1 SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
2 $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
3 $ RWORK, IWORK, INFO )
4 *
5 * -- LAPACK driver 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 * ..
14 * .. Array Arguments ..
15 INTEGER IWORK( * )
16 DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
17 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
18 $ U( LDU, * ), V( LDV, * ), WORK( * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZGGSVD computes the generalized singular value decomposition (GSVD)
25 * of an M-by-N complex matrix A and P-by-N complex matrix B:
26 *
27 * U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R )
28 *
29 * where U, V and Q are unitary matrices.
30 * Let K+L = the effective numerical rank of the
31 * matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
32 * triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
33 * matrices and of the following structures, respectively:
34 *
35 * If M-K-L >= 0,
36 *
37 * K L
38 * D1 = K ( I 0 )
39 * L ( 0 C )
40 * M-K-L ( 0 0 )
41 *
42 * K L
43 * D2 = L ( 0 S )
44 * P-L ( 0 0 )
45 *
46 * N-K-L K L
47 * ( 0 R ) = K ( 0 R11 R12 )
48 * L ( 0 0 R22 )
49 * where
50 *
51 * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
52 * S = diag( BETA(K+1), ... , BETA(K+L) ),
53 * C**2 + S**2 = I.
54 *
55 * R is stored in A(1:K+L,N-K-L+1:N) on exit.
56 *
57 * If M-K-L < 0,
58 *
59 * K M-K K+L-M
60 * D1 = K ( I 0 0 )
61 * M-K ( 0 C 0 )
62 *
63 * K M-K K+L-M
64 * D2 = M-K ( 0 S 0 )
65 * K+L-M ( 0 0 I )
66 * P-L ( 0 0 0 )
67 *
68 * N-K-L K M-K K+L-M
69 * ( 0 R ) = K ( 0 R11 R12 R13 )
70 * M-K ( 0 0 R22 R23 )
71 * K+L-M ( 0 0 0 R33 )
72 *
73 * where
74 *
75 * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
76 * S = diag( BETA(K+1), ... , BETA(M) ),
77 * C**2 + S**2 = I.
78 *
79 * (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
80 * ( 0 R22 R23 )
81 * in B(M-K+1:L,N+M-K-L+1:N) on exit.
82 *
83 * The routine computes C, S, R, and optionally the unitary
84 * transformation matrices U, V and Q.
85 *
86 * In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
87 * A and B implicitly gives the SVD of A*inv(B):
88 * A*inv(B) = U*(D1*inv(D2))*V**H.
89 * If ( A**H,B**H)**H has orthnormal columns, then the GSVD of A and B is also
90 * equal to the CS decomposition of A and B. Furthermore, the GSVD can
91 * be used to derive the solution of the eigenvalue problem:
92 * A**H*A x = lambda* B**H*B x.
93 * In some literature, the GSVD of A and B is presented in the form
94 * U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 )
95 * where U and V are orthogonal and X is nonsingular, and D1 and D2 are
96 * ``diagonal''. The former GSVD form can be converted to the latter
97 * form by taking the nonsingular matrix X as
98 *
99 * X = Q*( I 0 )
100 * ( 0 inv(R) )
101 *
102 * Arguments
103 * =========
104 *
105 * JOBU (input) CHARACTER*1
106 * = 'U': Unitary matrix U is computed;
107 * = 'N': U is not computed.
108 *
109 * JOBV (input) CHARACTER*1
110 * = 'V': Unitary matrix V is computed;
111 * = 'N': V is not computed.
112 *
113 * JOBQ (input) CHARACTER*1
114 * = 'Q': Unitary matrix Q is computed;
115 * = 'N': Q is not computed.
116 *
117 * M (input) INTEGER
118 * The number of rows of the matrix A. M >= 0.
119 *
120 * N (input) INTEGER
121 * The number of columns of the matrices A and B. N >= 0.
122 *
123 * P (input) INTEGER
124 * The number of rows of the matrix B. P >= 0.
125 *
126 * K (output) INTEGER
127 * L (output) INTEGER
128 * On exit, K and L specify the dimension of the subblocks
129 * described in Purpose.
130 * K + L = effective numerical rank of (A**H,B**H)**H.
131 *
132 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
133 * On entry, the M-by-N matrix A.
134 * On exit, A contains the triangular matrix R, or part of R.
135 * See Purpose for details.
136 *
137 * LDA (input) INTEGER
138 * The leading dimension of the array A. LDA >= max(1,M).
139 *
140 * B (input/output) COMPLEX*16 array, dimension (LDB,N)
141 * On entry, the P-by-N matrix B.
