1 SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
2 $ LDQ, PT, LDPT, C, LDC, WORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
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 AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
15 $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DGBBRD reduces a real general m-by-n band matrix A to upper
22 * bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
23 *
24 * The routine computes B, and optionally forms Q or P**T, or computes
25 * Q**T*C for a given matrix C.
26 *
27 * Arguments
28 * =========
29 *
30 * VECT (input) CHARACTER*1
31 * Specifies whether or not the matrices Q and P**T are to be
32 * formed.
33 * = 'N': do not form Q or P**T;
34 * = 'Q': form Q only;
35 * = 'P': form P**T only;
36 * = 'B': form both.
37 *
38 * M (input) INTEGER
39 * The number of rows of the matrix A. M >= 0.
40 *
41 * N (input) INTEGER
42 * The number of columns of the matrix A. N >= 0.
43 *
44 * NCC (input) INTEGER
45 * The number of columns of the matrix C. NCC >= 0.
46 *
47 * KL (input) INTEGER
48 * The number of subdiagonals of the matrix A. KL >= 0.
49 *
50 * KU (input) INTEGER
51 * The number of superdiagonals of the matrix A. KU >= 0.
52 *
53 * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
54 * On entry, the m-by-n band matrix A, stored in rows 1 to
55 * KL+KU+1. The j-th column of A is stored in the j-th column of
56 * the array AB as follows:
57 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
58 * On exit, A is overwritten by values generated during the
59 * reduction.
60 *
61 * LDAB (input) INTEGER
62 * The leading dimension of the array A. LDAB >= KL+KU+1.
63 *
64 * D (output) DOUBLE PRECISION array, dimension (min(M,N))
65 * The diagonal elements of the bidiagonal matrix B.
66 *
67 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
68 * The superdiagonal elements of the bidiagonal matrix B.
69 *
70 * Q (output) DOUBLE PRECISION array, dimension (LDQ,M)
71 * If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
72 * If VECT = 'N' or 'P', the array Q is not referenced.
73 *
74 * LDQ (input) INTEGER
75 * The leading dimension of the array Q.
76 * LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
77 *
78 * PT (output) DOUBLE PRECISION array, dimension (LDPT,N)
79 * If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
80 * If VECT = 'N' or 'Q', the array PT is not referenced.
81 *
82 * LDPT (input) INTEGER
83 * The leading dimension of the array PT.
84 * LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
85 *
86 * C (input/output) DOUBLE PRECISION array, dimension (LDC,NCC)
87 * On entry, an m-by-ncc matrix C.
88 * On exit, C is overwritten by Q**T*C.
89 * C is not referenced if NCC = 0.
90 *
91 * LDC (input) INTEGER
92 * The leading dimension of the array C.
93 * LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
94 *
95 * WORK (workspace) DOUBLE PRECISION array, dimension (2*max(M,N))
96 *
97 * INFO (output) INTEGER
98 * = 0: successful exit.
99 * < 0: if INFO = -i, the i-th argument had an illegal value.
100 *
101 * =====================================================================
102 *
103 * .. Parameters ..
104 DOUBLE PRECISION ZERO, ONE
105 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
106 * ..
107 * .. Local Scalars ..
108 LOGICAL WANTB, WANTC, WANTPT, WANTQ
109 INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
110 $ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
111 DOUBLE PRECISION RA, RB, RC, RS
112 * ..
113 * .. External Subroutines ..
114 EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT, XERBLA
115 * ..
116 * .. Intrinsic Functions ..
117 INTRINSIC MAX, MIN
118 * ..
119 * .. External Functions ..
120 LOGICAL LSAME
121 EXTERNAL LSAME
122 * ..
123 * .. Executable Statements ..
124 *
125 * Test the input parameters
126 *
127 WANTB = LSAME( VECT, 'B' )
128 WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
129 WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
130 WANTC = NCC.GT.0
131 KLU1 = KL + KU + 1
132 INFO = 0
133 IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
134 $ THEN
135 INFO = -1
136 ELSE IF( M.LT.0 ) THEN
137 INFO = -2
138 ELSE IF( N.LT.0 ) THEN
139 INFO = -3
140 ELSE IF( NCC.LT.0 ) THEN
141 INFO = -4
142 ELSE IF( KL.LT.0 ) THEN
143 INFO = -5
144 ELSE IF( KU.LT.0 ) THEN
145 INFO = -6
146 ELSE IF( LDAB.LT.KLU1 ) THEN
147 INFO = -8
148 ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
149 INFO = -12
150 ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
151 INFO = -14
152 ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
153 INFO = -16
154 END IF
155 IF( INFO.NE.0 ) THEN
156 CALL XERBLA( 'DGBBRD', -INFO )
157 RETURN
158 END IF
159 *
160 * Initialize Q and P**T to the unit matrix, if needed
161 *
162 IF( WANTQ )
163 $ CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
164 IF( WANTPT )
165 $ CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
166 *
167 * Quick return if possible.
