Annotation of rpl/lapack/lapack/dggsvp.f, revision 1.5
1.1 bertrand 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 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
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'*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'*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',B')'. Z' denotes the
43: * transpose of Z.
44: *
45: * This decomposition is the preprocessing step for computing the
46: * Generalized Singular Value Decomposition (GSVD), see subroutine
47: * DGGSVD.
48: *
49: * Arguments
50: * =========
51: *
52: * JOBU (input) CHARACTER*1
53: * = 'U': Orthogonal matrix U is computed;
54: * = 'N': U is not computed.
55: *
56: * JOBV (input) CHARACTER*1
57: * = 'V': Orthogonal matrix V is computed;
58: * = 'N': V is not computed.
59: *
60: * JOBQ (input) CHARACTER*1
61: * = 'Q': Orthogonal matrix Q is computed;
62: * = 'N': Q is not computed.
63: *
64: * M (input) INTEGER
65: * The number of rows of the matrix A. M >= 0.
66: *
67: * P (input) INTEGER
68: * The number of rows of the matrix B. P >= 0.
69: *
70: * N (input) INTEGER
71: * The number of columns of the matrices A and B. N >= 0.
72: *
73: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
74: * On entry, the M-by-N matrix A.
75: * On exit, A contains the triangular (or trapezoidal) matrix
76: * described in the Purpose section.
77: *
78: * LDA (input) INTEGER
79: * The leading dimension of the array A. LDA >= max(1,M).
80: *
81: * B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
82: * On entry, the P-by-N matrix B.
83: * On exit, B contains the triangular matrix described in
84: * the Purpose section.
85: *
86: * LDB (input) INTEGER
87: * The leading dimension of the array B. LDB >= max(1,P).
88: *
89: * TOLA (input) DOUBLE PRECISION
90: * TOLB (input) DOUBLE PRECISION
91: * TOLA and TOLB are the thresholds to determine the effective
92: * numerical rank of matrix B and a subblock of A. Generally,
93: * they are set to
94: * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
95: * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
96: * The size of TOLA and TOLB may affect the size of backward
97: * errors of the decomposition.
98: *
99: * K (output) INTEGER
100: * L (output) INTEGER
101: * On exit, K and L specify the dimension of the subblocks
102: * described in Purpose.
103: * K + L = effective numerical rank of (A',B')'.
104: *
105: * U (output) DOUBLE PRECISION array, dimension (LDU,M)
106: * If JOBU = 'U', U contains the orthogonal matrix U.
107: * If JOBU = 'N', U is not referenced.
108: *
109: * LDU (input) INTEGER
110: * The leading dimension of the array U. LDU >= max(1,M) if
111: * JOBU = 'U'; LDU >= 1 otherwise.
112: *
113: * V (output) DOUBLE PRECISION array, dimension (LDV,P)
114: * If JOBV = 'V', V contains the orthogonal matrix V.
115: * If JOBV = 'N', V is not referenced.
116: *
117: * LDV (input) INTEGER
118: * The leading dimension of the array V. LDV >= max(1,P) if
119: * JOBV = 'V'; LDV >= 1 otherwise.
120: *
121: * Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
122: * If JOBQ = 'Q', Q contains the orthogonal matrix Q.
123: * If JOBQ = 'N', Q is not referenced.
124: *
125: * LDQ (input) INTEGER
126: * The leading dimension of the array Q. LDQ >= max(1,N) if
127: * JOBQ = 'Q'; LDQ >= 1 otherwise.
128: *
129: * IWORK (workspace) INTEGER array, dimension (N)
130: *
131: * TAU (workspace) DOUBLE PRECISION array, dimension (N)
132: *
133: * WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P))
134: *
135: * INFO (output) INTEGER
136: * = 0: successful exit
137: * < 0: if INFO = -i, the i-th argument had an illegal value.
138: *
139: *
140: * Further Details
141: * ===============
142: *
143: * The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
144: * with column pivoting to detect the effective numerical rank of the
145: * a matrix. It may be replaced by a better rank determination strategy.
146: *
147: * =====================================================================
148: *
149: * .. Parameters ..
150: DOUBLE PRECISION ZERO, ONE
151: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
152: * ..
153: * .. Local Scalars ..
154: LOGICAL FORWRD, WANTQ, WANTU, WANTV
155: INTEGER I, J
156: * ..
157: * .. External Functions ..
158: LOGICAL LSAME
159: EXTERNAL LSAME
160: * ..
161: * .. External Subroutines ..
162: EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
163: $ DORG2R, DORM2R, DORMR2, XERBLA
164: * ..
165: * .. Intrinsic Functions ..
166: INTRINSIC ABS, MAX, MIN
167: * ..
168: * .. Executable Statements ..
169: *
170: * Test the input parameters
171: *
172: WANTU = LSAME( JOBU, 'U' )
173: WANTV = LSAME( JOBV, 'V' )
174: WANTQ = LSAME( JOBQ, 'Q' )
175: FORWRD = .TRUE.
