1: SUBROUTINE ZGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
2: $ LDQ, PT, LDPT, C, LDC, WORK, RWORK, INFO )
3: *
4: * -- LAPACK routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * .. Scalar Arguments ..
10: CHARACTER VECT
11: INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
12: * ..
13: * .. Array Arguments ..
14: DOUBLE PRECISION D( * ), E( * ), RWORK( * )
15: COMPLEX*16 AB( LDAB, * ), C( LDC, * ), PT( LDPT, * ),
16: $ Q( LDQ, * ), WORK( * )
17: * ..
18: *
19: * Purpose
20: * =======
21: *
22: * ZGBBRD reduces a complex general m-by-n band matrix A to real upper
23: * bidiagonal form B by a unitary transformation: Q' * A * P = B.
24: *
25: * The routine computes B, and optionally forms Q or P', or computes
26: * Q'*C for a given matrix C.
27: *
28: * Arguments
29: * =========
30: *
31: * VECT (input) CHARACTER*1
32: * Specifies whether or not the matrices Q and P' are to be
33: * formed.
34: * = 'N': do not form Q or P';
35: * = 'Q': form Q only;
36: * = 'P': form P' only;
37: * = 'B': form both.
38: *
39: * M (input) INTEGER
40: * The number of rows of the matrix A. M >= 0.
41: *
42: * N (input) INTEGER
43: * The number of columns of the matrix A. N >= 0.
44: *
45: * NCC (input) INTEGER
46: * The number of columns of the matrix C. NCC >= 0.
47: *
48: * KL (input) INTEGER
49: * The number of subdiagonals of the matrix A. KL >= 0.
50: *
51: * KU (input) INTEGER
52: * The number of superdiagonals of the matrix A. KU >= 0.
53: *
54: * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
55: * On entry, the m-by-n band matrix A, stored in rows 1 to
56: * KL+KU+1. The j-th column of A is stored in the j-th column of
57: * the array AB as follows:
58: * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
59: * On exit, A is overwritten by values generated during the
60: * reduction.
61: *
62: * LDAB (input) INTEGER
63: * The leading dimension of the array A. LDAB >= KL+KU+1.
64: *
65: * D (output) DOUBLE PRECISION array, dimension (min(M,N))
66: * The diagonal elements of the bidiagonal matrix B.
67: *
68: * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
69: * The superdiagonal elements of the bidiagonal matrix B.
70: *
71: * Q (output) COMPLEX*16 array, dimension (LDQ,M)
72: * If VECT = 'Q' or 'B', the m-by-m unitary matrix Q.
73: * If VECT = 'N' or 'P', the array Q is not referenced.
74: *
75: * LDQ (input) INTEGER
76: * The leading dimension of the array Q.
77: * LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
78: *
79: * PT (output) COMPLEX*16 array, dimension (LDPT,N)
80: * If VECT = 'P' or 'B', the n-by-n unitary matrix P'.
81: * If VECT = 'N' or 'Q', the array PT is not referenced.
82: *
83: * LDPT (input) INTEGER
84: * The leading dimension of the array PT.
85: * LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
86: *
87: * C (input/output) COMPLEX*16 array, dimension (LDC,NCC)
88: * On entry, an m-by-ncc matrix C.
89: * On exit, C is overwritten by Q'*C.
90: * C is not referenced if NCC = 0.
91: *
92: * LDC (input) INTEGER
93: * The leading dimension of the array C.
94: * LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
95: *
96: * WORK (workspace) COMPLEX*16 array, dimension (max(M,N))
97: *
98: * RWORK (workspace) DOUBLE PRECISION array, dimension (max(M,N))
99: *
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
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