1 SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
2 $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
3 $ IWORK, TAU, WORK, INFO )
4 *
5 * -- LAPACK routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBQ, JOBU, JOBV
12 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
13 DOUBLE PRECISION TOLA, TOLB
14 * ..
15 * .. Array Arguments ..
16 INTEGER IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
18 $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DGGSVP computes orthogonal matrices U, V and Q such that
25 *
26 * N-K-L K L
27 * U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
28 * L ( 0 0 A23 )
29 * M-K-L ( 0 0 0 )
30 *
31 * N-K-L K L
32 * = K ( 0 A12 A13 ) if M-K-L < 0;
33 * M-K ( 0 0 A23 )
34 *
35 * N-K-L K L
36 * V**T*B*Q = L ( 0 0 B13 )
37 * P-L ( 0 0 0 )
38 *
39 * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
40 * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
41 * otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
42 * numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
43 *
44 * This decomposition is the preprocessing step for computing the
45 * Generalized Singular Value Decomposition (GSVD), see subroutine
46 * DGGSVD.
47 *
48 * Arguments
49 * =========
50 *
51 * JOBU (input) CHARACTER*1
52 * = 'U': Orthogonal matrix U is computed;
53 * = 'N': U is not computed.
54 *
55 * JOBV (input) CHARACTER*1
56 * = 'V': Orthogonal matrix V is computed;
57 * = 'N': V is not computed.
58 *
59 * JOBQ (input) CHARACTER*1
60 * = 'Q': Orthogonal matrix Q is computed;
61 * = 'N': Q is not computed.
62 *
63 * M (input) INTEGER
64 * The number of rows of the matrix A. M >= 0.
65 *
66 * P (input) INTEGER
67 * The number of rows of the matrix B. P >= 0.
68 *
69 * N (input) INTEGER
70 * The number of columns of the matrices A and B. N >= 0.
71 *
72 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
73 * On entry, the M-by-N matrix A.
74 * On exit, A contains the triangular (or trapezoidal) matrix
75 * described in the Purpose section.
76 *
77 * LDA (input) INTEGER
78 * The leading dimension of the array A. LDA >= max(1,M).
79 *
80 * B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
81 * On entry, the P-by-N matrix B.
82 * On exit, B contains the triangular matrix described in
83 * the Purpose section.
84 *
85 * LDB (input) INTEGER
86 * The leading dimension of the array B. LDB >= max(1,P).
87 *
88 * TOLA (input) DOUBLE PRECISION
89 * TOLB (input) DOUBLE PRECISION
90 * TOLA and TOLB are the thresholds to determine the effective
91 * numerical rank of matrix B and a subblock of A. Generally,
92 * they are set to
93 * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
94 * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
95 * The size of TOLA and TOLB may affect the size of backward
96 * errors of the decomposition.
97 *
98 * K (output) INTEGER
99 * L (output) INTEGER
100 * On exit, K and L specify the dimension of the subblocks
101 * described in Purpose section.
102 * K + L = effective numerical rank of (A**T,B**T)**T.
103 *
104 * U (output) DOUBLE PRECISION array, dimension (LDU,M)
105 * If JOBU = 'U', U contains the orthogonal matrix U.
106 * If JOBU = 'N', U is not referenced.
107 *
108 * LDU (input) INTEGER
109 * The leading dimension of the array U. LDU >= max(1,M) if
110 * JOBU = 'U'; LDU >= 1 otherwise.
111 *
112 * V (output) DOUBLE PRECISION array, dimension (LDV,P)
113 * If JOBV = 'V', V contains the orthogonal matrix V.
114 * If JOBV = 'N', V is not referenced.
115 *
116 * LDV (input) INTEGER
117 * The leading dimension of the array V. LDV >= max(1,P) if
118 * JOBV = 'V'; LDV >= 1 otherwise.
119 *
120 * Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
121 * If JOBQ = 'Q', Q contains the orthogonal matrix Q.
122 * If JOBQ = 'N', Q is not referenced.
123 *
124 * LDQ (input) INTEGER
125 * The leading dimension of the array Q. LDQ >= max(1,N) if
126 * JOBQ = 'Q'; LDQ >= 1 otherwise.