168 *
169 IF( M.EQ.0 .OR. N.EQ.0 )
170 $ RETURN
171 *
172 MINMN = MIN( M, N )
173 *
174 IF( KL+KU.GT.1 ) THEN
175 *
176 * Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
177 * first to lower bidiagonal form and then transform to upper
178 * bidiagonal
179 *
180 IF( KU.GT.0 ) THEN
181 ML0 = 1
182 MU0 = 2
183 ELSE
184 ML0 = 2
185 MU0 = 1
186 END IF
187 *
188 * Wherever possible, plane rotations are generated and applied in
189 * vector operations of length NR over the index set J1:J2:KLU1.
190 *
191 * The sines of the plane rotations are stored in WORK(1:max(m,n))
192 * and the cosines in WORK(max(m,n)+1:2*max(m,n)).
193 *
194 MN = MAX( M, N )
195 KLM = MIN( M-1, KL )
196 KUN = MIN( N-1, KU )
197 KB = KLM + KUN
198 KB1 = KB + 1
199 INCA = KB1*LDAB
200 NR = 0
201 J1 = KLM + 2
202 J2 = 1 - KUN
203 *
204 DO 90 I = 1, MINMN
205 *
206 * Reduce i-th column and i-th row of matrix to bidiagonal form
207 *
208 ML = KLM + 1
209 MU = KUN + 1
210 DO 80 KK = 1, KB
211 J1 = J1 + KB
212 J2 = J2 + KB
213 *
214 * generate plane rotations to annihilate nonzero elements
215 * which have been created below the band
216 *
217 IF( NR.GT.0 )
218 $ CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
219 $ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
220 *
221 * apply plane rotations from the left
222 *
223 DO 10 L = 1, KB
224 IF( J2-KLM+L-1.GT.N ) THEN
225 NRT = NR - 1
226 ELSE
227 NRT = NR
228 END IF
229 IF( NRT.GT.0 )
230 $ CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
231 $ AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
232 $ WORK( MN+J1 ), WORK( J1 ), KB1 )
233 10 CONTINUE
234 *
235 IF( ML.GT.ML0 ) THEN
236 IF( ML.LE.M-I+1 ) THEN
237 *
238 * generate plane rotation to annihilate a(i+ml-1,i)
239 * within the band, and apply rotation from the left
240 *
241 CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
242 $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
243 $ RA )
244 AB( KU+ML-1, I ) = RA
245 IF( I.LT.N )
246 $ CALL DROT( MIN( KU+ML-2, N-I ),
247 $ AB( KU+ML-2, I+1 ), LDAB-1,
248 $ AB( KU+ML-1, I+1 ), LDAB-1,
249 $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
250 END IF
251 NR = NR + 1
252 J1 = J1 - KB1
253 END IF
254 *
255 IF( WANTQ ) THEN
256 *
257 * accumulate product of plane rotations in Q
258 *
259 DO 20 J = J1, J2, KB1
260 CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
261 $ WORK( MN+J ), WORK( J ) )
262 20 CONTINUE
263 END IF
264 *
265 IF( WANTC ) THEN
266 *
267 * apply plane rotations to C
268 *
269 DO 30 J = J1, J2, KB1
270 CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
271 $ WORK( MN+J ), WORK( J ) )
272 30 CONTINUE
273 END IF
274 *
275 IF( J2+KUN.GT.N ) THEN
276 *
277 * adjust J2 to keep within the bounds of the matrix
278 *
279 NR = NR - 1
280 J2 = J2 - KB1
281 END IF
282 *
283 DO 40 J = J1, J2, KB1
284 *
285 * create nonzero element a(j-1,j+ku) above the band
286 * and store it in WORK(n+1:2*n)
287 *
288 WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
289 AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
290 40 CONTINUE
291 *
292 * generate plane rotations to annihilate nonzero elements
293 * which have been generated above the band
294 *
295 IF( NR.GT.0 )
296 $ CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
297 $ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
298 $ KB1 )
299 *
300 * apply plane rotations from the right
301 *
302 DO 50 L = 1, KB
303 IF( J2+L-1.GT.M ) THEN
304 NRT = NR - 1
305 ELSE
306 NRT = NR
307 END IF
308 IF( NRT.GT.0 )
309 $ CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
310 $ AB( L, J1+KUN ), INCA,
311 $ WORK( MN+J1+KUN ), WORK( J1+KUN ),
312 $ KB1 )
313 50 CONTINUE
314 *
315 IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
