001:       SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
002:      $                   LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
003:      $                   RWORK, IWORK, INFO )
004: *
005: *  -- LAPACK driver routine (version 3.2) --
006: *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
007: *     November 2006
008: *
009: *     .. Scalar Arguments ..
010:       CHARACTER          JOBQ, JOBU, JOBV
011:       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
012: *     ..
013: *     .. Array Arguments ..
014:       INTEGER            IWORK( * )
015:       DOUBLE PRECISION   ALPHA( * ), BETA( * ), RWORK( * )
016:       COMPLEX*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
017:      $                   U( LDU, * ), V( LDV, * ), WORK( * )
018: *     ..
019: *
020: *  Purpose
021: *  =======
022: *
023: *  ZGGSVD computes the generalized singular value decomposition (GSVD)
024: *  of an M-by-N complex matrix A and P-by-N complex matrix B:
025: *
026: *        U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )
027: *
028: *  where U, V and Q are unitary matrices, and Z' means the conjugate
029: *  transpose of Z.  Let K+L = the effective numerical rank of the
030: *  matrix (A',B')', then R is a (K+L)-by-(K+L) nonsingular upper
031: *  triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
032: *  matrices and of the following structures, respectively:
033: *
034: *  If M-K-L >= 0,
035: *
036: *                      K  L
037: *         D1 =     K ( I  0 )
038: *                  L ( 0  C )
039: *              M-K-L ( 0  0 )
040: *
041: *                    K  L
042: *         D2 =   L ( 0  S )
043: *              P-L ( 0  0 )
044: *
045: *                  N-K-L  K    L
046: *    ( 0 R ) = K (  0   R11  R12 )
047: *              L (  0    0   R22 )
048: *  where
049: *
050: *    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
051: *    S = diag( BETA(K+1),  ... , BETA(K+L) ),
052: *    C**2 + S**2 = I.
053: *
054: *    R is stored in A(1:K+L,N-K-L+1:N) on exit.
055: *
056: *  If M-K-L < 0,
057: *
058: *                    K M-K K+L-M
059: *         D1 =   K ( I  0    0   )
060: *              M-K ( 0  C    0   )
061: *
062: *                      K M-K K+L-M
063: *         D2 =   M-K ( 0  S    0  )
064: *              K+L-M ( 0  0    I  )
065: *                P-L ( 0  0    0  )
066: *
067: *                     N-K-L  K   M-K  K+L-M
068: *    ( 0 R ) =     K ( 0    R11  R12  R13  )
069: *                M-K ( 0     0   R22  R23  )
070: *              K+L-M ( 0     0    0   R33  )
071: *
072: *  where
073: *
074: *    C = diag( ALPHA(K+1), ... , ALPHA(M) ),
075: *    S = diag( BETA(K+1),  ... , BETA(M) ),
076: *    C**2 + S**2 = I.
077: *
078: *    (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
079: *    ( 0  R22 R23 )
080: *    in B(M-K+1:L,N+M-K-L+1:N) on exit.
081: *
082: *  The routine computes C, S, R, and optionally the unitary
083: *  transformation matrices U, V and Q.
084: *
085: *  In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
086: *  A and B implicitly gives the SVD of A*inv(B):
087: *                       A*inv(B) = U*(D1*inv(D2))*V'.
088: *  If ( A',B')' has orthnormal columns, then the GSVD of A and B is also
089: *  equal to the CS decomposition of A and B. Furthermore, the GSVD can
090: *  be used to derive the solution of the eigenvalue problem:
091: *                       A'*A x = lambda* B'*B x.
092: *  In some literature, the GSVD of A and B is presented in the form
093: *                   U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
094: *  where U and V are orthogonal and X is nonsingular, and D1 and D2 are
095: *  ``diagonal''.  The former GSVD form can be converted to the latter
096: *  form by taking the nonsingular matrix X as
097: *
098: *                        X = Q*(  I   0    )
099: *                              (  0 inv(R) )
100: *
101: *  Arguments
102: *  =========
103: *
104: *  JOBU    (input) CHARACTER*1
105: *          = 'U':  Unitary matrix U is computed;
106: *          = 'N':  U is not computed.
107: *
108: *  JOBV    (input) CHARACTER*1
109: *          = 'V':  Unitary matrix V is computed;
110: *          = 'N':  V is not computed.