176: *
177: INFO = 0
178: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
179: INFO = -1
180: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
181: INFO = -2
182: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
183: INFO = -3
184: ELSE IF( M.LT.0 ) THEN
185: INFO = -4
186: ELSE IF( P.LT.0 ) THEN
187: INFO = -5
188: ELSE IF( N.LT.0 ) THEN
189: INFO = -6
190: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
191: INFO = -8
192: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
193: INFO = -10
194: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
195: INFO = -16
196: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
197: INFO = -18
198: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
199: INFO = -20
200: END IF
201: IF( INFO.NE.0 ) THEN
202: CALL XERBLA( 'DGGSVP', -INFO )
203: RETURN
204: END IF
205: *
206: * QR with column pivoting of B: B*P = V*( S11 S12 )
207: * ( 0 0 )
208: *
209: DO 10 I = 1, N
210: IWORK( I ) = 0
211: 10 CONTINUE
212: CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
213: *
214: * Update A := A*P
215: *
216: CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
217: *
218: * Determine the effective rank of matrix B.
219: *
220: L = 0
221: DO 20 I = 1, MIN( P, N )
222: IF( ABS( B( I, I ) ).GT.TOLB )
223: $ L = L + 1
224: 20 CONTINUE
225: *
226: IF( WANTV ) THEN
227: *
228: * Copy the details of V, and form V.
229: *
230: CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
231: IF( P.GT.1 )
232: $ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
233: $ LDV )
234: CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
235: END IF
236: *
237: * Clean up B
238: *
239: DO 40 J = 1, L - 1
240: DO 30 I = J + 1, L
241: B( I, J ) = ZERO
242: 30 CONTINUE
243: 40 CONTINUE
244: IF( P.GT.L )
245: $ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
246: *
247: IF( WANTQ ) THEN
248: *
249: * Set Q = I and Update Q := Q*P
250: *
251: CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
252: CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
253: END IF
254: *
255: IF( P.GE.L .AND. N.NE.L ) THEN
256: *
257: * RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
258: *
259: CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
260: *
261: * Update A := A*Z'
262: *
263: CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
264: $ LDA, WORK, INFO )
265: *
266: IF( WANTQ ) THEN
267: *
268: * Update Q := Q*Z'
269: *
270: CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
271: $ LDQ, WORK, INFO )
272: END IF
273: *
274: * Clean up B
275: *
276: CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
277: DO 60 J = N - L + 1, N
278: DO 50 I = J - N + L + 1, L
279: B( I, J ) = ZERO
280: 50 CONTINUE
281: 60 CONTINUE
282: *
283: END IF
284: *
285: * Let N-L L
286: * A = ( A11 A12 ) M,
287: *
288: * then the following does the complete QR decomposition of A11:
289: *
290: * A11 = U*( 0 T12 )*P1'
291: * ( 0 0 )
292: *
293: DO 70 I = 1, N - L
294: IWORK( I ) = 0
295: 70 CONTINUE
296: CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
297: *
298: * Determine the effective rank of A11
299: *
300: K = 0
301: DO 80 I = 1, MIN( M, N-L )
302: IF( ABS( A( I, I ) ).GT.TOLA )
303: $ K = K + 1
304: 80 CONTINUE
305: *
306: * Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N )
307: *
308: CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
309: $ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
310: *
311: IF( WANTU ) THEN
312: *
313: * Copy the details of U, and form U
314: *
315: CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
316: IF( M.GT.1 )
317: $ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
318: $ LDU )
319: CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
320: END IF
321: *
322: IF( WANTQ ) THEN
323: *
324: * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
325: *
326: CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
327: END IF
328: *
329: * Clean up A: set the strictly lower triangular part of
330: * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
331: *
332: DO 100 J = 1, K - 1
333: DO 90 I = J + 1, K
334: A( I, J ) = ZERO
335: 90 CONTINUE
336: 100 CONTINUE
337: IF( M.GT.K )
338: $ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
339: *
340: IF( N-L.GT.K ) THEN
341: *
342: * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
343: *
344: CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
345: *
346: IF( WANTQ ) THEN
347: *
348: * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1'
349: *
350: CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
351: $ Q, LDQ, WORK, INFO )
352: END IF
353: *
354: * Clean up A
355: *
356: CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
357: DO 120 J = N - L - K + 1, N - L
358: DO 110 I = J - N + L + K + 1, K
359: A( I, J ) = ZERO
360: 110 CONTINUE
361: 120 CONTINUE
362: *
363: END IF
364: *
365: IF( M.GT.K ) THEN
366: *
367: * QR factorization of A( K+1:M,N-L+1:N )
368: *
369: CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
370: *
371: IF( WANTU ) THEN
372: *
373: * Update U(:,K+1:M) := U(:,K+1:M)*U1
374: *
375: CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
376: $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
377: $ WORK, INFO )
378: END IF
379: *
380: * Clean up
381: *
382: DO 140 J = N - L + 1, N
383: DO 130 I = J - N + K + L + 1, M
384: A( I, J ) = ZERO
385: 130 CONTINUE
386: 140 CONTINUE
387: *
388: END IF
389: *
390: RETURN
391: *
392: * End of DGGSVP
393: *
394: END
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