316 IF( MU.LE.N-I+1 ) THEN
317 *
318 * generate plane rotation to annihilate a(i,i+mu-1)
319 * within the band, and apply rotation from the right
320 *
321 CALL DLARTG( AB( KU-MU+3, I+MU-2 ),
322 $ AB( KU-MU+2, I+MU-1 ),
323 $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
324 $ RA )
325 AB( KU-MU+3, I+MU-2 ) = RA
326 CALL DROT( MIN( KL+MU-2, M-I ),
327 $ AB( KU-MU+4, I+MU-2 ), 1,
328 $ AB( KU-MU+3, I+MU-1 ), 1,
329 $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
330 END IF
331 NR = NR + 1
332 J1 = J1 - KB1
333 END IF
334 *
335 IF( WANTPT ) THEN
336 *
337 * accumulate product of plane rotations in P**T
338 *
339 DO 60 J = J1, J2, KB1
340 CALL DROT( N, PT( J+KUN-1, 1 ), LDPT,
341 $ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
342 $ WORK( J+KUN ) )
343 60 CONTINUE
344 END IF
345 *
346 IF( J2+KB.GT.M ) THEN
347 *
348 * adjust J2 to keep within the bounds of the matrix
349 *
350 NR = NR - 1
351 J2 = J2 - KB1
352 END IF
353 *
354 DO 70 J = J1, J2, KB1
355 *
356 * create nonzero element a(j+kl+ku,j+ku-1) below the
357 * band and store it in WORK(1:n)
358 *
359 WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
360 AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
361 70 CONTINUE
362 *
363 IF( ML.GT.ML0 ) THEN
364 ML = ML - 1
365 ELSE
366 MU = MU - 1
367 END IF
368 80 CONTINUE
369 90 CONTINUE
370 END IF
371 *
372 IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
373 *
374 * A has been reduced to lower bidiagonal form
375 *
376 * Transform lower bidiagonal form to upper bidiagonal by applying
377 * plane rotations from the left, storing diagonal elements in D
378 * and off-diagonal elements in E
379 *
380 DO 100 I = 1, MIN( M-1, N )
381 CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
382 D( I ) = RA
383 IF( I.LT.N ) THEN
384 E( I ) = RS*AB( 1, I+1 )
385 AB( 1, I+1 ) = RC*AB( 1, I+1 )
386 END IF
387 IF( WANTQ )
388 $ CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
389 IF( WANTC )
390 $ CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
391 $ RS )
392 100 CONTINUE
393 IF( M.LE.N )
394 $ D( M ) = AB( 1, M )
395 ELSE IF( KU.GT.0 ) THEN
396 *
397 * A has been reduced to upper bidiagonal form
398 *
399 IF( M.LT.N ) THEN
400 *
401 * Annihilate a(m,m+1) by applying plane rotations from the
402 * right, storing diagonal elements in D and off-diagonal
403 * elements in E
404 *
405 RB = AB( KU, M+1 )
406 DO 110 I = M, 1, -1
407 CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA )
408 D( I ) = RA
409 IF( I.GT.1 ) THEN
410 RB = -RS*AB( KU, I )
411 E( I-1 ) = RC*AB( KU, I )
412 END IF
413 IF( WANTPT )
414 $ CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
415 $ RC, RS )
416 110 CONTINUE
417 ELSE
418 *
419 * Copy off-diagonal elements to E and diagonal elements to D
420 *
421 DO 120 I = 1, MINMN - 1
422 E( I ) = AB( KU, I+1 )
423 120 CONTINUE
424 DO 130 I = 1, MINMN
425 D( I ) = AB( KU+1, I )
426 130 CONTINUE
427 END IF
428 ELSE
429 *
430 * A is diagonal. Set elements of E to zero and copy diagonal
431 * elements to D.
432 *
433 DO 140 I = 1, MINMN - 1
434 E( I ) = ZERO
435 140 CONTINUE
436 DO 150 I = 1, MINMN
437 D( I ) = AB( 1, I )
438 150 CONTINUE
439 END IF
440 RETURN
441 *
442 * End of DGBBRD
443 *
444 END
2 $ LDQ, PT, LDPT, C, LDC, WORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
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 AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
15 $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DGBBRD reduces a real general m-by-n band matrix A to upper
22 * bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
23 *
24 * The routine computes B, and optionally forms Q or P**T, or computes
25 * Q**T*C for a given matrix C.
