for the magnitude of the largest
223 * element in (X(j) - XTRUE) divided by the magnitude of the
224 * largest element in X(j). The estimate is as reliable as
225 * the estimate for RCOND, and is almost always a slight
226 * overestimate of the true error.
227 *
228 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
229 * The componentwise relative backward error of each solution
230 * vector X(j) (i.e., the smallest relative change in
231 * any element of A or B that makes X(j) an exact solution).
232 *
233 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
234 *
235 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (N)
236 * On exit, RWORK(1) contains the reciprocal pivot growth
237 * factor norm(A)/norm(U). The "max absolute element" norm is
238 * used. If RWORK(1) is much less than 1, then the stability
239 * of the LU factorization of the (equilibrated) matrix A
240 * could be poor. This also means that the solution X, condition
241 * estimator RCOND, and forward error bound FERR could be
242 * unreliable. If factorization fails with 0<INFO<=N, then
243 * RWORK(1) contains the reciprocal pivot growth factor for the
244 * leading INFO columns of A.
245 *
246 * INFO (output) INTEGER
247 * = 0: successful exit
248 * < 0: if INFO = -i, the i-th argument had an illegal value
249 * > 0: if INFO = i, and i is
250 * <= N: U(i,i) is exactly zero. The factorization
251 * has been completed, but the factor U is exactly
252 * singular, so the solution and error bounds
253 * could not be computed. RCOND = 0 is returned.
254 * = N+1: U is nonsingular, but RCOND is less than machine
255 * precision, meaning that the matrix is singular
256 * to working precision. Nevertheless, the
257 * solution and error bounds are computed because
258 * there are a number of situations where the
259 * computed solution can be more accurate than the
260 * value of RCOND would suggest.
261 *
262 * =====================================================================
263 * Moved setting of INFO = N+1 so INFO does not subsequently get
264 * overwritten. Sven, 17 Mar 05.
265 * =====================================================================
266 *
267 * .. Parameters ..
268 DOUBLE PRECISION ZERO, ONE
269 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
270 * ..
271 * .. Local Scalars ..
272 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
273 CHARACTER NORM
274 INTEGER I, INFEQU, J, J1, J2
275 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
276 $ ROWCND, RPVGRW, SMLNUM
277 * ..
278 * .. External Functions ..
279 LOGICAL LSAME
280 DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB
281 EXTERNAL LSAME, DLAMCH, ZLANGB, ZLANTB
282 * ..
283 * .. External Subroutines ..
284 EXTERNAL XERBLA, ZCOPY, ZGBCON, ZGBEQU, ZGBRFS, ZGBTRF,
285 $ ZGBTRS, ZLACPY, ZLAQGB
286 * ..
287 * .. Intrinsic Functions ..
288 INTRINSIC ABS, MAX, MIN
289 * ..
290 * .. Executable Statements ..
291 *
292 INFO = 0
293 NOFACT = LSAME( FACT, 'N' )
294 EQUIL = LSAME( FACT, 'E' )
295 NOTRAN = LSAME( TRANS, 'N' )
296 IF( NOFACT .OR. EQUIL ) THEN
297 EQUED = 'N'
298 ROWEQU = .FALSE.
299 COLEQU = .FALSE.
300 ELSE
301 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
302 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
303 SMLNUM = DLAMCH( 'Safe minimum' )
304 BIGNUM = ONE / SMLNUM
305 END IF
306 *
307 * Test the input parameters.
308 *
309 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
310 $ THEN
311 INFO = -1
312 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
313 $ LSAME( TRANS, 'C' ) ) THEN
314 INFO = -2
315 ELSE IF( N.LT.0 ) THEN
316 INFO = -3
317 ELSE IF( KL.LT.0 ) THEN
318 INFO = -4
319 ELSE IF( KU.LT.0 ) THEN
320 INFO = -5
321 ELSE IF( NRHS.LT.0 ) THEN
322 INFO = -6
323 ELSE IF( LDAB.LT.KL+KU+1 ) THEN
324 INFO = -8
325 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
326 INFO = -10
327 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
328 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
329 INFO = -12
330 ELSE
331 IF( ROWEQU ) THEN
332 RCMIN = BIGNUM
333 RCMAX = ZERO
334 DO 10 J = 1, N
335 RCMIN = MIN( RCMIN, R( J ) )
336 RCMAX = MAX( RCMAX, R( J ) )
337 10 CONTINUE
338 IF( RCMIN.LE.ZERO ) THEN
339 INFO = -13
340 ELSE IF( N.GT.0 ) THEN
341 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
342 ELSE
343 ROWCND = ONE
344 END IF
345 END IF
346 IF( COLEQU .AND. INFO.EQ.0 ) THEN
347 RCMIN = BIGNUM
348 RCMAX = ZERO
349 DO 20 J = 1, N
350 RCMIN = MIN( RCMIN, C( J ) )
351 RCMAX = MAX( RCMAX, C( J ) )
352 20 CONTINUE
353 IF( RCMIN.LE.ZERO ) THEN
354 INFO = -14
355 ELSE IF( N.GT.0 ) THEN
356 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
357 ELSE
358 COLCND = ONE
359 END IF
360 END IF
361 IF( INFO.EQ.0 ) THEN
362 IF( LDB.LT.MAX( 1, N ) ) THEN
363 INFO = -16
364 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
365 INFO = -18
366 END IF
367 END IF
368 END IF
369 *
370 IF( INFO.NE.0 ) THEN
371 CALL XERBLA( 'ZGBSVX', -INFO )
372 RETURN
373 END IF
374 *
375 IF( EQUIL ) THEN
376 *
377 * Compute row and column scalings to equilibrate the matrix A.
378 *
379 CALL ZGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
380 $ AMAX, INFEQU )
381 IF( INFEQU.EQ.0 ) THEN
382 *
383 * Equilibrate the matrix.
384 *
385 CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
386 $ AMAX, EQUED )
387 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
388 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
389 END IF
390 END IF
391 *
392 * Scale the right hand side.
393 *
394 IF( NOTRAN ) THEN
395 IF( ROWEQU ) THEN
396 DO 40 J = 1, NRHS
397 DO 30 I = 1, N
398 B( I, J ) = R( I )*B( I, J )
399 30 CONTINUE
400 40 CONTINUE
401 END IF
402 ELSE IF( COLEQU ) THEN
403 DO 60 J = 1, NRHS
404 DO 50 I = 1, N
405 B( I, J ) = C( I )*B( I, J )
406 50 CONTINUE
407 60 CONTINUE
408 END IF
409 *
410 IF( NOFACT .OR. EQUIL ) THEN
411 *
412 * Compute the LU factorization of the band matrix A.
413 *
414 DO 70 J = 1, N
415 J1 = MAX( J-KU, 1 )
416 J2 = MIN( J+KL, N )
417 CALL ZCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
418 $ AFB( KL+KU+1-J+J1, J ), 1 )
419 70 CONTINUE
420 *
421 CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
422 *
423 * Return if INFO is non-zero.
424 *
425 IF( INFO.GT.0 ) THEN
426 *
427 * Compute the reciprocal pivot growth factor of the
428 * leading rank-deficient INFO columns of A.
429 *
430 ANORM = ZERO
431 DO 90 J = 1, INFO
432 DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
433 ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
434 80 CONTINUE
435 90 CONTINUE
436 RPVGRW = ZLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
437 $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
438 $ RWORK )
439 IF( RPVGRW.EQ.ZERO ) THEN
440 RPVGRW = ONE
441 ELSE
442 RPVGRW = ANORM / RPVGRW
443 END IF
444 RWORK( 1 ) = RPVGRW
445 RCOND = ZERO
446 RETURN
447 END IF
448 END IF
449 *
450 * Compute the norm of the matrix A and the
451 * reciprocal pivot growth factor RPVGRW.
452 *
453 IF( NOTRAN ) THEN
454 NORM = '1'
455 ELSE
456 NORM = 'I'
457 END IF
458 ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
459 RPVGRW = ZLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
460 IF( RPVGRW.EQ.ZERO ) THEN
461 RPVGRW = ONE
462 ELSE
463 RPVGRW = ZLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW
464 END IF
465 *
466 * Compute the reciprocal of the condition number of A.
467 *
468 CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
469 $ WORK, RWORK, INFO )
470 *
471 * Compute the solution matrix X.
472 *
473 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
474 CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
475 $ INFO )
476 *
477 * Use iterative refinement to improve the computed solution and
478 * compute error bounds and backward error estimates for it.
479 *
480 CALL ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
481 $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
482 *
483 * Transform the solution matrix X to a solution of the original
484 * system.
485 *
486 IF( NOTRAN ) THEN
487 IF( COLEQU ) THEN
488 DO 110 J = 1, NRHS
489 DO 100 I = 1, N
490 X( I, J ) = C( I )*X( I, J )
491 100 CONTINUE
492 110 CONTINUE
493 DO 120 J = 1, NRHS
494 FERR( J ) = FERR( J ) / COLCND
495 120 CONTINUE
496 END IF
497 ELSE IF( ROWEQU ) THEN
498 DO 140 J = 1, NRHS
499 DO 130 I = 1, N
500 X( I, J ) = R( I )*X( I, J )
501 130 CONTINUE
502 140 CONTINUE
503 DO 150 J = 1, NRHS
504 FERR( J ) = FERR( J ) / ROWCND
505 150 CONTINUE
506 END IF
507 *
508 * Set INFO = N+1 if the matrix is singular to working precision.
509 *
510 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
511 $ INFO = N + 1
512 *
513 RWORK( 1 ) = RPVGRW
514 RETURN
515 *
516 * End of ZGBSVX
517 *
518 END