00001 SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, 00002 $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, 00003 $ IWORK, INFO ) 00004 * 00005 * -- LAPACK driver routine (version 3.2) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * November 2006 00009 * 00010 * .. Scalar Arguments .. 00011 CHARACTER JOBQ, JOBU, JOBV 00012 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P 00013 * .. 00014 * .. Array Arguments .. 00015 INTEGER IWORK( * ) 00016 DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), 00017 $ BETA( * ), Q( LDQ, * ), U( LDU, * ), 00018 $ V( LDV, * ), WORK( * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * DGGSVD computes the generalized singular value decomposition (GSVD) 00025 * of an M-by-N real matrix A and P-by-N real matrix B: 00026 * 00027 * U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) 00028 * 00029 * where U, V and Q are orthogonal matrices, and Z' is the transpose 00030 * of Z. Let K+L = the effective numerical rank of the matrix (A',B')', 00031 * then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and 00032 * D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the 00033 * following structures, respectively: 00034 * 00035 * If M-K-L >= 0, 00036 * 00037 * K L 00038 * D1 = K ( I 0 ) 00039 * L ( 0 C ) 00040 * M-K-L ( 0 0 ) 00041 * 00042 * K L 00043 * D2 = L ( 0 S ) 00044 * P-L ( 0 0 ) 00045 * 00046 * N-K-L K L 00047 * ( 0 R ) = K ( 0 R11 R12 ) 00048 * L ( 0 0 R22 ) 00049 * 00050 * where 00051 * 00052 * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), 00053 * S = diag( BETA(K+1), ... , BETA(K+L) ), 00054 * C**2 + S**2 = I. 00055 * 00056 * R is stored in A(1:K+L,N-K-L+1:N) on exit. 00057 * 00058 * If M-K-L < 0, 00059 * 00060 * K M-K K+L-M 00061 * D1 = K ( I 0 0 ) 00062 * M-K ( 0 C 0 ) 00063 * 00064 * K M-K K+L-M 00065 * D2 = M-K ( 0 S 0 ) 00066 * K+L-M ( 0 0 I ) 00067 * P-L ( 0 0 0 ) 00068 * 00069 * N-K-L K M-K K+L-M 00070 * ( 0 R ) = K ( 0 R11 R12 R13 ) 00071 * M-K ( 0 0 R22 R23 ) 00072 * K+L-M ( 0 0 0 R33 ) 00073 * 00074 * where 00075 * 00076 * C = diag( ALPHA(K+1), ... , ALPHA(M) ), 00077 * S = diag( BETA(K+1), ... , BETA(M) ), 00078 * C**2 + S**2 = I. 00079 * 00080 * (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored 00081 * ( 0 R22 R23 ) 00082 * in B(M-K+1:L,N+M-K-L+1:N) on exit. 00083 * 00084 * The routine computes C, S, R, and optionally the orthogonal 00085 * transformation matrices U, V and Q. 00086 * 00087 * In particular, if B is an N-by-N nonsingular matrix, then the GSVD of 00088 * A and B implicitly gives the SVD of A*inv(B): 00089 * A*inv(B) = U*(D1*inv(D2))*V'. 00090 * If ( A',B')' has orthonormal columns, then the GSVD of A and B is 00091 * also equal to the CS decomposition of A and B. Furthermore, the GSVD 00092 * can be used to derive the solution of the eigenvalue problem: 00093 * A'*A x = lambda* B'*B x. 00094 * In some literature, the GSVD of A and B is presented in the form 00095 * U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) 00096 * where U and V are orthogonal and X is nonsingular, D1 and D2 are 00097 * ``diagonal''. The former GSVD form can be converted to the latter 00098 * form by taking the nonsingular matrix X as 00099 * 00100 * X = Q*( I 0 ) 00101 * ( 0 inv(R) ). 00102 * 00103 * Arguments 00104 * ========= 00105 * 00106 * JOBU (input) CHARACTER*1 00107 * = 'U': Orthogonal matrix U is computed; 00108 * = 'N': U is not computed. 00109 * 00110 * JOBV (input) CHARACTER*1 00111 * = 'V': Orthogonal matrix V is computed; 00112 * = 'N': V is not computed. 00113 * 00114 * JOBQ (input) CHARACTER*1 00115 * = 'Q': Orthogonal matrix Q is computed; 00116 * = 'N': Q is not computed. 