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