142 * On exit, B contains part of the triangular matrix R if
143 * M-K-L < 0. See Purpose for details.
144 *
145 * LDB (input) INTEGER
146 * The leading dimension of the array B. LDB >= max(1,P).
147 *
148 * ALPHA (output) DOUBLE PRECISION array, dimension (N)
149 * BETA (output) DOUBLE PRECISION array, dimension (N)
150 * On exit, ALPHA and BETA contain the generalized singular
151 * value pairs of A and B;
152 * ALPHA(1:K) = 1,
153 * BETA(1:K) = 0,
154 * and if M-K-L >= 0,
155 * ALPHA(K+1:K+L) = C,
156 * BETA(K+1:K+L) = S,
157 * or if M-K-L < 0,
158 * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
159 * BETA(K+1:M) = S, BETA(M+1:K+L) = 1
160 * and
161 * ALPHA(K+L+1:N) = 0
162 * BETA(K+L+1:N) = 0
163 *
164 * U (output) COMPLEX*16 array, dimension (LDU,M)
165 * If JOBU = 'U', U contains the M-by-M unitary matrix U.
166 * If JOBU = 'N', U is not referenced.
167 *
168 * LDU (input) INTEGER
169 * The leading dimension of the array U. LDU >= max(1,M) if
170 * JOBU = 'U'; LDU >= 1 otherwise.
171 *
172 * V (output) COMPLEX*16 array, dimension (LDV,P)
173 * If JOBV = 'V', V contains the P-by-P unitary matrix V.
174 * If JOBV = 'N', V is not referenced.
175 *
176 * LDV (input) INTEGER
177 * The leading dimension of the array V. LDV >= max(1,P) if
178 * JOBV = 'V'; LDV >= 1 otherwise.
179 *
180 * Q (output) COMPLEX*16 array, dimension (LDQ,N)
181 * If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
182 * If JOBQ = 'N', Q is not referenced.
183 *
184 * LDQ (input) INTEGER
185 * The leading dimension of the array Q. LDQ >= max(1,N) if
186 * JOBQ = 'Q'; LDQ >= 1 otherwise.
187 *
188 * WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)
189 *
190 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
191 *
192 * IWORK (workspace/output) INTEGER array, dimension (N)
193 * On exit, IWORK stores the sorting information. More
194 * precisely, the following loop will sort ALPHA
195 * for I = K+1, min(M,K+L)
196 * swap ALPHA(I) and ALPHA(IWORK(I))
197 * endfor
198 * such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
199 *
200 * INFO (output) INTEGER
201 * = 0: successful exit.
202 * < 0: if INFO = -i, the i-th argument had an illegal value.
203 * > 0: if INFO = 1, the Jacobi-type procedure failed to
204 * converge. For further details, see subroutine ZTGSJA.
205 *
206 * Internal Parameters
207 * ===================
208 *
209 * TOLA DOUBLE PRECISION
210 * TOLB DOUBLE PRECISION
211 * TOLA and TOLB are the thresholds to determine the effective
212 * rank of (A**H,B**H)**H. Generally, they are set to
213 * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
214 * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
215 * The size of TOLA and TOLB may affect the size of backward
216 * errors of the decomposition.
217 *
218 * Further Details
219 * ===============
220 *
221 * 2-96 Based on modifications by
222 * Ming Gu and Huan Ren, Computer Science Division, University of
223 * California at Berkeley, USA
224 *
225 * =====================================================================
226 *
227 * .. Local Scalars ..
228 LOGICAL WANTQ, WANTU, WANTV
229 INTEGER I, IBND, ISUB, J, NCYCLE
230 DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
231 * ..
232 * .. External Functions ..
233 LOGICAL LSAME
234 DOUBLE PRECISION DLAMCH, ZLANGE
235 EXTERNAL LSAME, DLAMCH, ZLANGE
236 * ..
237 * .. External Subroutines ..
238 EXTERNAL DCOPY, XERBLA, ZGGSVP, ZTGSJA
239 * ..
240 * .. Intrinsic Functions ..
241 INTRINSIC MAX, MIN
242 * ..
243 * .. Executable Statements ..
244 *
245 * Decode and test the input parameters
246 *
247 WANTU = LSAME( JOBU, 'U' )
248 WANTV = LSAME( JOBV, 'V' )
249 WANTQ = LSAME( JOBQ, 'Q' )
250 *
251 INFO = 0
252 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
253 INFO = -1
254 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
255 INFO = -2
256 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
257 INFO = -3
258 ELSE IF( M.LT.0 ) THEN
259 INFO = -4
260 ELSE IF( N.LT.0 ) THEN
261 INFO = -5
262 ELSE IF( P.LT.0 ) THEN
263 INFO = -6
264 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
265 INFO = -10
266 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
267 INFO = -12
268 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
269 INFO = -16
270 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
271 INFO = -18
272 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
273 INFO = -20
274 END IF
275 IF( INFO.NE.0 ) THEN
276 CALL XERBLA( 'ZGGSVD', -INFO )
277 RETURN
278 END IF
279 *
280 * Compute the Frobenius norm of matrices A and B
281 *
282 ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
283 BNORM = ZLANGE( '1', P, N, B, LDB, RWORK )
284 *
285 * Get machine precision and set up threshold for determining
286 * the effective numerical rank of the matrices A and B.
287 *
288 ULP = DLAMCH( 'Precision' )
289 UNFL = DLAMCH( 'Safe Minimum' )
290 TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
291 TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
292 *
293 CALL ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
294 $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
295 $ WORK, WORK( N+1 ), INFO )
296 *
297 * Compute the GSVD of two upper "triangular" matrices
298 *
299 CALL ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
300 $ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
301 $ WORK, NCYCLE, INFO )
302 *
303 * Sort the singular values and store the pivot indices in IWORK
304 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
305 *
306 CALL DCOPY( N, ALPHA, 1, RWORK, 1 )
307 IBND = MIN( L, M-K )
308 DO 20 I = 1, IBND
309 *
310 * Scan for largest ALPHA(K+I)
311 *
312 ISUB = I
313 SMAX = RWORK( K+I )
314 DO 10 J = I + 1, IBND
315 TEMP = RWORK( K+J )
316 IF( TEMP.GT.SMAX ) THEN
317 ISUB = J
318 SMAX = TEMP
319 END IF
320 10 CONTINUE
321 IF( ISUB.NE.I ) THEN
322 RWORK( K+ISUB ) = RWORK( K+I )
323 RWORK( K+I ) = SMAX
324 IWORK( K+I ) = K + ISUB
325 ELSE
326 IWORK( K+I ) = K + I
327 END IF
328 20 CONTINUE
329 *
330 RETURN
331 *
332 * End of ZGGSVD
333 *
334 END
2 $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
3 $ RWORK, IWORK, INFO )
4 *
5 * -- LAPACK driver 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 * ..
14 * .. Array Arguments ..
15 INTEGER IWORK( * )
16 DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
17 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
18 $ U( LDU, * ), V( LDV, * ), WORK( * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZGGSVD computes the generalized singular value decomposition (GSVD)
25 * of an M-by-N complex matrix A and P-by-N complex matrix B:
26 *
27 * U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R )
28 *
29 * where U, V and Q are unitary matrices.
30 * Let K+L = the effective numerical rank of the
31 * matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
32 * triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
33 * matrices and of the following structures, respectively:
34 *
35 * If M-K-L >= 0,
36 *
37 * K L
38 * D1 = K ( I 0 )
39 * L ( 0 C )
40 * M-K-L ( 0 0 )
41 *
42 * K L
43 * D2 = L ( 0 S )
44 * P-L ( 0 0 )
45 *
46 * N-K-L K L
47 * ( 0 R ) = K ( 0 R11 R12 )
48 * L ( 0 0 R22 )
49 * where
50 *
51 * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
52 * S = diag( BETA(K+1), ... , BETA(K+L) ),
53 * C**2 + S**2 = I.
54 *
55 * R is stored in A(1:K+L,N-K-L+1:N) on exit.
56 *
57 * If M-K-L < 0,
58 *
59 * K M-K K+L-M
60 * D1 = K ( I 0 0 )
61 * M-K ( 0 C 0 )
62 *
63 * K M-K K+L-M
64 * D2 = M-K ( 0 S 0 )
65 * K+L-M ( 0 0 I )
66 * P-L ( 0 0 0 )
67 *
68 * N-K-L K M-K K+L-M
69 * ( 0 R ) = K ( 0 R11 R12 R13 )
70 * M-K ( 0 0 R22 R23 )
71 * K+L-M ( 0 0 0 R33 )
72 *
73 * where
74 *
75 * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
76 * S = diag( BETA(K+1), ... , BETA(M) ),
77 * C**2 + S**2 = I.
78 *
79 * (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
80 * ( 0 R22 R23 )
81 * in B(M-K+1:L,N+M-K-L+1:N) on exit.