00117 * 00118 * M (input) INTEGER 00119 * The number of rows of the matrix A. M >= 0. 00120 * 00121 * N (input) INTEGER 00122 * The number of columns of the matrices A and B. N >= 0. 00123 * 00124 * P (input) INTEGER 00125 * The number of rows of the matrix B. P >= 0. 00126 * 00127 * K (output) INTEGER 00128 * L (output) INTEGER 00129 * On exit, K and L specify the dimension of the subblocks 00130 * described in the Purpose section. 00131 * K + L = effective numerical rank of (A',B')'. 00132 * 00133 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 00134 * On entry, the M-by-N matrix A. 00135 * On exit, A contains the triangular matrix R, or part of R. 00136 * See Purpose for details. 00137 * 00138 * LDA (input) INTEGER 00139 * The leading dimension of the array A. LDA >= max(1,M). 00140 * 00141 * B (input/output) DOUBLE PRECISION array, dimension (LDB,N) 00142 * On entry, the P-by-N matrix B. 00143 * On exit, B contains the triangular matrix R if M-K-L < 0. 00144 * See Purpose for details. 00145 * 00146 * LDB (input) INTEGER 00147 * The leading dimension of the array B. LDB >= max(1,P). 00148 * 00149 * ALPHA (output) DOUBLE PRECISION array, dimension (N) 00150 * BETA (output) DOUBLE PRECISION array, dimension (N) 00151 * On exit, ALPHA and BETA contain the generalized singular 00152 * value pairs of A and B; 00153 * ALPHA(1:K) = 1, 00154 * BETA(1:K) = 0, 00155 * and if M-K-L >= 0, 00156 * ALPHA(K+1:K+L) = C, 00157 * BETA(K+1:K+L) = S, 00158 * or if M-K-L < 0, 00159 * ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 00160 * BETA(K+1:M) =S, BETA(M+1:K+L) =1 00161 * and 00162 * ALPHA(K+L+1:N) = 0 00163 * BETA(K+L+1:N) = 0 00164 * 00165 * U (output) DOUBLE PRECISION array, dimension (LDU,M) 00166 * If JOBU = 'U', U contains the M-by-M orthogonal matrix U. 00167 * If JOBU = 'N', U is not referenced. 00168 * 00169 * LDU (input) INTEGER 00170 * The leading dimension of the array U. LDU >= max(1,M) if 00171 * JOBU = 'U'; LDU >= 1 otherwise. 00172 * 00173 * V (output) DOUBLE PRECISION array, dimension (LDV,P) 00174 * If JOBV = 'V', V contains the P-by-P orthogonal matrix V. 00175 * If JOBV = 'N', V is not referenced. 00176 * 00177 * LDV (input) INTEGER 00178 * The leading dimension of the array V. LDV >= max(1,P) if 00179 * JOBV = 'V'; LDV >= 1 otherwise. 00180 * 00181 * Q (output) DOUBLE PRECISION array, dimension (LDQ,N) 00182 * If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. 00183 * If JOBQ = 'N', Q is not referenced. 00184 * 00185 * LDQ (input) INTEGER 00186 * The leading dimension of the array Q. LDQ >= max(1,N) if 00187 * JOBQ = 'Q'; LDQ >= 1 otherwise. 00188 * 00189 * WORK (workspace) DOUBLE PRECISION array, 00190 * dimension (max(3*N,M,P)+N) 00191 * 00192 * IWORK (workspace/output) INTEGER array, dimension (N) 00193 * On exit, IWORK stores the sorting information. More 00194 * precisely, the following loop will sort ALPHA 00195 * for I = K+1, min(M,K+L) 00196 * swap ALPHA(I) and ALPHA(IWORK(I)) 00197 * endfor 00198 * such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). 00199 * 00200 * INFO (output) INTEGER 00201 * = 0: successful exit 00202 * < 0: if INFO = -i, the i-th argument had an illegal value. 00203 * > 0: if INFO = 1, the Jacobi-type procedure failed to 00204 * converge. For further details, see subroutine DTGSJA. 00205 * 00206 * Internal Parameters 00207 * =================== 00208 * 00209 * TOLA DOUBLE PRECISION 00210 * TOLB DOUBLE PRECISION 00211 * TOLA and TOLB are the thresholds to determine the effective 00212 * rank of (A',B')'. Generally, they are set to 00213 * TOLA = MAX(M,N)*norm(A)*MAZHEPS, 00214 * TOLB = MAX(P,N)*norm(B)*MAZHEPS. 