Annotation of rpl/lapack/lapack/dgbbrd.f, revision 1.17
1.9 bertrand 1: *> \brief \b DGBBRD
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 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">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 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 )
1.15 bertrand 23: *
1.9 bertrand 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: * ..
1.15 bertrand 32: *
1.9 bertrand 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: *
1.15 bertrand 177: *> \author Univ. of Tennessee
178: *> \author Univ. of California Berkeley
179: *> \author Univ. of Colorado Denver
180: *> \author NAG Ltd.
1.9 bertrand 181: *
1.15 bertrand 182: *> \date December 2016
1.9 bertrand 183: *
184: *> \ingroup doubleGBcomputational
185: *
186: * =====================================================================
1.1 bertrand 187: SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
188: $ LDQ, PT, LDPT, C, LDC, WORK, INFO )
189: *
1.15 bertrand 190: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 191: * -- LAPACK is a software package provided by Univ. of Tennessee, --
192: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.15 bertrand 193: * December 2016
1.1 bertrand 194: *
195: * .. Scalar Arguments ..
196: CHARACTER VECT
197: INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
198: * ..
199: * .. Array Arguments ..
200: DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
201: $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
202: * ..
203: *
204: * =====================================================================
205: *
206: * .. Parameters ..
207: DOUBLE PRECISION ZERO, ONE
208: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
209: * ..
210: * .. Local Scalars ..
211: LOGICAL WANTB, WANTC, WANTPT, WANTQ
212: INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
213: $ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
214: DOUBLE PRECISION RA, RB, RC, RS
215: * ..
216: * .. External Subroutines ..
217: EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT, XERBLA
218: * ..
219: * .. Intrinsic Functions ..
220: INTRINSIC MAX, MIN
221: * ..
222: * .. External Functions ..
223: LOGICAL LSAME
224: EXTERNAL LSAME
225: * ..
226: * .. Executable Statements ..
227: *
228: * Test the input parameters
229: *
230: WANTB = LSAME( VECT, 'B' )
231: WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
232: WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
233: WANTC = NCC.GT.0
234: KLU1 = KL + KU + 1
235: INFO = 0
236: IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
237: $ THEN
238: INFO = -1
239: ELSE IF( M.LT.0 ) THEN
240: INFO = -2
241: ELSE IF( N.LT.0 ) THEN
242: INFO = -3
243: ELSE IF( NCC.LT.0 ) THEN
244: INFO = -4
245: ELSE IF( KL.LT.0 ) THEN
246: INFO = -5
247: ELSE IF( KU.LT.0 ) THEN
248: INFO = -6
249: ELSE IF( LDAB.LT.KLU1 ) THEN
250: INFO = -8
251: ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
252: INFO = -12
253: ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
254: INFO = -14
255: ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
256: INFO = -16
257: END IF
258: IF( INFO.NE.0 ) THEN
259: CALL XERBLA( 'DGBBRD', -INFO )
260: RETURN
261: END IF
262: *
1.8 bertrand 263: * Initialize Q and P**T to the unit matrix, if needed
1.1 bertrand 264: *
265: IF( WANTQ )
266: $ CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
267: IF( WANTPT )
268: $ CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
269: *
270: * Quick return if possible.
271: *
272: IF( M.EQ.0 .OR. N.EQ.0 )
273: $ RETURN
274: *
275: MINMN = MIN( M, N )
276: *
277: IF( KL+KU.GT.1 ) THEN
278: *
279: * Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
280: * first to lower bidiagonal form and then transform to upper
281: * bidiagonal
282: *
283: IF( KU.GT.0 ) THEN
284: ML0 = 1
285: MU0 = 2
286: ELSE
287: ML0 = 2
288: MU0 = 1
289: END IF
290: *
291: * Wherever possible, plane rotations are generated and applied in
292: * vector operations of length NR over the index set J1:J2:KLU1.
293: *
294: * The sines of the plane rotations are stored in WORK(1:max(m,n))
295: * and the cosines in WORK(max(m,n)+1:2*max(m,n)).