82 *
83 * The routine computes C, S, R, and optionally the unitary
84 * transformation matrices U, V and Q.
85 *
86 * In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
87 * A and B implicitly gives the SVD of A*inv(B):
88 * A*inv(B) = U*(D1*inv(D2))*V**H.
89 * If ( A**H,B**H)**H has orthnormal columns, then the GSVD of A and B is also
90 * equal to the CS decomposition of A and B. Furthermore, the GSVD can
91 * be used to derive the solution of the eigenvalue problem:
92 * A**H*A x = lambda* B**H*B x.
93 * In some literature, the GSVD of A and B is presented in the form
94 * U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 )
95 * where U and V are orthogonal and X is nonsingular, and D1 and D2 are
96 * ``diagonal''. The former GSVD form can be converted to the latter
97 * form by taking the nonsingular matrix X as
98 *
99 * X = Q*( I 0 )
100 * ( 0 inv(R) )
101 *
102 * Arguments
103 * =========
104 *
105 * JOBU (input) CHARACTER*1
106 * = 'U': Unitary matrix U is computed;
107 * = 'N': U is not computed.
108 *
109 * JOBV (input) CHARACTER*1
110 * = 'V': Unitary matrix V is computed;
111 * = 'N': V is not computed.
112 *
113 * JOBQ (input) CHARACTER*1
114 * = 'Q': Unitary matrix Q is computed;
115 * = 'N': Q is not computed.
116 *
117 * M (input) INTEGER
118 * The number of rows of the matrix A. M >= 0.
119 *
120 * N (input) INTEGER
121 * The number of columns of the matrices A and B. N >= 0.
122 *
123 * P (input) INTEGER
124 * The number of rows of the matrix B. P >= 0.
125 *
126 * K (output) INTEGER
127 * L (output) INTEGER
128 * On exit, K and L specify the dimension of the subblocks
129 * described in Purpose.
130 * K + L = effective numerical rank of (A**H,B**H)**H.
131 *
132 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
133 * On entry, the M-by-N matrix A.
134 * On exit, A contains the triangular matrix R, or part of R.
135 * See Purpose for details.
136 *
137 * LDA (input) INTEGER
138 * The leading dimension of the array A. LDA >= max(1,M).
139 *
140 * B (input/output) COMPLEX*16 array, dimension (LDB,N)
141 * On entry, the P-by-N matrix B.
142 * On exit, B contains part of the triangular matrix R if
143 * M-K-L < 0. See Purpose for details.
144 *
145 * LDB (input) INTEGER
146 * The leading dimension of the array B. LDB >= max(1,P).
147 *
148 * ALPHA (output) DOUBLE PRECISION array, dimension (N)
149 * BETA (output) DOUBLE PRECISION array, dimension (N)
150 * On exit, ALPHA and BETA contain the generalized singular
151 * value pairs of A and B;
152 * ALPHA(1:K) = 1,
153 * BETA(1:K) = 0,
154 * and if M-K-L >= 0,
155 * ALPHA(K+1:K+L) = C,
156 * BETA(K+1:K+L) = S,
157 * or if M-K-L < 0,
158 * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
159 * BETA(K+1:M) = S, BETA(M+1:K+L) = 1
160 * and
161 * ALPHA(K+L+1:N) = 0
162 * BETA(K+L+1:N) = 0
163 *
164 * U (output) COMPLEX*16 array, dimension (LDU,M)
165 * If JOBU = 'U', U contains the M-by-M unitary matrix U.
166 * If JOBU = 'N', U is not referenced.
167 *
168 * LDU (input) INTEGER
169 * The leading dimension of the array U. LDU >= max(1,M) if
170 * JOBU = 'U'; LDU >= 1 otherwise.
171 *
172 * V (output) COMPLEX*16 array, dimension (LDV,P)
173 * If JOBV = 'V', V contains the P-by-P unitary matrix V.
174 * If JOBV = 'N', V is not referenced.
175 *
176 * LDV (input) INTEGER
177 * The leading dimension of the array V. LDV >= max(1,P) if
178 * JOBV = 'V'; LDV >= 1 otherwise.
179 *
180 * Q (output) COMPLEX*16 array, dimension (LDQ,N)
181 * If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
182 * If JOBQ = 'N', Q is not referenced.
183 *
184 * LDQ (input) INTEGER
185 * The leading dimension of the array Q. LDQ >= max(1,N) if
186 * JOBQ = 'Q'; LDQ >= 1 otherwise.