00215 * The size of TOLA and TOLB may affect the size of backward 00216 * errors of the decomposition. 00217 * 00218 * Further Details 00219 * =============== 00220 * 00221 * 2-96 Based on modifications by 00222 * Ming Gu and Huan Ren, Computer Science Division, University of 00223 * California at Berkeley, USA 00224 * 00225 * ===================================================================== 00226 * 00227 * .. Local Scalars .. 00228 LOGICAL WANTQ, WANTU, WANTV 00229 INTEGER I, IBND, ISUB, J, NCYCLE 00230 DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL 00231 * .. 00232 * .. External Functions .. 00233 LOGICAL LSAME 00234 DOUBLE PRECISION DLAMCH, DLANGE 00235 EXTERNAL LSAME, DLAMCH, DLANGE 00236 * .. 00237 * .. External Subroutines .. 00238 EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA 00239 * .. 00240 * .. Intrinsic Functions .. 00241 INTRINSIC MAX, MIN 00242 * .. 00243 * .. Executable Statements .. 00244 * 00245 * Test the input parameters 00246 * 00247 WANTU = LSAME( JOBU, 'U' ) 00248 WANTV = LSAME( JOBV, 'V' ) 00249 WANTQ = LSAME( JOBQ, 'Q' ) 00250 * 00251 INFO = 0 00252 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN 00253 INFO = -1 00254 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN 00255 INFO = -2 00256 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN 00257 INFO = -3 00258 ELSE IF( M.LT.0 ) THEN 00259 INFO = -4 00260 ELSE IF( N.LT.0 ) THEN 00261 INFO = -5 00262 ELSE IF( P.LT.0 ) THEN 00263 INFO = -6 00264 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00265 INFO = -10 00266 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN 00267 INFO = -12 00268 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN 00269 INFO = -16 00270 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN 00271 INFO = -18 00272 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN 00273 INFO = -20 00274 END IF 00275 IF( INFO.NE.0 ) THEN 00276 CALL XERBLA( 'DGGSVD', -INFO ) 00277 RETURN 00278 END IF 00279 * 00280 * Compute the Frobenius norm of matrices A and B 00281 * 00282 ANORM = DLANGE( '1', M, N, A, LDA, WORK ) 00283 BNORM = DLANGE( '1', P, N, B, LDB, WORK ) 00284 * 00285 * Get machine precision and set up threshold for determining 00286 * the effective numerical rank of the matrices A and B. 00287 * 00288 ULP = DLAMCH( 'Precision' ) 00289 UNFL = DLAMCH( 'Safe Minimum' ) 00290 TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP 00291 TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP 00292 * 00293 * Preprocessing 00294 * 00295 CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, 00296 $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK, 00297 $ WORK( N+1 ), INFO ) 00298 * 00299 * Compute the GSVD of two upper "triangular" matrices 00300 * 00301 CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, 00302 $ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, 00303 $ WORK, NCYCLE, INFO ) 00304 * 00305 * Sort the singular values and store the pivot indices in IWORK 00306 * Copy ALPHA to WORK, then sort ALPHA in WORK 00307 * 00308 CALL DCOPY( N, ALPHA, 1, WORK, 1 ) 00309 IBND = MIN( L, M-K ) 00310 DO 20 I = 1, IBND 00311 * 00312 * Scan for largest ALPHA(K+I) 00313 * 00314 ISUB = I 00315 SMAX = WORK( K+I ) 00316 DO 10 J = I + 1, IBND 00317 TEMP = WORK( K+J ) 00318 IF( TEMP.GT.SMAX ) THEN 00319 ISUB = J 00320 SMAX = TEMP 00321 END IF 00322 10 CONTINUE 00323 IF( ISUB.NE.I ) THEN 00324 WORK( K+ISUB ) = WORK( K+I ) 00325 WORK( K+I ) = SMAX 00326 IWORK( K+I ) = K + ISUB 00327 ELSE 00328 IWORK( K+I ) = K + I 00329 END IF 00330 20 CONTINUE 00331 * 00332 RETURN 00333 * 00334 * End of DGGSVD 00335 * 00336 END
1.7.2