111: *
112: *  JOBQ    (input) CHARACTER*1
113: *          = 'Q':  Unitary matrix Q is computed;
114: *          = 'N':  Q is not computed.
115: *
116: *  M       (input) INTEGER
117: *          The number of rows of the matrix A.  M >= 0.
118: *
119: *  N       (input) INTEGER
120: *          The number of columns of the matrices A and B.  N >= 0.
121: *
122: *  P       (input) INTEGER
123: *          The number of rows of the matrix B.  P >= 0.
124: *
125: *  K       (output) INTEGER
126: *  L       (output) INTEGER
127: *          On exit, K and L specify the dimension of the subblocks
128: *          described in Purpose.
129: *          K + L = effective numerical rank of (A',B')'.
130: *
131: *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
132: *          On entry, the M-by-N matrix A.
133: *          On exit, A contains the triangular matrix R, or part of R.
134: *          See Purpose for details.
135: *
136: *  LDA     (input) INTEGER
137: *          The leading dimension of the array A. LDA >= max(1,M).
138: *
139: *  B       (input/output) COMPLEX*16 array, dimension (LDB,N)
140: *          On entry, the P-by-N matrix B.
141: *          On exit, B contains part of the triangular matrix R if
142: *          M-K-L < 0.  See Purpose for details.
143: *
144: *  LDB     (input) INTEGER
145: *          The leading dimension of the array B. LDB >= max(1,P).
146: *
147: *  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
148: *  BETA    (output) DOUBLE PRECISION array, dimension (N)
149: *          On exit, ALPHA and BETA contain the generalized singular
150: *          value pairs of A and B;
151: *            ALPHA(1:K) = 1,
152: *            BETA(1:K)  = 0,
153: *          and if M-K-L >= 0,
154: *            ALPHA(K+1:K+L) = C,
155: *            BETA(K+1:K+L)  = S,
156: *          or if M-K-L < 0,
157: *            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
158: *            BETA(K+1:M) = S, BETA(M+1:K+L) = 1
159: *          and
160: *            ALPHA(K+L+1:N) = 0
161: *            BETA(K+L+1:N)  = 0
162: *
163: *  U       (output) COMPLEX*16 array, dimension (LDU,M)
164: *          If JOBU = 'U', U contains the M-by-M unitary matrix U.
165: *          If JOBU = 'N', U is not referenced.
166: *
167: *  LDU     (input) INTEGER
168: *          The leading dimension of the array U. LDU >= max(1,M) if
169: *          JOBU = 'U'; LDU >= 1 otherwise.
170: *
171: *  V       (output) COMPLEX*16 array, dimension (LDV,P)
172: *          If JOBV = 'V', V contains the P-by-P unitary matrix V.
173: *          If JOBV = 'N', V is not referenced.
174: *
175: *  LDV     (input) INTEGER
176: *          The leading dimension of the array V. LDV >= max(1,P) if
177: *          JOBV = 'V'; LDV >= 1 otherwise.
178: *
179: *  Q       (output) COMPLEX*16 array, dimension (LDQ,N)
180: *          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
181: *          If JOBQ = 'N', Q is not referenced.
182: *
183: *  LDQ     (input) INTEGER
184: *          The leading dimension of the array Q. LDQ >= max(1,N) if
185: *          JOBQ = 'Q'; LDQ >= 1 otherwise.
186: *
187: *  WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)
188: *
189: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
190: *
191: *  IWORK   (workspace/output) INTEGER array, dimension (N)
192: *          On exit, IWORK stores the sorting information. More
193: *          precisely, the following loop will sort ALPHA
194: *             for I = K+1, min(M,K+L)
195: *                 swap ALPHA(I) and ALPHA(IWORK(I))
196: *             endfor
197: *          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
198: *
199: *  INFO    (output) INTEGER
200: *          = 0:  successful exit.
201: *          < 0:  if INFO = -i, the i-th argument had an illegal value.
202: *          > 0:  if INFO = 1, the Jacobi-type procedure failed to
203: *                converge.  For further details, see subroutine ZTGSJA.