127 *
128 * IWORK (workspace) INTEGER array, dimension (N)
129 *
130 * TAU (workspace) DOUBLE PRECISION array, dimension (N)
131 *
132 * WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P))
133 *
134 * INFO (output) INTEGER
135 * = 0: successful exit
136 * < 0: if INFO = -i, the i-th argument had an illegal value.
137 *
138 *
139 * Further Details
140 * ===============
141 *
142 * The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
143 * with column pivoting to detect the effective numerical rank of the
144 * a matrix. It may be replaced by a better rank determination strategy.
145 *
146 * =====================================================================
147 *
148 * .. Parameters ..
149 DOUBLE PRECISION ZERO, ONE
150 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
151 * ..
152 * .. Local Scalars ..
153 LOGICAL FORWRD, WANTQ, WANTU, WANTV
154 INTEGER I, J
155 * ..
156 * .. External Functions ..
157 LOGICAL LSAME
158 EXTERNAL LSAME
159 * ..
160 * .. External Subroutines ..
161 EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
162 $ DORG2R, DORM2R, DORMR2, XERBLA
163 * ..
164 * .. Intrinsic Functions ..
165 INTRINSIC ABS, MAX, MIN
166 * ..
167 * .. Executable Statements ..
168 *
169 * Test the input parameters
170 *
171 WANTU = LSAME( JOBU, 'U' )
172 WANTV = LSAME( JOBV, 'V' )
173 WANTQ = LSAME( JOBQ, 'Q' )
174 FORWRD = .TRUE.
175 *
176 INFO = 0
177 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
178 INFO = -1
179 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
180 INFO = -2
181 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
182 INFO = -3
183 ELSE IF( M.LT.0 ) THEN
184 INFO = -4
185 ELSE IF( P.LT.0 ) THEN
186 INFO = -5
187 ELSE IF( N.LT.0 ) THEN
188 INFO = -6
189 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
190 INFO = -8
191 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
192 INFO = -10
193 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
194 INFO = -16
195 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
196 INFO = -18
197 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
198 INFO = -20
199 END IF
200 IF( INFO.NE.0 ) THEN
201 CALL XERBLA( 'DGGSVP', -INFO )
202 RETURN
203 END IF
204 *
205 * QR with column pivoting of B: B*P = V*( S11 S12 )
206 * ( 0 0 )
207 *
208 DO 10 I = 1, N
209 IWORK( I ) = 0
210 10 CONTINUE
211 CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
212 *
213 * Update A := A*P
214 *
215 CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
216 *
217 * Determine the effective rank of matrix B.
218 *
219 L = 0
220 DO 20 I = 1, MIN( P, N )
221 IF( ABS( B( I, I ) ).GT.TOLB )
222 $ L = L + 1
223 20 CONTINUE
224 *
225 IF( WANTV ) THEN
226 *
227 * Copy the details of V, and form V.
228 *
229 CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
230 IF( P.GT.1 )
231 $ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
232 $ LDV )
233 CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
234 END IF
235 *
236 * Clean up B
237 *
238 DO 40 J = 1, L - 1
239 DO 30 I = J + 1, L
240 B( I, J ) = ZERO
241 30 CONTINUE
242 40 CONTINUE
243 IF( P.GT.L )
244 $ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
245 *
246 IF( WANTQ ) THEN
247 *
248 * Set Q = I and Update Q := Q*P
249 *
250 CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
251 CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
252 END IF
253 *
254 IF( P.GE.L .AND. N.NE.L ) THEN
255 *
256 * RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
257 *
258 CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
259 *
260 * Update A := A*Z**T
261 *
262 CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
263 $ LDA, WORK, INFO )
264 *
265 IF( WANTQ ) THEN
266 *
267 * Update Q := Q*Z**T
268 *
269 CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
270 $ LDQ, WORK, INFO )
271 END IF
272 *
273 * Clean up B
274 *
275 CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
276 DO 60 J = N - L + 1, N
277 DO 50 I = J - N + L + 1, L
278 B( I, J ) = ZERO
279 50 CONTINUE
280 60 CONTINUE
281 *
282 END IF
283 *
284 * Let N-L L
285 * A = ( A11 A12 ) M,
286 *
287 * then the following does the complete QR decomposition of A11:
288 *
289 * A11 = U*( 0 T12 )*P1**T
290 * ( 0 0 )
291 *
292 DO 70 I = 1, N - L
293 IWORK( I ) = 0
294 70 CONTINUE
295 CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
296 *