296: *
297: MN = MAX( M, N )
298: KLM = MIN( M-1, KL )
299: KUN = MIN( N-1, KU )
300: KB = KLM + KUN
301: KB1 = KB + 1
302: INCA = KB1*LDAB
303: NR = 0
304: J1 = KLM + 2
305: J2 = 1 - KUN
306: *
307: DO 90 I = 1, MINMN
308: *
309: * Reduce i-th column and i-th row of matrix to bidiagonal form
310: *
311: ML = KLM + 1
312: MU = KUN + 1
313: DO 80 KK = 1, KB
314: J1 = J1 + KB
315: J2 = J2 + KB
316: *
317: * generate plane rotations to annihilate nonzero elements
318: * which have been created below the band
319: *
320: IF( NR.GT.0 )
321: $ CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
322: $ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
323: *
324: * apply plane rotations from the left
325: *
326: DO 10 L = 1, KB
327: IF( J2-KLM+L-1.GT.N ) THEN
328: NRT = NR - 1
329: ELSE
330: NRT = NR
331: END IF
332: IF( NRT.GT.0 )
333: $ CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
334: $ AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
335: $ WORK( MN+J1 ), WORK( J1 ), KB1 )
336: 10 CONTINUE
337: *
338: IF( ML.GT.ML0 ) THEN
339: IF( ML.LE.M-I+1 ) THEN
340: *
341: * generate plane rotation to annihilate a(i+ml-1,i)
342: * within the band, and apply rotation from the left
343: *
344: CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
345: $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
346: $ RA )
347: AB( KU+ML-1, I ) = RA
348: IF( I.LT.N )
349: $ CALL DROT( MIN( KU+ML-2, N-I ),
350: $ AB( KU+ML-2, I+1 ), LDAB-1,
351: $ AB( KU+ML-1, I+1 ), LDAB-1,
352: $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
353: END IF
354: NR = NR + 1
355: J1 = J1 - KB1
356: END IF
357: *
358: IF( WANTQ ) THEN
359: *
360: * accumulate product of plane rotations in Q
361: *
362: DO 20 J = J1, J2, KB1
363: CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
364: $ WORK( MN+J ), WORK( J ) )
365: 20 CONTINUE
366: END IF
367: *
368: IF( WANTC ) THEN
369: *
370: * apply plane rotations to C
371: *
372: DO 30 J = J1, J2, KB1
373: CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
374: $ WORK( MN+J ), WORK( J ) )
375: 30 CONTINUE
376: END IF
377: *
378: IF( J2+KUN.GT.N ) THEN
379: *
380: * adjust J2 to keep within the bounds of the matrix
381: *
382: NR = NR - 1
383: J2 = J2 - KB1
384: END IF
385: *
386: DO 40 J = J1, J2, KB1
387: *
388: * create nonzero element a(j-1,j+ku) above the band
389: * and store it in WORK(n+1:2*n)
390: *
391: WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
392: AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
393: 40 CONTINUE
394: *
395: * generate plane rotations to annihilate nonzero elements
396: * which have been generated above the band
397: *
398: IF( NR.GT.0 )
399: $ CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
400: $ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
401: $ KB1 )
402: *
403: * apply plane rotations from the right
404: *
405: DO 50 L = 1, KB
406: IF( J2+L-1.GT.M ) THEN
407: NRT = NR - 1
408: ELSE
409: NRT = NR
410: END IF
411: IF( NRT.GT.0 )
412: $ CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
413: $ AB( L, J1+KUN ), INCA,
414: $ WORK( MN+J1+KUN ), WORK( J1+KUN ),
415: $ KB1 )
416: 50 CONTINUE
417: *
418: IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
419: IF( MU.LE.N-I+1 ) THEN
420: *
421: * generate plane rotation to annihilate a(i,i+mu-1)