204: *
205: *  Internal Parameters
206: *  ===================
207: *
208: *  TOLA    DOUBLE PRECISION
209: *  TOLB    DOUBLE PRECISION
210: *          TOLA and TOLB are the thresholds to determine the effective
211: *          rank of (A',B')'. Generally, they are set to
212: *                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,
213: *                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
214: *          The size of TOLA and TOLB may affect the size of backward
215: *          errors of the decomposition.
216: *
217: *  Further Details
218: *  ===============
219: *
220: *  2-96 Based on modifications by
221: *     Ming Gu and Huan Ren, Computer Science Division, University of
222: *     California at Berkeley, USA
223: *
224: *  =====================================================================
225: *
226: *     .. Local Scalars ..
227:       LOGICAL            WANTQ, WANTU, WANTV
228:       INTEGER            I, IBND, ISUB, J, NCYCLE
229:       DOUBLE PRECISION   ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
230: *     ..
231: *     .. External Functions ..
232:       LOGICAL            LSAME
233:       DOUBLE PRECISION   DLAMCH, ZLANGE
234:       EXTERNAL           LSAME, DLAMCH, ZLANGE
235: *     ..
236: *     .. External Subroutines ..
237:       EXTERNAL           DCOPY, XERBLA, ZGGSVP, ZTGSJA
238: *     ..
239: *     .. Intrinsic Functions ..
240:       INTRINSIC          MAX, MIN
241: *     ..
242: *     .. Executable Statements ..
243: *
244: *     Decode and test the input parameters
245: *
246:       WANTU = LSAME( JOBU, 'U' )
247:       WANTV = LSAME( JOBV, 'V' )
248:       WANTQ = LSAME( JOBQ, 'Q' )
249: *
250:       INFO = 0
251:       IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
252:          INFO = -1
253:       ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
254:          INFO = -2
255:       ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
256:          INFO = -3
257:       ELSE IF( M.LT.0 ) THEN
258:          INFO = -4
259:       ELSE IF( N.LT.0 ) THEN
260:          INFO = -5
261:       ELSE IF( P.LT.0 ) THEN
262:          INFO = -6
263:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
264:          INFO = -10
265:       ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
266:          INFO = -12
267:       ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
268:          INFO = -16
269:       ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
270:          INFO = -18
271:       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
272:          INFO = -20
273:       END IF
274:       IF( INFO.NE.0 ) THEN
275:          CALL XERBLA( 'ZGGSVD', -INFO )
276:          RETURN
277:       END IF
278: *
279: *     Compute the Frobenius norm of matrices A and B
280: *
281:       ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
282:       BNORM = ZLANGE( '1', P, N, B, LDB, RWORK )
283: *
284: *     Get machine precision and set up threshold for determining
285: *     the effective numerical rank of the matrices A and B.
286: *
287:       ULP = DLAMCH( 'Precision' )
288:       UNFL = DLAMCH( 'Safe Minimum' )
289:       TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
290:       TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
291: *
292:       CALL ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
293:      $             TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
294:      $             WORK, WORK( N+1 ), INFO )
295: *
296: *     Compute the GSVD of two upper "triangular" matrices
297: *
298:       CALL ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
299:      $             TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
300:      $             WORK, NCYCLE, INFO )
301: *
302: *     Sort the singular values and store the pivot indices in IWORK
303: *     Copy ALPHA to RWORK, then sort ALPHA in RWORK
304: *
305:       CALL DCOPY( N, ALPHA, 1, RWORK, 1 )
306:       IBND = MIN( L, M-K )
307:       DO 20 I = 1, IBND
308: *
309: *        Scan for largest ALPHA(K+I)
310: *
311:          ISUB = I
312:          SMAX = RWORK( K+I )
313:          DO 10 J = I + 1, IBND
314:             TEMP = RWORK( K+J )
315:             IF( TEMP.GT.SMAX ) THEN
316:                ISUB = J
317:                SMAX = TEMP
318:             END IF
319:    10    CONTINUE
320:          IF( ISUB.NE.I ) THEN
321:             RWORK( K+ISUB ) = RWORK( K+I )
322:             RWORK( K+I ) = SMAX
323:             IWORK( K+I ) = K + ISUB
324:          ELSE
325:             IWORK( K+I ) = K + I
326:          END IF
327:    20 CONTINUE
328: *
329:       RETURN
330: *
331: *     End of ZGGSVD
332: *
333:       END
334: