001:       SUBROUTINE ZGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
002:      $                   LDQ, PT, LDPT, C, LDC, WORK, RWORK, INFO )
003: *
004: *  -- LAPACK routine (version 3.2) --
005: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
006: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
007: *     November 2006
008: *
009: *     .. Scalar Arguments ..
010:       CHARACTER          VECT
011:       INTEGER            INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
012: *     ..
013: *     .. Array Arguments ..
014:       DOUBLE PRECISION   D( * ), E( * ), RWORK( * )
015:       COMPLEX*16         AB( LDAB, * ), C( LDC, * ), PT( LDPT, * ),
016:      $                   Q( LDQ, * ), WORK( * )
017: *     ..
018: *
019: *  Purpose
020: *  =======
021: *
022: *  ZGBBRD reduces a complex general m-by-n band matrix A to real upper
023: *  bidiagonal form B by a unitary transformation: Q' * A * P = B.
024: *
025: *  The routine computes B, and optionally forms Q or P', or computes
026: *  Q'*C for a given matrix C.
027: *
028: *  Arguments
029: *  =========
030: *
031: *  VECT    (input) CHARACTER*1
032: *          Specifies whether or not the matrices Q and P' are to be
033: *          formed.
034: *          = 'N': do not form Q or P';
035: *          = 'Q': form Q only;
036: *          = 'P': form P' only;
037: *          = 'B': form both.
038: *
039: *  M       (input) INTEGER
040: *          The number of rows of the matrix A.  M >= 0.
041: *
042: *  N       (input) INTEGER
043: *          The number of columns of the matrix A.  N >= 0.
044: *
045: *  NCC     (input) INTEGER
046: *          The number of columns of the matrix C.  NCC >= 0.
047: *
048: *  KL      (input) INTEGER
049: *          The number of subdiagonals of the matrix A. KL >= 0.
050: *
051: *  KU      (input) INTEGER
052: *          The number of superdiagonals of the matrix A. KU >= 0.
053: *
054: *  AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
055: *          On entry, the m-by-n band matrix A, stored in rows 1 to
056: *          KL+KU+1. The j-th column of A is stored in the j-th column of
057: *          the array AB as follows:
058: *          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
059: *          On exit, A is overwritten by values generated during the
060: *          reduction.
061: *
062: *  LDAB    (input) INTEGER
063: *          The leading dimension of the array A. LDAB >= KL+KU+1.
064: *
065: *  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
066: *          The diagonal elements of the bidiagonal matrix B.
067: *
068: *  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
069: *          The superdiagonal elements of the bidiagonal matrix B.
070: *
071: *  Q       (output) COMPLEX*16 array, dimension (LDQ,M)
072: *          If VECT = 'Q' or 'B', the m-by-m unitary matrix Q.
073: *          If VECT = 'N' or 'P', the array Q is not referenced.
074: *
075: *  LDQ     (input) INTEGER
076: *          The leading dimension of the array Q.
077: *          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
078: *
079: *  PT      (output) COMPLEX*16 array, dimension (LDPT,N)
080: *          If VECT = 'P' or 'B', the n-by-n unitary matrix P'.
081: *          If VECT = 'N' or 'Q', the array PT is not referenced.
082: *
083: *  LDPT    (input) INTEGER
084: *          The leading dimension of the array PT.
085: *          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
086: *
087: *  C       (input/output) COMPLEX*16 array, dimension (LDC,NCC)
088: *          On entry, an m-by-ncc matrix C.
089: *          On exit, C is overwritten by Q'*C.
090: *          C is not referenced if NCC = 0.
091: *
092: *  LDC     (input) INTEGER
093: *          The leading dimension of the array C.
094: *          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
095: *
096: *  WORK    (workspace) COMPLEX*16 array, dimension (max(M,N))
097: *
098: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (max(M,N))
099: *
100: *  INFO    (output) INTEGER
101: *          = 0:  successful exit.
102: *          < 0:  if INFO = -i, the i-th argument had an illegal value.
103: *
104: *  =====================================================================
105: *
106: *     .. Parameters ..
107:       DOUBLE PRECISION   ZERO
108:       PARAMETER          ( ZERO = 0.0D+0 )
109:       COMPLEX*16         CZERO, CONE
110:       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
111:      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
112: *     ..
113: *     .. Local Scalars ..
114:       LOGICAL            WANTB, WANTC, WANTPT, WANTQ
115:       INTEGER            I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
116:      $                   KUN, L, MINMN, ML, ML0, MU, MU0, NR, NRT
117:       DOUBLE PRECISION   ABST, RC
118:       COMPLEX*16         RA, RB, RS, T
119: *     ..
120: *     .. External Subroutines ..
121:       EXTERNAL           XERBLA, ZLARGV, ZLARTG, ZLARTV, ZLASET, ZROT,
122:      $                   ZSCAL
123: *     ..
124: *     .. Intrinsic Functions ..
125:       INTRINSIC          ABS, DCONJG, MAX, MIN
126: *     ..
127: *     .. External Functions ..
128:       LOGICAL            LSAME
129:       EXTERNAL           LSAME
130: *     ..
131: *     .. Executable Statements ..
132: *
133: *     Test the input parameters
134: *
135:       WANTB = LSAME( VECT, 'B' )
136:       WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
137:       WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
138:       WANTC = NCC.GT.0
139:       KLU1 = KL + KU + 1
140:       INFO = 0
141:       IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
142:      $     THEN
143:          INFO = -1
144:       ELSE IF( M.LT.0 ) THEN
145:          INFO = -2
146:       ELSE IF( N.LT.0 ) THEN
147:          INFO = -3
148:       ELSE IF( NCC.LT.0 ) THEN
149:          INFO = -4
150:       ELSE IF( KL.LT.0 ) THEN
151:          INFO = -5
152:       ELSE IF( KU.LT.0 ) THEN
153:          INFO = -6
154:       ELSE IF( LDAB.LT.KLU1 ) THEN
155:          INFO = -8
156:       ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
157:          INFO = -12
158:       ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
159:          INFO = -14
160:       ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
161:          INFO = -16
162:       END IF
163:       IF( INFO.NE.0 ) THEN
164:          CALL XERBLA( 'ZGBBRD', -INFO )
165:          RETURN
166:       END IF
167: *
168: *     Initialize Q and P' to the unit matrix, if needed
169: *
170:       IF( WANTQ )
171:      $   CALL ZLASET( 'Full', M, M, CZERO, CONE, Q, LDQ )
172:       IF( WANTPT )
173:      $   CALL ZLASET( 'Full', N, N, CZERO, CONE, PT, LDPT )
174: *
175: *     Quick return if possible.
176: *
177:       IF( M.EQ.0 .OR. N.EQ.0 )
178:      $   RETURN
179: *
180:       MINMN = MIN( M, N )
181: *
182:       IF( KL+KU.GT.1 ) THEN
183: *
184: *        Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
185: *        first to lower bidiagonal form and then transform to upper
186: *        bidiagonal
187: *
188:          IF( KU.GT.0 ) THEN
189:             ML0 = 1
190:             MU0 = 2
191:          ELSE
192:             ML0 = 2
193:             MU0 = 1
194:          END IF
195: *
196: *        Wherever possible, plane rotations are generated and applied in
197: *        vector operations of length NR over the index set J1:J2:KLU1.
198: *
199: *        The complex sines of the plane rotations are stored in WORK,
200: *        and the real cosines in RWORK.
201: *
202:          KLM = MIN( M-1, KL )
203:          KUN = MIN( N-1, KU )
204:          KB = KLM + KUN
205:          KB1 = KB + 1
206:          INCA = KB1*LDAB
207:          NR = 0
208:          J1 = KLM + 2
209:          J2 = 1 - KUN
210: *
211:          DO 90 I = 1, MINMN
212: *
213: *           Reduce i-th column and i-th row of matrix to bidiagonal form
214: *
215:             ML = KLM + 1
216:             MU = KUN + 1
217:             DO 80 KK = 1, KB
218:                J1 = J1 + KB
219:                J2 = J2 + KB
220: *
221: *              generate plane rotations to annihilate nonzero elements
222: *              which have been created below the band
223: *
224:                IF( NR.GT.0 )
225:      $            CALL ZLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
226:      $                         WORK( J1 ), KB1, RWORK( J1 ), KB1 )
227: *
228: *              apply plane rotations from the left
229: *
230:                DO 10 L = 1, KB
231:                   IF( J2-KLM+L-1.GT.N ) THEN
232:                      NRT = NR - 1
233:                   ELSE
234:                      NRT = NR
235:                   END IF
236:                   IF( NRT.GT.0 )
237:      $               CALL ZLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
238:      $                            AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