297 * Determine the effective rank of A11
298 *
299 K = 0
300 DO 80 I = 1, MIN( M, N-L )
301 IF( ABS( A( I, I ) ).GT.TOLA )
302 $ K = K + 1
303 80 CONTINUE
304 *
305 * Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
306 *
307 CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
308 $ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
309 *
310 IF( WANTU ) THEN
311 *
312 * Copy the details of U, and form U
313 *
314 CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
315 IF( M.GT.1 )
316 $ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
317 $ LDU )
318 CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
319 END IF
320 *
321 IF( WANTQ ) THEN
322 *
323 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
324 *
325 CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
326 END IF
327 *
328 * Clean up A: set the strictly lower triangular part of
329 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
330 *
331 DO 100 J = 1, K - 1
332 DO 90 I = J + 1, K
333 A( I, J ) = ZERO
334 90 CONTINUE
335 100 CONTINUE
336 IF( M.GT.K )
337 $ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
338 *
339 IF( N-L.GT.K ) THEN
340 *
341 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
342 *
343 CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
344 *
345 IF( WANTQ ) THEN
346 *
347 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
348 *
349 CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
350 $ Q, LDQ, WORK, INFO )
351 END IF
352 *
353 * Clean up A
354 *
355 CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
356 DO 120 J = N - L - K + 1, N - L
357 DO 110 I = J - N + L + K + 1, K
358 A( I, J ) = ZERO
359 110 CONTINUE
360 120 CONTINUE
361 *
362 END IF
363 *
364 IF( M.GT.K ) THEN
365 *
366 * QR factorization of A( K+1:M,N-L+1:N )
367 *
368 CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
369 *
370 IF( WANTU ) THEN
371 *
372 * Update U(:,K+1:M) := U(:,K+1:M)*U1
373 *
374 CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
375 $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
376 $ WORK, INFO )
377 END IF
378 *
379 * Clean up
380 *
381 DO 140 J = N - L + 1, N
382 DO 130 I = J - N + K + L + 1, M
383 A( I, J ) = ZERO
384 130 CONTINUE
385 140 CONTINUE
386 *
387 END IF
388 *
389 RETURN
390 *
391 * End of DGGSVP
392 *
393 END
2 $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
3 $ IWORK, TAU, WORK, INFO )
4 *
5 * -- LAPACK routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBQ, JOBU, JOBV
12 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
13 DOUBLE PRECISION TOLA, TOLB
14 * ..
15 * .. Array Arguments ..
16 INTEGER IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
18 $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DGGSVP computes orthogonal matrices U, V and Q such that
25 *
26 * N-K-L K L
27 * U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
28 * L ( 0 0 A23 )
29 * M-K-L ( 0 0 0 )
30 *
31 * N-K-L K L
32 * = K ( 0 A12 A13 ) if M-K-L < 0;
33 * M-K ( 0 0 A23 )
34 *
35 * N-K-L K L
36 * V**T*B*Q = L ( 0 0 B13 )
37 * P-L ( 0 0 0 )
38 *
39 * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
40 * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
41 * otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
42 * numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
43 *
44 * This decomposition is the preprocessing step for computing the
45 * Generalized Singular Value Decomposition (GSVD), see subroutine
46 * DGGSVD.
47 *
48 * Arguments
49 * =========
50 *
51 * JOBU (input) CHARACTER*1
52 * = 'U': Orthogonal matrix U is computed;
53 * = 'N': U is not computed.
54 *
55 * JOBV (input) CHARACTER*1
56 * = 'V': Orthogonal matrix V is computed;
57 * = 'N': V is not computed.
58 *
59 * JOBQ (input) CHARACTER*1
60 * = 'Q': Orthogonal matrix Q is computed;
61 * = 'N': Q is not computed.
62 *
63 * M (input) INTEGER
64 * The number of rows of the matrix A. M >= 0.
65 *
66 * P (input) INTEGER
67 * The number of rows of the matrix B. P >= 0.
68 *
69 * N (input) INTEGER
70 * The number of columns of the matrices A and B. N >= 0.
71 *
72 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
73 * On entry, the M-by-N matrix A.
74 * On exit, A contains the triangular (or trapezoidal) matrix
75 * described in the Purpose section.
76 *
77 * LDA (input) INTEGER
78 * The leading dimension of the array A. LDA >= max(1,M).