187 *
188 * WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)
189 *
190 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
191 *
192 * IWORK (workspace/output) INTEGER array, dimension (N)
193 * On exit, IWORK stores the sorting information. More
194 * precisely, the following loop will sort ALPHA
195 * for I = K+1, min(M,K+L)
196 * swap ALPHA(I) and ALPHA(IWORK(I))
197 * endfor
198 * such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
199 *
200 * INFO (output) INTEGER
201 * = 0: successful exit.
202 * < 0: if INFO = -i, the i-th argument had an illegal value.
203 * > 0: if INFO = 1, the Jacobi-type procedure failed to
204 * converge. For further details, see subroutine ZTGSJA.
205 *
206 * Internal Parameters
207 * ===================
208 *
209 * TOLA DOUBLE PRECISION
210 * TOLB DOUBLE PRECISION
211 * TOLA and TOLB are the thresholds to determine the effective
212 * rank of (A**H,B**H)**H. Generally, they are set to
213 * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
214 * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
215 * The size of TOLA and TOLB may affect the size of backward
216 * errors of the decomposition.
217 *
218 * Further Details
219 * ===============
220 *
221 * 2-96 Based on modifications by
222 * Ming Gu and Huan Ren, Computer Science Division, University of
223 * California at Berkeley, USA
224 *
225 * =====================================================================
226 *
227 * .. Local Scalars ..
228 LOGICAL WANTQ, WANTU, WANTV
229 INTEGER I, IBND, ISUB, J, NCYCLE
230 DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
231 * ..
232 * .. External Functions ..
233 LOGICAL LSAME
234 DOUBLE PRECISION DLAMCH, ZLANGE
235 EXTERNAL LSAME, DLAMCH, ZLANGE
236 * ..
237 * .. External Subroutines ..
238 EXTERNAL DCOPY, XERBLA, ZGGSVP, ZTGSJA
239 * ..
240 * .. Intrinsic Functions ..
241 INTRINSIC MAX, MIN
242 * ..
243 * .. Executable Statements ..
244 *
245 * Decode and test the input parameters
246 *
247 WANTU = LSAME( JOBU, 'U' )
248 WANTV = LSAME( JOBV, 'V' )
249 WANTQ = LSAME( JOBQ, 'Q' )
250 *
251 INFO = 0
252 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
253 INFO = -1
254 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
255 INFO = -2
256 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
257 INFO = -3
258 ELSE IF( M.LT.0 ) THEN
259 INFO = -4
260 ELSE IF( N.LT.0 ) THEN
261 INFO = -5
262 ELSE IF( P.LT.0 ) THEN
263 INFO = -6
264 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
265 INFO = -10
266 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
267 INFO = -12
268 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
269 INFO = -16
270 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
271 INFO = -18
272 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
273 INFO = -20
274 END IF
275 IF( INFO.NE.0 ) THEN
276 CALL XERBLA( 'ZGGSVD', -INFO )
277 RETURN
278 END IF
279 *
280 * Compute the Frobenius norm of matrices A and B
281 *
282 ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
283 BNORM = ZLANGE( '1', P, N, B, LDB, RWORK )
284 *
285 * Get machine precision and set up threshold for determining
286 * the effective numerical rank of the matrices A and B.
287 *
288 ULP = DLAMCH( 'Precision' )
289 UNFL = DLAMCH( 'Safe Minimum' )
290 TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
291 TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
292 *
293 CALL ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
294 $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
295 $ WORK, WORK( N+1 ), INFO )
296 *
297 * Compute the GSVD of two upper "triangular" matrices
298 *
299 CALL ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
300 $ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
301 $ WORK, NCYCLE, INFO )
302 *
303 * Sort the singular values and store the pivot indices in IWORK
304 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
305 *
306 CALL DCOPY( N, ALPHA, 1, RWORK, 1 )
307 IBND = MIN( L, M-K )
308 DO 20 I = 1, IBND
309 *
310 * Scan for largest ALPHA(K+I)
311 *
312 ISUB = I
313 SMAX = RWORK( K+I )
314 DO 10 J = I + 1, IBND
315 TEMP = RWORK( K+J )
316 IF( TEMP.GT.SMAX ) THEN
317 ISUB = J
318 SMAX = TEMP
319 END IF
320 10 CONTINUE
321 IF( ISUB.NE.I ) THEN
322 RWORK( K+ISUB ) = RWORK( K+I )
323 RWORK( K+I ) = SMAX
324 IWORK( K+I ) = K + ISUB
325 ELSE
326 IWORK( K+I ) = K + I
327 END IF
328 20 CONTINUE
329 *
330 RETURN
331 *
332 * End of ZGGSVD
333 *
334 END