239:      $                            RWORK( J1 ), WORK( J1 ), KB1 )
240:    10          CONTINUE
241: *
242:                IF( ML.GT.ML0 ) THEN
243:                   IF( ML.LE.M-I+1 ) THEN
244: *
245: *                    generate plane rotation to annihilate a(i+ml-1,i)
246: *                    within the band, and apply rotation from the left
247: *
248:                      CALL ZLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
249:      $                            RWORK( I+ML-1 ), WORK( I+ML-1 ), RA )
250:                      AB( KU+ML-1, I ) = RA
251:                      IF( I.LT.N )
252:      $                  CALL ZROT( MIN( KU+ML-2, N-I ),
253:      $                             AB( KU+ML-2, I+1 ), LDAB-1,
254:      $                             AB( KU+ML-1, I+1 ), LDAB-1,
255:      $                             RWORK( I+ML-1 ), WORK( I+ML-1 ) )
256:                   END IF
257:                   NR = NR + 1
258:                   J1 = J1 - KB1
259:                END IF
260: *
261:                IF( WANTQ ) THEN
262: *
263: *                 accumulate product of plane rotations in Q
264: *
265:                   DO 20 J = J1, J2, KB1
266:                      CALL ZROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
267:      $                          RWORK( J ), DCONJG( WORK( J ) ) )
268:    20             CONTINUE
269:                END IF
270: *
271:                IF( WANTC ) THEN
272: *
273: *                 apply plane rotations to C
274: *
275:                   DO 30 J = J1, J2, KB1
276:                      CALL ZROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
277:      $                          RWORK( J ), WORK( J ) )
278:    30             CONTINUE
279:                END IF
280: *
281:                IF( J2+KUN.GT.N ) THEN
282: *
283: *                 adjust J2 to keep within the bounds of the matrix
284: *
285:                   NR = NR - 1
286:                   J2 = J2 - KB1
287:                END IF
288: *
289:                DO 40 J = J1, J2, KB1
290: *
291: *                 create nonzero element a(j-1,j+ku) above the band
292: *                 and store it in WORK(n+1:2*n)
293: *
294:                   WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
295:                   AB( 1, J+KUN ) = RWORK( J )*AB( 1, J+KUN )
296:    40          CONTINUE
297: *
298: *              generate plane rotations to annihilate nonzero elements
299: *              which have been generated above the band
300: *
301:                IF( NR.GT.0 )
302:      $            CALL ZLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
303:      $                         WORK( J1+KUN ), KB1, RWORK( J1+KUN ),
304:      $                         KB1 )
305: *
306: *              apply plane rotations from the right
307: *
308:                DO 50 L = 1, KB
309:                   IF( J2+L-1.GT.M ) THEN
310:                      NRT = NR - 1
311:                   ELSE
312:                      NRT = NR
313:                   END IF
314:                   IF( NRT.GT.0 )
315:      $               CALL ZLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
316:      $                            AB( L, J1+KUN ), INCA,
317:      $                            RWORK( J1+KUN ), WORK( J1+KUN ), KB1 )
318:    50          CONTINUE
319: *
320:                IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
321:                   IF( MU.LE.N-I+1 ) THEN
322: *
323: *                    generate plane rotation to annihilate a(i,i+mu-1)
324: *                    within the band, and apply rotation from the right
325: *
326:                      CALL ZLARTG( AB( KU-MU+3, I+MU-2 ),
327:      $                            AB( KU-MU+2, I+MU-1 ),
328:      $                            RWORK( I+MU-1 ), WORK( I+MU-1 ), RA )
329:                      AB( KU-MU+3, I+MU-2 ) = RA
330:                      CALL ZROT( MIN( KL+MU-2, M-I ),
331:      $                          AB( KU-MU+4, I+MU-2 ), 1,
332:      $                          AB( KU-MU+3, I+MU-1 ), 1,
333:      $                          RWORK( I+MU-1 ), WORK( I+MU-1 ) )
334:                   END IF
335:                   NR = NR + 1
336:                   J1 = J1 - KB1
337:                END IF
338: *
339:                IF( WANTPT ) THEN
340: *
341: *                 accumulate product of plane rotations in P'
342: *
343:                   DO 60 J = J1, J2, KB1