422: * within the band, and apply rotation from the right
423: *
424: CALL DLARTG( AB( KU-MU+3, I+MU-2 ),
425: $ AB( KU-MU+2, I+MU-1 ),
426: $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
427: $ RA )
428: AB( KU-MU+3, I+MU-2 ) = RA
429: CALL DROT( MIN( KL+MU-2, M-I ),
430: $ AB( KU-MU+4, I+MU-2 ), 1,
431: $ AB( KU-MU+3, I+MU-1 ), 1,
432: $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
433: END IF
434: NR = NR + 1
435: J1 = J1 - KB1
436: END IF
437: *
438: IF( WANTPT ) THEN
439: *
1.8 bertrand 440: * accumulate product of plane rotations in P**T
1.1 bertrand 441: *
442: DO 60 J = J1, J2, KB1
443: CALL DROT( N, PT( J+KUN-1, 1 ), LDPT,
444: $ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
445: $ WORK( J+KUN ) )
446: 60 CONTINUE
447: END IF
448: *
449: IF( J2+KB.GT.M ) THEN
450: *
451: * adjust J2 to keep within the bounds of the matrix
452: *
453: NR = NR - 1
454: J2 = J2 - KB1
455: END IF
456: *
457: DO 70 J = J1, J2, KB1
458: *
459: * create nonzero element a(j+kl+ku,j+ku-1) below the
460: * band and store it in WORK(1:n)
461: *
462: WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
463: AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
464: 70 CONTINUE
465: *
466: IF( ML.GT.ML0 ) THEN
467: ML = ML - 1
468: ELSE
469: MU = MU - 1
470: END IF
471: 80 CONTINUE
472: 90 CONTINUE
473: END IF
474: *
475: IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
476: *
477: * A has been reduced to lower bidiagonal form
478: *
479: * Transform lower bidiagonal form to upper bidiagonal by applying
480: * plane rotations from the left, storing diagonal elements in D
481: * and off-diagonal elements in E
482: *
483: DO 100 I = 1, MIN( M-1, N )
484: CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
485: D( I ) = RA
486: IF( I.LT.N ) THEN
487: E( I ) = RS*AB( 1, I+1 )
488: AB( 1, I+1 ) = RC*AB( 1, I+1 )
489: END IF
490: IF( WANTQ )
491: $ CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
492: IF( WANTC )
493: $ CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
494: $ RS )
495: 100 CONTINUE
496: IF( M.LE.N )
497: $ D( M ) = AB( 1, M )
498: ELSE IF( KU.GT.0 ) THEN
499: *
500: * A has been reduced to upper bidiagonal form
501: *
502: IF( M.LT.N ) THEN
503: *
504: * Annihilate a(m,m+1) by applying plane rotations from the
505: * right, storing diagonal elements in D and off-diagonal
506: * elements in E
507: *
508: RB = AB( KU, M+1 )
509: DO 110 I = M, 1, -1
510: CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA )
511: D( I ) = RA
512: IF( I.GT.1 ) THEN
513: RB = -RS*AB( KU, I )
514: E( I-1 ) = RC*AB( KU, I )
515: END IF
516: IF( WANTPT )
517: $ CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
518: $ RC, RS )
519: 110 CONTINUE
520: ELSE
521: *
522: * Copy off-diagonal elements to E and diagonal elements to D
523: *
524: DO 120 I = 1, MINMN - 1
525: E( I ) = AB( KU, I+1 )
526: 120 CONTINUE
527: DO 130 I = 1, MINMN
528: D( I ) = AB( KU+1, I )
529: 130 CONTINUE
530: END IF
531: ELSE
532: *
533: * A is diagonal. Set elements of E to zero and copy diagonal
534: * elements to D.
535: *
536: DO 140 I = 1, MINMN - 1
537: E( I ) = ZERO
538: 140 CONTINUE
539: DO 150 I = 1, MINMN
540: D( I ) = AB( 1, I )
541: 150 CONTINUE
542: END IF
543: RETURN
544: *
545: * End of DGBBRD
546: *
547: END
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