79 *
80 * B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
81 * On entry, the P-by-N matrix B.
82 * On exit, B contains the triangular matrix described in
83 * the Purpose section.
84 *
85 * LDB (input) INTEGER
86 * The leading dimension of the array B. LDB >= max(1,P).
87 *
88 * TOLA (input) DOUBLE PRECISION
89 * TOLB (input) DOUBLE PRECISION
90 * TOLA and TOLB are the thresholds to determine the effective
91 * numerical rank of matrix B and a subblock of A. Generally,
92 * they are set to
93 * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
94 * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
95 * The size of TOLA and TOLB may affect the size of backward
96 * errors of the decomposition.
97 *
98 * K (output) INTEGER
99 * L (output) INTEGER
100 * On exit, K and L specify the dimension of the subblocks
101 * described in Purpose section.
102 * K + L = effective numerical rank of (A**T,B**T)**T.
103 *
104 * U (output) DOUBLE PRECISION array, dimension (LDU,M)
105 * If JOBU = 'U', U contains the orthogonal matrix U.
106 * If JOBU = 'N', U is not referenced.
107 *
108 * LDU (input) INTEGER
109 * The leading dimension of the array U. LDU >= max(1,M) if
110 * JOBU = 'U'; LDU >= 1 otherwise.
111 *
112 * V (output) DOUBLE PRECISION array, dimension (LDV,P)
113 * If JOBV = 'V', V contains the orthogonal matrix V.
114 * If JOBV = 'N', V is not referenced.
115 *
116 * LDV (input) INTEGER
117 * The leading dimension of the array V. LDV >= max(1,P) if
118 * JOBV = 'V'; LDV >= 1 otherwise.
119 *
120 * Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
121 * If JOBQ = 'Q', Q contains the orthogonal matrix Q.
122 * If JOBQ = 'N', Q is not referenced.
123 *
124 * LDQ (input) INTEGER
125 * The leading dimension of the array Q. LDQ >= max(1,N) if
126 * JOBQ = 'Q'; LDQ >= 1 otherwise.
127 *
128 * IWORK (workspace) INTEGER array, dimension (N)
129 *
130 * TAU (workspace) DOUBLE PRECISION array, dimension (N)
131 *
132 * WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P))
133 *
134 * INFO (output) INTEGER
135 * = 0: successful exit
136 * < 0: if INFO = -i, the i-th argument had an illegal value.
137 *
138 *
139 * Further Details
140 * ===============
141 *
142 * The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
143 * with column pivoting to detect the effective numerical rank of the
144 * a matrix. It may be replaced by a better rank determination strategy.
145 *
146 * =====================================================================
147 *
148 * .. Parameters ..
149 DOUBLE PRECISION ZERO, ONE
150 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
151 * ..
152 * .. Local Scalars ..
153 LOGICAL FORWRD, WANTQ, WANTU, WANTV
154 INTEGER I, J
155 * ..
156 * .. External Functions ..
157 LOGICAL LSAME
158 EXTERNAL LSAME
159 * ..
160 * .. External Subroutines ..
161 EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
162 $ DORG2R, DORM2R, DORMR2, XERBLA
163 * ..
164 * .. Intrinsic Functions ..
165 INTRINSIC ABS, MAX, MIN
166 * ..
167 * .. Executable Statements ..
168 *
169 * Test the input parameters
170 *
171 WANTU = LSAME( JOBU, 'U' )
172 WANTV = LSAME( JOBV, 'V' )
173 WANTQ = LSAME( JOBQ, 'Q' )
174 FORWRD = .TRUE.
175 *
176 INFO = 0
177 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
178 INFO = -1
179 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
180 INFO = -2
181 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
182 INFO = -3
183 ELSE IF( M.LT.0 ) THEN
184 INFO = -4
185 ELSE IF( P.LT.0 ) THEN
186 INFO = -5
187 ELSE IF( N.LT.0 ) THEN
188 INFO = -6
189 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
190 INFO = -8
191 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
192 INFO = -10
193 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
194 INFO = -16
195 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
196 INFO = -18
197 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
198 INFO = -20
199 END IF
200 IF( INFO.NE.0 ) THEN
201 CALL XERBLA( 'DGGSVP', -INFO )
202 RETURN
203 END IF
204 *
205 * QR with column pivoting of B: B*P = V*( S11 S12 )
206 * ( 0 0 )
207 *
208 DO 10 I = 1, N
209 IWORK( I ) = 0
210 10 CONTINUE
211 CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
212 *
213 * Update A := A*P
214 *
215 CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
216 *
217 * Determine the effective rank of matrix B.