26 *
27 * Arguments
28 * =========
29 *
30 * VECT (input) CHARACTER*1
31 * Specifies whether or not the matrices Q and P**T are to be
32 * formed.
33 * = 'N': do not form Q or P**T;
34 * = 'Q': form Q only;
35 * = 'P': form P**T only;
36 * = 'B': form both.
37 *
38 * M (input) INTEGER
39 * The number of rows of the matrix A. M >= 0.
40 *
41 * N (input) INTEGER
42 * The number of columns of the matrix A. N >= 0.
43 *
44 * NCC (input) INTEGER
45 * The number of columns of the matrix C. NCC >= 0.
46 *
47 * KL (input) INTEGER
48 * The number of subdiagonals of the matrix A. KL >= 0.
49 *
50 * KU (input) INTEGER
51 * The number of superdiagonals of the matrix A. KU >= 0.
52 *
53 * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
54 * On entry, the m-by-n band matrix A, stored in rows 1 to
55 * KL+KU+1. The j-th column of A is stored in the j-th column of
56 * the array AB as follows:
57 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
58 * On exit, A is overwritten by values generated during the
59 * reduction.
60 *
61 * LDAB (input) INTEGER
62 * The leading dimension of the array A. LDAB >= KL+KU+1.
63 *
64 * D (output) DOUBLE PRECISION array, dimension (min(M,N))
65 * The diagonal elements of the bidiagonal matrix B.
66 *
67 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
68 * The superdiagonal elements of the bidiagonal matrix B.
69 *
70 * Q (output) DOUBLE PRECISION array, dimension (LDQ,M)
71 * If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
72 * If VECT = 'N' or 'P', the array Q is not referenced.
73 *
74 * LDQ (input) INTEGER
75 * The leading dimension of the array Q.
76 * LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
77 *
78 * PT (output) DOUBLE PRECISION array, dimension (LDPT,N)
79 * If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
80 * If VECT = 'N' or 'Q', the array PT is not referenced.
81 *
82 * LDPT (input) INTEGER
83 * The leading dimension of the array PT.
84 * LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
85 *
86 * C (input/output) DOUBLE PRECISION array, dimension (LDC,NCC)
87 * On entry, an m-by-ncc matrix C.
88 * On exit, C is overwritten by Q**T*C.
89 * C is not referenced if NCC = 0.
90 *
91 * LDC (input) INTEGER
92 * The leading dimension of the array C.
93 * LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
94 *
95 * WORK (workspace) DOUBLE PRECISION array, dimension (2*max(M,N))
96 *
97 * INFO (output) INTEGER
98 * = 0: successful exit.
99 * < 0: if INFO = -i, the i-th argument had an illegal value.
100 *
101 * =====================================================================
102 *
103 * .. Parameters ..
104 DOUBLE PRECISION ZERO, ONE
105 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
106 * ..
107 * .. Local Scalars ..
108 LOGICAL WANTB, WANTC, WANTPT, WANTQ
109 INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
110 $ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
111 DOUBLE PRECISION RA, RB, RC, RS
112 * ..
113 * .. External Subroutines ..
114 EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT, XERBLA
115 * ..
116 * .. Intrinsic Functions ..
117 INTRINSIC MAX, MIN
118 * ..
119 * .. External Functions ..
120 LOGICAL LSAME
121 EXTERNAL LSAME
122 * ..
123 * .. Executable Statements ..
124 *
125 * Test the input parameters
126 *
127 WANTB = LSAME( VECT, 'B' )
128 WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
129 WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
130 WANTC = NCC.GT.0
131 KLU1 = KL + KU + 1
132 INFO = 0
133 IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
134 $ THEN
135 INFO = -1
136 ELSE IF( M.LT.0 ) THEN
137 INFO = -2
138 ELSE IF( N.LT.0 ) THEN
139 INFO = -3
140 ELSE IF( NCC.LT.0 ) THEN
141 INFO = -4
142 ELSE IF( KL.LT.0 ) THEN
143 INFO = -5
144 ELSE IF( KU.LT.0 ) THEN
145 INFO = -6
146 ELSE IF( LDAB.LT.KLU1 ) THEN
147 INFO = -8
148 ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
149 INFO = -12
150 ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
151 INFO = -14
152 ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
153 INFO = -16
154 END IF
155 IF( INFO.NE.0 ) THEN
156 CALL XERBLA( 'DGBBRD', -INFO )
157 RETURN
158 END IF
159 *
160 * Initialize Q and P**T to the unit matrix, if needed
161 *
162 IF( WANTQ )
163 $ CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
164 IF( WANTPT )
165 $ CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
166 *
167 * Quick return if possible.