344:                      CALL ZROT( N, PT( J+KUN-1, 1 ), LDPT,
345:      $                          PT( J+KUN, 1 ), LDPT, RWORK( J+KUN ),
346:      $                          DCONJG( WORK( J+KUN ) ) )
347:    60             CONTINUE
348:                END IF
349: *
350:                IF( J2+KB.GT.M ) THEN
351: *
352: *                 adjust J2 to keep within the bounds of the matrix
353: *
354:                   NR = NR - 1
355:                   J2 = J2 - KB1
356:                END IF
357: *
358:                DO 70 J = J1, J2, KB1
359: *
360: *                 create nonzero element a(j+kl+ku,j+ku-1) below the
361: *                 band and store it in WORK(1:n)
362: *
363:                   WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
364:                   AB( KLU1, J+KUN ) = RWORK( J+KUN )*AB( KLU1, J+KUN )
365:    70          CONTINUE
366: *
367:                IF( ML.GT.ML0 ) THEN
368:                   ML = ML - 1
369:                ELSE
370:                   MU = MU - 1
371:                END IF
372:    80       CONTINUE
373:    90    CONTINUE
374:       END IF
375: *
376:       IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
377: *
378: *        A has been reduced to complex lower bidiagonal form
379: *
380: *        Transform lower bidiagonal form to upper bidiagonal by applying
381: *        plane rotations from the left, overwriting superdiagonal
382: *        elements on subdiagonal elements
383: *
384:          DO 100 I = 1, MIN( M-1, N )
385:             CALL ZLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
386:             AB( 1, I ) = RA
387:             IF( I.LT.N ) THEN
388:                AB( 2, I ) = RS*AB( 1, I+1 )
389:                AB( 1, I+1 ) = RC*AB( 1, I+1 )
390:             END IF
391:             IF( WANTQ )
392:      $         CALL ZROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC,
393:      $                    DCONJG( RS ) )
394:             IF( WANTC )
395:      $         CALL ZROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
396:      $                    RS )
397:   100    CONTINUE
398:       ELSE
399: *
400: *        A has been reduced to complex upper bidiagonal form or is
401: *        diagonal
402: *
403:          IF( KU.GT.0 .AND. M.LT.N ) THEN
404: *
405: *           Annihilate a(m,m+1) by applying plane rotations from the
406: *           right
407: *
408:             RB = AB( KU, M+1 )
409:             DO 110 I = M, 1, -1
410:                CALL ZLARTG( AB( KU+1, I ), RB, RC, RS, RA )
411:                AB( KU+1, I ) = RA
412:                IF( I.GT.1 ) THEN
413:                   RB = -DCONJG( RS )*AB( KU, I )
414:                   AB( KU, I ) = RC*AB( KU, I )
415:                END IF
416:                IF( WANTPT )
417:      $            CALL ZROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
418:      $                       RC, DCONJG( RS ) )
419:   110       CONTINUE
420:          END IF
421:       END IF
422: *
423: *     Make diagonal and superdiagonal elements real, storing them in D
424: *     and E
425: *
426:       T = AB( KU+1, 1 )
427:       DO 120 I = 1, MINMN
428:          ABST = ABS( T )
429:          D( I ) = ABST
430:          IF( ABST.NE.ZERO ) THEN
431:             T = T / ABST
432:          ELSE
433:             T = CONE
434:          END IF
435:          IF( WANTQ )
436:      $      CALL ZSCAL( M, T, Q( 1, I ), 1 )
437:          IF( WANTC )
438:      $      CALL ZSCAL( NCC, DCONJG( T ), C( I, 1 ), LDC )
439:          IF( I.LT.MINMN ) THEN
440:             IF( KU.EQ.0 .AND. KL.EQ.0 ) THEN
441:                E( I ) = ZERO
442:                T = AB( 1, I+1 )
443:             ELSE
444:                IF( KU.EQ.0 ) THEN
445:                   T = AB( 2, I )*DCONJG( T )
446:                ELSE
447:                   T = AB( KU, I+1 )*DCONJG( T )
448:                END IF
449:                ABST = ABS( T )
450:                E( I ) = ABST
451:                IF( ABST.NE.ZERO ) THEN
452:                   T = T / ABST
453:                ELSE
454:                   T = CONE
455:                END IF
456:                IF( WANTPT )
457:      $            CALL ZSCAL( N, T, PT( I+1, 1 ), LDPT )
458:                T = AB( KU+1, I+1 )*DCONJG( T )
459:             END IF
460:          END IF
461:   120 CONTINUE
462:       RETURN
463: *
464: *     End of ZGBBRD
465: *
466:       END
467: