001:       SUBROUTINE CGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
002:      $                   LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
003:      $                   RCOND, FERR, BERR, WORK, RWORK, INFO )
004: *
005: *  -- LAPACK driver routine (version 3.2) --
006: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
007: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
008: *     November 2006
009: *
010: *     .. Scalar Arguments ..
011:       CHARACTER          EQUED, FACT, TRANS
012:       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
013:       REAL               RCOND
014: *     ..
015: *     .. Array Arguments ..
016:       INTEGER            IPIV( * )
017:       REAL               BERR( * ), C( * ), FERR( * ), R( * ),
018:      $                   RWORK( * )
019:       COMPLEX            AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
020:      $                   WORK( * ), X( LDX, * )
021: *     ..
022: *
023: *  Purpose
024: *  =======
025: *
026: *  CGBSVX uses the LU factorization to compute the solution to a complex
027: *  system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
028: *  where A is a band matrix of order N with KL subdiagonals and KU
029: *  superdiagonals, and X and B are N-by-NRHS matrices.
030: *
031: *  Error bounds on the solution and a condition estimate are also
032: *  provided.
033: *
034: *  Description
035: *  ===========
036: *
037: *  The following steps are performed by this subroutine:
038: *
039: *  1. If FACT = 'E', real scaling factors are computed to equilibrate
040: *     the system:
041: *        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
042: *        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
043: *        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
044: *     Whether or not the system will be equilibrated depends on the
045: *     scaling of the matrix A, but if equilibration is used, A is
046: *     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
047: *     or diag(C)*B (if TRANS = 'T' or 'C').
048: *
049: *  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
050: *     matrix A (after equilibration if FACT = 'E') as
051: *        A = L * U,
052: *     where L is a product of permutation and unit lower triangular
053: *     matrices with KL subdiagonals, and U is upper triangular with
054: *     KL+KU superdiagonals.
055: *
056: *  3. If some U(i,i)=0, so that U is exactly singular, then the routine
057: *     returns with INFO = i. Otherwise, the factored form of A is used
058: *     to estimate the condition number of the matrix A.  If the
059: *     reciprocal of the condition number is less than machine precision,
060: *     INFO = N+1 is returned as a warning, but the routine still goes on
061: *     to solve for X and compute error bounds as described below.
062: *
063: *  4. The system of equations is solved for X using the factored form
064: *     of A.
065: *
066: *  5. Iterative refinement is applied to improve the computed solution
067: *     matrix and calculate error bounds and backward error estimates
068: *     for it.
069: *
070: *  6. If equilibration was used, the matrix X is premultiplied by
071: *     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
072: *     that it solves the original system before equilibration.
073: *
074: *  Arguments
075: *  =========
076: *
077: *  FACT    (input) CHARACTER*1
078: *          Specifies whether or not the factored form of the matrix A is
079: *          supplied on entry, and if not, whether the matrix A should be
080: *          equilibrated before it is factored.
081: *          = 'F':  On entry, AFB and IPIV contain the factored form of
082: *                  A.  If EQUED is not 'N', the matrix A has been
083: *                  equilibrated with scaling factors given by R and C.
084: *                  AB, AFB, and IPIV are not modified.
085: *          = 'N':  The matrix A will be copied to AFB and factored.
086: *          = 'E':  The matrix A will be equilibrated if necessary, then
087: *                  copied to AFB and factored.
088: *
089: *  TRANS   (input) CHARACTER*1
090: *          Specifies the form of the system of equations.
091: *          = 'N':  A * X = B     (No transpose)
092: *          = 'T':  A**T * X = B  (Transpose)
093: *          = 'C':  A**H * X = B  (Conjugate transpose)
094: *
095: *  N       (input) INTEGER
096: *          The number of linear equations, i.e., the order of the
097: *          matrix A.  N >= 0.
098: *
099: *  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 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 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 CGBTRF.  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 CGBTRF; 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) REAL 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) REAL 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 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 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) REAL
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) REAL 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) REAL 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 array, dimension (2*N)
234: *
235: *  RWORK   (workspace/output) REAL 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:       REAL               ZERO, ONE
269:       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
270: *     ..
271: *     .. Local Scalars ..
272:       LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
273:       CHARACTER          NORM
274:       INTEGER            I, INFEQU, J, J1, J2
275:       REAL               AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
276:      $                   ROWCND, RPVGRW, SMLNUM
277: *     ..
278: *     .. External Functions ..
279:       LOGICAL            LSAME
280:       REAL               CLANGB, CLANTB, SLAMCH
281:       EXTERNAL           LSAME, CLANGB, CLANTB, SLAMCH
282: *     ..
283: *     .. External Subroutines ..
284:       EXTERNAL           CCOPY, CGBCON, CGBEQU, CGBRFS, CGBTRF, CGBTRS,
285:      $                   CLACPY, CLAQGB, XERBLA
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 = SLAMCH( '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( 'CGBSVX', -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 CGBEQU( 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 CLAQGB( 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 CCOPY( 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 CGBTRF( 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 = CLANTB( '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 = CLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
459:       RPVGRW = CLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
460:       IF( RPVGRW.EQ.ZERO ) THEN
461:          RPVGRW = ONE
462:       ELSE
463:          RPVGRW = CLANGB( '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 CGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
469:      $             WORK, RWORK, INFO )
470: *
471: *     Compute the solution matrix X.
472: *
473:       CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
474:       CALL CGBTRS( 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 CGBRFS( 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.SLAMCH( 'Epsilon' ) )
511:      $   INFO = N + 1
512: *
513:       RWORK( 1 ) = RPVGRW
514:       RETURN
515: *
516: *     End of CGBSVX
517: *
518:       END
519: