1: *> \brief \b DGBBRD
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGBBRD + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbbrd.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbbrd.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbbrd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
22: * LDQ, PT, LDPT, C, LDC, WORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER VECT
26: * INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
30: * $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DGBBRD reduces a real general m-by-n band matrix A to upper
40: *> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
41: *>
42: *> The routine computes B, and optionally forms Q or P**T, or computes
43: *> Q**T*C for a given matrix C.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] VECT
50: *> \verbatim
51: *> VECT is CHARACTER*1
52: *> Specifies whether or not the matrices Q and P**T are to be
53: *> formed.
54: *> = 'N': do not form Q or P**T;
55: *> = 'Q': form Q only;
56: *> = 'P': form P**T only;
57: *> = 'B': form both.
58: *> \endverbatim
59: *>
60: *> \param[in] M
61: *> \verbatim
62: *> M is INTEGER
63: *> The number of rows of the matrix A. M >= 0.
64: *> \endverbatim
65: *>
66: *> \param[in] N
67: *> \verbatim
68: *> N is INTEGER
69: *> The number of columns of the matrix A. N >= 0.
70: *> \endverbatim
71: *>
72: *> \param[in] NCC
73: *> \verbatim
74: *> NCC is INTEGER
75: *> The number of columns of the matrix C. NCC >= 0.
76: *> \endverbatim
77: *>
78: *> \param[in] KL
79: *> \verbatim
80: *> KL is INTEGER
81: *> The number of subdiagonals of the matrix A. KL >= 0.
82: *> \endverbatim
83: *>
84: *> \param[in] KU
85: *> \verbatim
86: *> KU is INTEGER
87: *> The number of superdiagonals of the matrix A. KU >= 0.
88: *> \endverbatim
89: *>
90: *> \param[in,out] AB
91: *> \verbatim
92: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
93: *> On entry, the m-by-n band matrix A, stored in rows 1 to
94: *> KL+KU+1. The j-th column of A is stored in the j-th column of
95: *> the array AB as follows:
96: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
97: *> On exit, A is overwritten by values generated during the
98: *> reduction.
99: *> \endverbatim
100: *>
101: *> \param[in] LDAB
102: *> \verbatim
103: *> LDAB is INTEGER
104: *> The leading dimension of the array A. LDAB >= KL+KU+1.
105: *> \endverbatim
106: *>
107: *> \param[out] D
108: *> \verbatim
109: *> D is DOUBLE PRECISION array, dimension (min(M,N))
110: *> The diagonal elements of the bidiagonal matrix B.
111: *> \endverbatim
112: *>
113: *> \param[out] E
114: *> \verbatim
115: *> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
116: *> The superdiagonal elements of the bidiagonal matrix B.
117: *> \endverbatim
118: *>
119: *> \param[out] Q
120: *> \verbatim
121: *> Q is DOUBLE PRECISION array, dimension (LDQ,M)
122: *> If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
123: *> If VECT = 'N' or 'P', the array Q is not referenced.
124: *> \endverbatim
125: *>
126: *> \param[in] LDQ
127: *> \verbatim
128: *> LDQ is INTEGER
129: *> The leading dimension of the array Q.
130: *> LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
131: *> \endverbatim
132: *>
133: *> \param[out] PT
134: *> \verbatim
135: *> PT is DOUBLE PRECISION array, dimension (LDPT,N)
136: *> If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
137: *> If VECT = 'N' or 'Q', the array PT is not referenced.
138: *> \endverbatim
139: *>
140: *> \param[in] LDPT
141: *> \verbatim
142: *> LDPT is INTEGER
143: *> The leading dimension of the array PT.
144: *> LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
145: *> \endverbatim
146: *>
147: *> \param[in,out] C
148: *> \verbatim
149: *> C is DOUBLE PRECISION array, dimension (LDC,NCC)
150: *> On entry, an m-by-ncc matrix C.
151: *> On exit, C is overwritten by Q**T*C.
152: *> C is not referenced if NCC = 0.
153: *> \endverbatim
154: *>
155: *> \param[in] LDC
156: *> \verbatim
157: *> LDC is INTEGER
158: *> The leading dimension of the array C.
159: *> LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
160: *> \endverbatim
161: *>
162: *> \param[out] WORK
163: *> \verbatim
164: *> WORK is DOUBLE PRECISION array, dimension (2*max(M,N))
165: *> \endverbatim
166: *>
167: *> \param[out] INFO
168: *> \verbatim
169: *> INFO is INTEGER
170: *> = 0: successful exit.
171: *> < 0: if INFO = -i, the i-th argument had an illegal value.
172: *> \endverbatim
173: *
174: * Authors:
175: * ========
176: *
177: *> \author Univ. of Tennessee
178: *> \author Univ. of California Berkeley
179: *> \author Univ. of Colorado Denver
180: *> \author NAG Ltd.
181: *
182: *> \ingroup doubleGBcomputational
183: *
184: * =====================================================================
185: SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
186: $ LDQ, PT, LDPT, C, LDC, WORK, INFO )
187: *
188: * -- LAPACK computational routine --
189: * -- LAPACK is a software package provided by Univ. of Tennessee, --
190: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
191: *
192: * .. Scalar Arguments ..
193: CHARACTER VECT
194: INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
195: * ..
196: * .. Array Arguments ..
197: DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
198: $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
199: * ..
200: *
201: * =====================================================================
202: *
203: * .. Parameters ..
204: DOUBLE PRECISION ZERO, ONE
205: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
206: * ..
207: * .. Local Scalars ..
208: LOGICAL WANTB, WANTC, WANTPT, WANTQ
209: INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
210: $ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
211: DOUBLE PRECISION RA, RB, RC, RS
212: * ..
213: * .. External Subroutines ..
214: EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT, XERBLA
215: * ..
216: * .. Intrinsic Functions ..
217: INTRINSIC MAX, MIN
218: * ..
219: * .. External Functions ..
220: LOGICAL LSAME
221: EXTERNAL LSAME
222: * ..
223: * .. Executable Statements ..
224: *
225: * Test the input parameters
226: *
227: WANTB = LSAME( VECT, 'B' )
228: WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
229: WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
230: WANTC = NCC.GT.0
231: KLU1 = KL + KU + 1
232: INFO = 0
233: IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
234: $ THEN
235: INFO = -1
236: ELSE IF( M.LT.0 ) THEN
237: INFO = -2
238: ELSE IF( N.LT.0 ) THEN
239: INFO = -3
240: ELSE IF( NCC.LT.0 ) THEN
241: INFO = -4
242: ELSE IF( KL.LT.0 ) THEN
243: INFO = -5
244: ELSE IF( KU.LT.0 ) THEN
245: INFO = -6
246: ELSE IF( LDAB.LT.KLU1 ) THEN
247: INFO = -8
248: ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
249: INFO = -12
250: ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
251: INFO = -14
252: ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
253: INFO = -16
254: END IF
255: IF( INFO.NE.0 ) THEN
256: CALL XERBLA( 'DGBBRD', -INFO )
257: RETURN
258: END IF
259: *
260: * Initialize Q and P**T to the unit matrix, if needed
261: *
262: IF( WANTQ )
263: $ CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
264: IF( WANTPT )
265: $ CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
266: *
267: * Quick return if possible.
268: *
269: IF( M.EQ.0 .OR. N.EQ.0 )
270: $ RETURN
271: *
272: MINMN = MIN( M, N )
273: *
274: IF( KL+KU.GT.1 ) THEN
275: *
276: * Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
277: * first to lower bidiagonal form and then transform to upper
278: * bidiagonal
279: *
280: IF( KU.GT.0 ) THEN
281: ML0 = 1
282: MU0 = 2
283: ELSE
284: ML0 = 2
285: MU0 = 1
286: END IF
287: *
288: * Wherever possible, plane rotations are generated and applied in
289: * vector operations of length NR over the index set J1:J2:KLU1.
290: *
291: * The sines of the plane rotations are stored in WORK(1:max(m,n))
292: * and the cosines in WORK(max(m,n)+1:2*max(m,n)).
293: *
294: MN = MAX( M, N )
295: KLM = MIN( M-1, KL )
296: KUN = MIN( N-1, KU )
297: KB = KLM + KUN
298: KB1 = KB + 1
299: INCA = KB1*LDAB
300: NR = 0
301: J1 = KLM + 2
302: J2 = 1 - KUN
303: *
304: DO 90 I = 1, MINMN
305: *
306: * Reduce i-th column and i-th row of matrix to bidiagonal form
307: *
308: ML = KLM + 1
309: MU = KUN + 1
310: DO 80 KK = 1, KB
311: J1 = J1 + KB
312: J2 = J2 + KB
313: *
314: * generate plane rotations to annihilate nonzero elements
315: * which have been created below the band
316: *
317: IF( NR.GT.0 )
318: $ CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
319: $ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
320: *
321: * apply plane rotations from the left
322: *
323: DO 10 L = 1, KB
324: IF( J2-KLM+L-1.GT.N ) THEN
325: NRT = NR - 1
326: ELSE
327: NRT = NR
328: END IF
329: IF( NRT.GT.0 )
330: $ CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
331: $ AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
332: $ WORK( MN+J1 ), WORK( J1 ), KB1 )
333: 10 CONTINUE
334: *
335: IF( ML.GT.ML0 ) THEN
336: IF( ML.LE.M-I+1 ) THEN
337: *
338: * generate plane rotation to annihilate a(i+ml-1,i)
339: * within the band, and apply rotation from the left
340: *
341: CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
342: $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
343: $ RA )
344: AB( KU+ML-1, I ) = RA
345: IF( I.LT.N )
346: $ CALL DROT( MIN( KU+ML-2, N-I ),
347: $ AB( KU+ML-2, I+1 ), LDAB-1,
348: $ AB( KU+ML-1, I+1 ), LDAB-1,
349: $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
350: END IF
351: NR = NR + 1
352: J1 = J1 - KB1
353: END IF
354: *
355: IF( WANTQ ) THEN
356: *
357: * accumulate product of plane rotations in Q
358: *
359: DO 20 J = J1, J2, KB1
360: CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
361: $ WORK( MN+J ), WORK( J ) )
362: 20 CONTINUE
363: END IF
364: *
365: IF( WANTC ) THEN
366: *
367: * apply plane rotations to C
368: *
369: DO 30 J = J1, J2, KB1
370: CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
371: $ WORK( MN+J ), WORK( J ) )
372: 30 CONTINUE
373: END IF
374: *
375: IF( J2+KUN.GT.N ) THEN
376: *
377: * adjust J2 to keep within the bounds of the matrix
378: *
379: NR = NR - 1
380: J2 = J2 - KB1
381: END IF
382: *
383: DO 40 J = J1, J2, KB1
384: *
385: * create nonzero element a(j-1,j+ku) above the band
386: * and store it in WORK(n+1:2*n)
387: *
388: WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
389: AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
390: 40 CONTINUE
391: *
392: * generate plane rotations to annihilate nonzero elements
393: * which have been generated above the band
394: *
395: IF( NR.GT.0 )
396: $ CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
397: $ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
398: $ KB1 )
399: *
400: * apply plane rotations from the right
401: *
402: DO 50 L = 1, KB
403: IF( J2+L-1.GT.M ) THEN
404: NRT = NR - 1
405: ELSE
406: NRT = NR
407: END IF
408: IF( NRT.GT.0 )
409: $ CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
410: $ AB( L, J1+KUN ), INCA,
411: $ WORK( MN+J1+KUN ), WORK( J1+KUN ),
412: $ KB1 )
413: 50 CONTINUE
414: *
415: IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
416: IF( MU.LE.N-I+1 ) THEN
417: *
418: * generate plane rotation to annihilate a(i,i+mu-1)
419: * within the band, and apply rotation from the right
420: *
421: CALL DLARTG( AB( KU-MU+3, I+MU-2 ),
422: $ AB( KU-MU+2, I+MU-1 ),
423: $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
424: $ RA )
425: AB( KU-MU+3, I+MU-2 ) = RA
426: CALL DROT( MIN( KL+MU-2, M-I ),
427: $ AB( KU-MU+4, I+MU-2 ), 1,
428: $ AB( KU-MU+3, I+MU-1 ), 1,
429: $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
430: END IF
431: NR = NR + 1
432: J1 = J1 - KB1
433: END IF
434: *
435: IF( WANTPT ) THEN
436: *
437: * accumulate product of plane rotations in P**T
438: *
439: DO 60 J = J1, J2, KB1
440: CALL DROT( N, PT( J+KUN-1, 1 ), LDPT,
441: $ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
442: $ WORK( J+KUN ) )
443: 60 CONTINUE
444: END IF
445: *
446: IF( J2+KB.GT.M ) THEN
447: *
448: * adjust J2 to keep within the bounds of the matrix
449: *
450: NR = NR - 1
451: J2 = J2 - KB1
452: END IF
453: *
454: DO 70 J = J1, J2, KB1
455: *
456: * create nonzero element a(j+kl+ku,j+ku-1) below the
457: * band and store it in WORK(1:n)
458: *
459: WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
460: AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
461: 70 CONTINUE
462: *
463: IF( ML.GT.ML0 ) THEN
464: ML = ML - 1
465: ELSE
466: MU = MU - 1
467: END IF
468: 80 CONTINUE
469: 90 CONTINUE
470: END IF
471: *
472: IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
473: *
474: * A has been reduced to lower bidiagonal form
475: *
476: * Transform lower bidiagonal form to upper bidiagonal by applying
477: * plane rotations from the left, storing diagonal elements in D
478: * and off-diagonal elements in E
479: *
480: DO 100 I = 1, MIN( M-1, N )
481: CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
482: D( I ) = RA
483: IF( I.LT.N ) THEN
484: E( I ) = RS*AB( 1, I+1 )
485: AB( 1, I+1 ) = RC*AB( 1, I+1 )
486: END IF
487: IF( WANTQ )
488: $ CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
489: IF( WANTC )
490: $ CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
491: $ RS )
492: 100 CONTINUE
493: IF( M.LE.N )
494: $ D( M ) = AB( 1, M )
495: ELSE IF( KU.GT.0 ) THEN
496: *
497: * A has been reduced to upper bidiagonal form
498: *
499: IF( M.LT.N ) THEN
500: *
501: * Annihilate a(m,m+1) by applying plane rotations from the
502: * right, storing diagonal elements in D and off-diagonal
503: * elements in E
504: *
505: RB = AB( KU, M+1 )
506: DO 110 I = M, 1, -1
507: CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA )
508: D( I ) = RA
509: IF( I.GT.1 ) THEN
510: RB = -RS*AB( KU, I )
511: E( I-1 ) = RC*AB( KU, I )
512: END IF
513: IF( WANTPT )
514: $ CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
515: $ RC, RS )
516: 110 CONTINUE
517: ELSE
518: *
519: * Copy off-diagonal elements to E and diagonal elements to D
520: *
521: DO 120 I = 1, MINMN - 1
522: E( I ) = AB( KU, I+1 )
523: 120 CONTINUE
524: DO 130 I = 1, MINMN
525: D( I ) = AB( KU+1, I )
526: 130 CONTINUE
527: END IF
528: ELSE
529: *
530: * A is diagonal. Set elements of E to zero and copy diagonal
531: * elements to D.
532: *
533: DO 140 I = 1, MINMN - 1
534: E( I ) = ZERO
535: 140 CONTINUE
536: DO 150 I = 1, MINMN
537: D( I ) = AB( 1, I )
538: 150 CONTINUE
539: END IF
540: RETURN
541: *
542: * End of DGBBRD
543: *
544: END
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