168 *
169 IF( M.EQ.0 .OR. N.EQ.0 )
170 $ RETURN
171 *
172 MINMN = MIN( M, N )
173 *
174 IF( KL+KU.GT.1 ) THEN
175 *
176 * Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
177 * first to lower bidiagonal form and then transform to upper
178 * bidiagonal
179 *
180 IF( KU.GT.0 ) THEN
181 ML0 = 1
182 MU0 = 2
183 ELSE
184 ML0 = 2
185 MU0 = 1
186 END IF
187 *
188 * Wherever possible, plane rotations are generated and applied in
189 * vector operations of length NR over the index set J1:J2:KLU1.
190 *
191 * The sines of the plane rotations are stored in WORK(1:max(m,n))
192 * and the cosines in WORK(max(m,n)+1:2*max(m,n)).
193 *
194 MN = MAX( M, N )
195 KLM = MIN( M-1, KL )
196 KUN = MIN( N-1, KU )
197 KB = KLM + KUN
198 KB1 = KB + 1
199 INCA = KB1*LDAB
200 NR = 0
201 J1 = KLM + 2
202 J2 = 1 - KUN
203 *
204 DO 90 I = 1, MINMN
205 *
206 * Reduce i-th column and i-th row of matrix to bidiagonal form
207 *
208 ML = KLM + 1
209 MU = KUN + 1
210 DO 80 KK = 1, KB
211 J1 = J1 + KB
212 J2 = J2 + KB
213 *
214 * generate plane rotations to annihilate nonzero elements
215 * which have been created below the band
216 *
217 IF( NR.GT.0 )
218 $ CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
219 $ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
220 *
221 * apply plane rotations from the left
222 *
223 DO 10 L = 1, KB
224 IF( J2-KLM+L-1.GT.N ) THEN
225 NRT = NR - 1
226 ELSE
227 NRT = NR
228 END IF
229 IF( NRT.GT.0 )
230 $ CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
231 $ AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
232 $ WORK( MN+J1 ), WORK( J1 ), KB1 )
233 10 CONTINUE
234 *
235 IF( ML.GT.ML0 ) THEN
236 IF( ML.LE.M-I+1 ) THEN
237 *
238 * generate plane rotation to annihilate a(i+ml-1,i)
239 * within the band, and apply rotation from the left
240 *
241 CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
242 $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
243 $ RA )
244 AB( KU+ML-1, I ) = RA
245 IF( I.LT.N )
246 $ CALL DROT( MIN( KU+ML-2, N-I ),
247 $ AB( KU+ML-2, I+1 ), LDAB-1,
248 $ AB( KU+ML-1, I+1 ), LDAB-1,
249 $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
250 END IF
251 NR = NR + 1
252 J1 = J1 - KB1
253 END IF
254 *
255 IF( WANTQ ) THEN
256 *
257 * accumulate product of plane rotations in Q
258 *
259 DO 20 J = J1, J2, KB1
260 CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
261 $ WORK( MN+J ), WORK( J ) )
262 20 CONTINUE
263 END IF
264 *
265 IF( WANTC ) THEN
266 *
267 * apply plane rotations to C
268 *
269 DO 30 J = J1, J2, KB1
270 CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
271 $ WORK( MN+J ), WORK( J ) )
272 30 CONTINUE
273 END IF
274 *
275 IF( J2+KUN.GT.N ) THEN
276 *
277 * adjust J2 to keep within the bounds of the matrix
278 *
279 NR = NR - 1
280 J2 = J2 - KB1
281 END IF
282 *
283 DO 40 J = J1, J2, KB1
284 *
285 * create nonzero element a(j-1,j+ku) above the band
286 * and store it in WORK(n+1:2*n)
287 *
288 WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
289 AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
290 40 CONTINUE
291 *
292 * generate plane rotations to annihilate nonzero elements
293 * which have been generated above the band
294 *
295 IF( NR.GT.0 )
296 $ CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
297 $ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
298 $ KB1 )
299 *
300 * apply plane rotations from the right
301 *
302 DO 50 L = 1, KB
303 IF( J2+L-1.GT.M ) THEN
304 NRT = NR - 1
305 ELSE
306 NRT = NR
307 END IF
308 IF( NRT.GT.0 )
309 $ CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
310 $ AB( L, J1+KUN ), INCA,
311 $ WORK( MN+J1+KUN ), WORK( J1+KUN ),
312 $ KB1 )
313 50 CONTINUE
314 *
315 IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
316 IF( MU.LE.N-I+1 ) THEN
317 *
318 * generate plane rotation to annihilate a(i,i+mu-1)
319 * within the band, and apply rotation from the right
320 *
321 CALL DLARTG( AB( KU-MU+3, I+MU-2 ),
322 $ AB( KU-MU+2, I+MU-1 ),
323 $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
324 $ RA )
325 AB( KU-MU+3, I+MU-2 ) = RA
326 CALL DROT( MIN( KL+MU-2, M-I ),
327 $ AB( KU-MU+4, I+MU-2 ), 1,
328 $ AB( KU-MU+3, I+MU-1 ), 1,
329 $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
330 END IF
331 NR = NR + 1
332 J1 = J1 - KB1
333 END IF
334 *
335 IF( WANTPT ) THEN
336 *
337 * accumulate product of plane rotations in P**T
338 *
339 DO 60 J = J1, J2, KB1
340 CALL DROT( N, PT( J+KUN-1, 1 ), LDPT,
341 $ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
342 $ WORK( J+KUN ) )
343 60 CONTINUE
344 END IF
345 *
346 IF( J2+KB.GT.M ) THEN
347 *
348 * adjust J2 to keep within the bounds of the matrix
349 *
350 NR = NR - 1
351 J2 = J2 - KB1
352 END IF
353 *
354 DO 70 J = J1, J2, KB1
355 *
356 * create nonzero element a(j+kl+ku,j+ku-1) below the
357 * band and store it in WORK(1:n)
358 *
359 WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
360 AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
361 70 CONTINUE
362 *
363 IF( ML.GT.ML0 ) THEN
364 ML = ML - 1
365 ELSE
366 MU = MU - 1
367 END IF
368 80 CONTINUE
369 90 CONTINUE
370 END IF
371 *
372 IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
373 *
374 * A has been reduced to lower bidiagonal form
375 *
376 * Transform lower bidiagonal form to upper bidiagonal by applying
377 * plane rotations from the left, storing diagonal elements in D
378 * and off-diagonal elements in E
379 *
380 DO 100 I = 1, MIN( M-1, N )
381 CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
382 D( I ) = RA
383 IF( I.LT.N ) THEN
384 E( I ) = RS*AB( 1, I+1 )
385 AB( 1, I+1 ) = RC*AB( 1, I+1 )
386 END IF
387 IF( WANTQ )
388 $ CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
389 IF( WANTC )
390 $ CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
391 $ RS )
392 100 CONTINUE
393 IF( M.LE.N )
394 $ D( M ) = AB( 1, M )
395 ELSE IF( KU.GT.0 ) THEN
396 *
397 * A has been reduced to upper bidiagonal form
398 *
399 IF( M.LT.N ) THEN
400 *
401 * Annihilate a(m,m+1) by applying plane rotations from the
402 * right, storing diagonal elements in D and off-diagonal
403 * elements in E
404 *
405 RB = AB( KU, M+1 )
406 DO 110 I = M, 1, -1
407 CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA )
408 D( I ) = RA
409 IF( I.GT.1 ) THEN
410 RB = -RS*AB( KU, I )
411 E( I-1 ) = RC*AB( KU, I )
412 END IF
413 IF( WANTPT )
414 $ CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
415 $ RC, RS )
416 110 CONTINUE
417 ELSE
418 *
419 * Copy off-diagonal elements to E and diagonal elements to D
420 *
421 DO 120 I = 1, MINMN - 1
422 E( I ) = AB( KU, I+1 )
423 120 CONTINUE
424 DO 130 I = 1, MINMN
425 D( I ) = AB( KU+1, I )
426 130 CONTINUE
427 END IF
428 ELSE
429 *
430 * A is diagonal. Set elements of E to zero and copy diagonal
431 * elements to D.
432 *
433 DO 140 I = 1, MINMN - 1
434 E( I ) = ZERO
435 140 CONTINUE
436 DO 150 I = 1, MINMN
437 D( I ) = AB( 1, I )
438 150 CONTINUE
439 END IF
440 RETURN
441 *
442 * End of DGBBRD
443 *
444 END