218 *
219 L = 0
220 DO 20 I = 1, MIN( P, N )
221 IF( ABS( B( I, I ) ).GT.TOLB )
222 $ L = L + 1
223 20 CONTINUE
224 *
225 IF( WANTV ) THEN
226 *
227 * Copy the details of V, and form V.
228 *
229 CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
230 IF( P.GT.1 )
231 $ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
232 $ LDV )
233 CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
234 END IF
235 *
236 * Clean up B
237 *
238 DO 40 J = 1, L - 1
239 DO 30 I = J + 1, L
240 B( I, J ) = ZERO
241 30 CONTINUE
242 40 CONTINUE
243 IF( P.GT.L )
244 $ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
245 *
246 IF( WANTQ ) THEN
247 *
248 * Set Q = I and Update Q := Q*P
249 *
250 CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
251 CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
252 END IF
253 *
254 IF( P.GE.L .AND. N.NE.L ) THEN
255 *
256 * RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
257 *
258 CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
259 *
260 * Update A := A*Z**T
261 *
262 CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
263 $ LDA, WORK, INFO )
264 *
265 IF( WANTQ ) THEN
266 *
267 * Update Q := Q*Z**T
268 *
269 CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
270 $ LDQ, WORK, INFO )
271 END IF
272 *
273 * Clean up B
274 *
275 CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
276 DO 60 J = N - L + 1, N
277 DO 50 I = J - N + L + 1, L
278 B( I, J ) = ZERO
279 50 CONTINUE
280 60 CONTINUE
281 *
282 END IF
283 *
284 * Let N-L L
285 * A = ( A11 A12 ) M,
286 *
287 * then the following does the complete QR decomposition of A11:
288 *
289 * A11 = U*( 0 T12 )*P1**T
290 * ( 0 0 )
291 *
292 DO 70 I = 1, N - L
293 IWORK( I ) = 0
294 70 CONTINUE
295 CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
296 *
297 * Determine the effective rank of A11
298 *
299 K = 0
300 DO 80 I = 1, MIN( M, N-L )
301 IF( ABS( A( I, I ) ).GT.TOLA )
302 $ K = K + 1
303 80 CONTINUE
304 *
305 * Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
306 *
307 CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
308 $ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
309 *
310 IF( WANTU ) THEN
311 *
312 * Copy the details of U, and form U
313 *
314 CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
315 IF( M.GT.1 )
316 $ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
317 $ LDU )
318 CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
319 END IF
320 *
321 IF( WANTQ ) THEN
322 *
323 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
324 *
325 CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
326 END IF
327 *
328 * Clean up A: set the strictly lower triangular part of
329 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
330 *
331 DO 100 J = 1, K - 1
332 DO 90 I = J + 1, K
333 A( I, J ) = ZERO
334 90 CONTINUE
335 100 CONTINUE
336 IF( M.GT.K )
337 $ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
338 *
339 IF( N-L.GT.K ) THEN
340 *
341 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
342 *
343 CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
344 *
345 IF( WANTQ ) THEN
346 *
347 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
348 *
349 CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
350 $ Q, LDQ, WORK, INFO )
351 END IF
352 *
353 * Clean up A
354 *
355 CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
356 DO 120 J = N - L - K + 1, N - L
357 DO 110 I = J - N + L + K + 1, K
358 A( I, J ) = ZERO
359 110 CONTINUE
360 120 CONTINUE
361 *
362 END IF
363 *
364 IF( M.GT.K ) THEN
365 *
366 * QR factorization of A( K+1:M,N-L+1:N )
367 *
368 CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
369 *
370 IF( WANTU ) THEN
371 *
372 * Update U(:,K+1:M) := U(:,K+1:M)*U1
373 *
374 CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
375 $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
376 $ WORK, INFO )
377 END IF
378 *
379 * Clean up
380 *
381 DO 140 J = N - L + 1, N
382 DO 130 I = J - N + K + L + 1, M
383 A( I, J ) = ZERO
384 130 CONTINUE
385 140 CONTINUE
386 *
387 END IF
388 *
389 RETURN
390 *
391 * End of DGGSVP
392 *
393 END