00001 /* dggsvd.f -- translated by f2c (version 20061008). 00002 You must link the resulting object file with libf2c: 00003 on Microsoft Windows system, link with libf2c.lib; 00004 on Linux or Unix systems, link with .../path/to/libf2c.a -lm 00005 or, if you install libf2c.a in a standard place, with -lf2c -lm 00006 -- in that order, at the end of the command line, as in 00007 cc *.o -lf2c -lm 00008 Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., 00009 00010 http://www.netlib.org/f2c/libf2c.zip 00011 */ 00012 00013 #include "f2c.h" 00014 #include "blaswrap.h" 00015 00016 /* Table of constant values */ 00017 00018 static integer c__1 = 1; 00019 00020 /* Subroutine */ int dggsvd_(char *jobu, char *jobv, char *jobq, integer *m, 00021 integer *n, integer *p, integer *k, integer *l, doublereal *a, 00022 integer *lda, doublereal *b, integer *ldb, doublereal *alpha, 00023 doublereal *beta, doublereal *u, integer *ldu, doublereal *v, integer 00024 *ldv, doublereal *q, integer *ldq, doublereal *work, integer *iwork, 00025 integer *info) 00026 { 00027 /* System generated locals */ 00028 integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1, 00029 u_offset, v_dim1, v_offset, i__1, i__2; 00030 00031 /* Local variables */ 00032 integer i__, j; 00033 doublereal ulp; 00034 integer ibnd; 00035 doublereal tola; 00036 integer isub; 00037 doublereal tolb, unfl, temp, smax; 00038 extern logical lsame_(char *, char *); 00039 doublereal anorm, bnorm; 00040 extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 00041 doublereal *, integer *); 00042 logical wantq, wantu, wantv; 00043 extern doublereal dlamch_(char *), dlange_(char *, integer *, 00044 integer *, doublereal *, integer *, doublereal *); 00045 extern /* Subroutine */ int dtgsja_(char *, char *, char *, integer *, 00046 integer *, integer *, integer *, integer *, doublereal *, integer 00047 *, doublereal *, integer *, doublereal *, doublereal *, 00048 doublereal *, doublereal *, doublereal *, integer *, doublereal *, 00049 integer *, doublereal *, integer *, doublereal *, integer *, 00050 integer *); 00051 integer ncycle; 00052 extern /* Subroutine */ int xerbla_(char *, integer *), dggsvp_( 00053 char *, char *, char *, integer *, integer *, integer *, 00054 doublereal *, integer *, doublereal *, integer *, doublereal *, 00055 doublereal *, integer *, integer *, doublereal *, integer *, 00056 doublereal *, integer *, doublereal *, integer *, integer *, 00057 doublereal *, doublereal *, integer *); 00058 00059 00060 /* -- LAPACK driver routine (version 3.2) -- */ 00061 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00062 /* November 2006 */ 00063 00064 /* .. Scalar Arguments .. */ 00065 /* .. */ 00066 /* .. Array Arguments .. */ 00067 /* .. */ 00068 00069 /* Purpose */ 00070 /* ======= */ 00071 00072 /* DGGSVD computes the generalized singular value decomposition (GSVD) */ 00073 /* of an M-by-N real matrix A and P-by-N real matrix B: */ 00074 00075 /* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) */ 00076 00077 /* where U, V and Q are orthogonal matrices, and Z' is the transpose */ 00078 /* of Z. Let K+L = the effective numerical rank of the matrix (A',B')', */ 00079 /* then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and */ 00080 /* D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the */ 00081 /* following structures, respectively: */ 00082 00083 /* If M-K-L >= 0, */ 00084 00085 /* K L */ 00086 /* D1 = K ( I 0 ) */ 00087 /* L ( 0 C ) */ 00088 /* M-K-L ( 0 0 ) */ 00089 00090 /* K L */ 00091 /* D2 = L ( 0 S ) */ 00092 /* P-L ( 0 0 ) */ 00093 00094 /* N-K-L K L */ 00095 /* ( 0 R ) = K ( 0 R11 R12 ) */ 00096 /* L ( 0 0 R22 ) */ 00097 00098 /* where */ 00099 00100 /* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), */ 00101 /* S = diag( BETA(K+1), ... , BETA(K+L) ), */ 00102 /* C**2 + S**2 = I. */ 00103 00104 /* R is stored in A(1:K+L,N-K-L+1:N) on exit. */ 00105 00106 /* If M-K-L < 0, */ 00107 00108 /* K M-K K+L-M */ 00109 /* D1 = K ( I 0 0 ) */ 00110 /* M-K ( 0 C 0 ) */ 00111 00112 /* K M-K K+L-M */ 00113 /* D2 = M-K ( 0 S 0 ) */ 00114 /* K+L-M ( 0 0 I ) */ 00115 /* P-L ( 0 0 0 ) */ 00116 00117 /* N-K-L K M-K K+L-M */ 00118 /* ( 0 R ) = K ( 0 R11 R12 R13 ) */ 00119 /* M-K ( 0 0 R22 R23 ) */ 00120 /* K+L-M ( 0 0 0 R33 ) */ 00121 00122 /* where */ 00123 00124 /* C = diag( ALPHA(K+1), ... , ALPHA(M) ), */ 00125 /* S = diag( BETA(K+1), ... , BETA(M) ), */ 00126 /* C**2 + S**2 = I. */ 00127 00128 /* (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored */ 00129 /* ( 0 R22 R23 ) */ 00130 /* in B(M-K+1:L,N+M-K-L+1:N) on exit. */ 00131 00132 /* The routine computes C, S, R, and optionally the orthogonal */ 00133 /* transformation matrices U, V and Q. */ 00134 00135 /* In particular, if B is an N-by-N nonsingular matrix, then the GSVD of */ 00136 /* A and B implicitly gives the SVD of A*inv(B): */ 00137 /* A*inv(B) = U*(D1*inv(D2))*V'. */ 00138 /* If ( A',B')' has orthonormal columns, then the GSVD of A and B is */ 00139 /* also equal to the CS decomposition of A and B. Furthermore, the GSVD */ 00140 /* can be used to derive the solution of the eigenvalue problem: */ 00141 /* A'*A x = lambda* B'*B x. */ 00142 /* In some literature, the GSVD of A and B is presented in the form */ 00143 /* U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) */ 00144 /* where U and V are orthogonal and X is nonsingular, D1 and D2 are */ 00145 /* ``diagonal''. The former GSVD form can be converted to the latter */ 00146 /* form by taking the nonsingular matrix X as */ 00147 00148 /* X = Q*( I 0 ) */ 00149 /* ( 0 inv(R) ). */ 00150 00151 /* Arguments */ 00152 /* ========= */ 00153 00154 /* JOBU (input) CHARACTER*1 */ 00155 /* = 'U': Orthogonal matrix U is computed; */ 00156 /* = 'N': U is not computed. */ 00157 00158 /* JOBV (input) CHARACTER*1 */ 00159 /* = 'V': Orthogonal matrix V is computed; */ 00160 /* = 'N': V is not computed. */ 00161 00162 /* JOBQ (input) CHARACTER*1 */ 00163 /* = 'Q': Orthogonal matrix Q is computed; */ 00164 /* = 'N': Q is not computed. */ 00165 00166 /* M (input) INTEGER */ 00167 /* The number of rows of the matrix A. M >= 0. */ 00168 00169 /* N (input) INTEGER */ 00170 /* The number of columns of the matrices A and B. N >= 0. */ 00171 00172 /* P (input) INTEGER */ 00173 /* The number of rows of the matrix B. P >= 0. */ 00174 00175 /* K (output) INTEGER */ 00176 /* L (output) INTEGER */ 00177 /* On exit, K and L specify the dimension of the subblocks */ 00178 /* described in the Purpose section. */ 00179 /* K + L = effective numerical rank of (A',B')'. */ 00180 00181 /* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) */ 00182 /* On entry, the M-by-N matrix A. */ 00183 /* On exit, A contains the triangular matrix R, or part of R. */ 00184 /* See Purpose for details. */ 00185 00186 /* LDA (input) INTEGER */ 00187 /* The leading dimension of the array A. LDA >= max(1,M). */ 00188 00189 /* B (input/output) DOUBLE PRECISION array, dimension (LDB,N) */ 00190 /* On entry, the P-by-N matrix B. */ 00191 /* On exit, B contains the triangular matrix R if M-K-L < 0. */ 00192 /* See Purpose for details. */ 00193 00194 /* LDB (input) INTEGER */ 00195 /* The leading dimension of the array B. LDB >= max(1,P). */ 00196 00197 /* ALPHA (output) DOUBLE PRECISION array, dimension (N) */ 00198 /* BETA (output) DOUBLE PRECISION array, dimension (N) */ 00199 /* On exit, ALPHA and BETA contain the generalized singular */ 00200 /* value pairs of A and B; */ 00201 /* ALPHA(1:K) = 1, */ 00202 /* BETA(1:K) = 0, */ 00203 /* and if M-K-L >= 0, */ 00204 /* ALPHA(K+1:K+L) = C, */ 00205 /* BETA(K+1:K+L) = S, */ 00206 /* or if M-K-L < 0, */ 00207 /* ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 */ 00208 /* BETA(K+1:M) =S, BETA(M+1:K+L) =1 */ 00209 /* and */ 00210 /* ALPHA(K+L+1:N) = 0 */ 00211 /* BETA(K+L+1:N) = 0 */ 00212 00213 /* U (output) DOUBLE PRECISION array, dimension (LDU,M) */ 00214 /* If JOBU = 'U', U contains the M-by-M orthogonal matrix U. */ 00215 /* If JOBU = 'N', U is not referenced. */ 00216 00217 /* LDU (input) INTEGER */ 00218 /* The leading dimension of the array U. LDU >= max(1,M) if */ 00219 /* JOBU = 'U'; LDU >= 1 otherwise. */ 00220 00221 /* V (output) DOUBLE PRECISION array, dimension (LDV,P) */ 00222 /* If JOBV = 'V', V contains the P-by-P orthogonal matrix V. */ 00223 /* If JOBV = 'N', V is not referenced. */ 00224 00225 /* LDV (input) INTEGER */ 00226 /* The leading dimension of the array V. LDV >= max(1,P) if */ 00227 /* JOBV = 'V'; LDV >= 1 otherwise. */ 00228 00229 /* Q (output) DOUBLE PRECISION array, dimension (LDQ,N) */ 00230 /* If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. */ 00231 /* If JOBQ = 'N', Q is not referenced. */ 00232 00233 /* LDQ (input) INTEGER */ 00234 /* The leading dimension of the array Q. LDQ >= max(1,N) if */ 00235 /* JOBQ = 'Q'; LDQ >= 1 otherwise. */ 00236 00237 /* WORK (workspace) DOUBLE PRECISION array, */ 00238 /* dimension (max(3*N,M,P)+N) */ 00239 00240 /* IWORK (workspace/output) INTEGER array, dimension (N) */ 00241 /* On exit, IWORK stores the sorting information. More */ 00242 /* precisely, the following loop will sort ALPHA */ 00243 /* for I = K+1, min(M,K+L) */ 00244 /* swap ALPHA(I) and ALPHA(IWORK(I)) */ 00245 /* endfor */ 00246 /* such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). */ 00247 00248 /* INFO (output) INTEGER */ 00249 /* = 0: successful exit */ 00250 /* < 0: if INFO = -i, the i-th argument had an illegal value. */ 00251 /* > 0: if INFO = 1, the Jacobi-type procedure failed to */ 00252 /* converge. For further details, see subroutine DTGSJA. */ 00253 00254 /* Internal Parameters */ 00255 /* =================== */ 00256 00257 /* TOLA DOUBLE PRECISION */ 00258 /* TOLB DOUBLE PRECISION */ 00259 /* TOLA and TOLB are the thresholds to determine the effective */ 00260 /* rank of (A',B')'. Generally, they are set to */ 00261 /* TOLA = MAX(M,N)*norm(A)*MAZHEPS, */ 00262 /* TOLB = MAX(P,N)*norm(B)*MAZHEPS. */ 00263 /* The size of TOLA and TOLB may affect the size of backward */ 00264 /* errors of the decomposition. */ 00265 00266 /* Further Details */ 00267 /* =============== */ 00268 00269 /* 2-96 Based on modifications by */ 00270 /* Ming Gu and Huan Ren, Computer Science Division, University of */ 00271 /* California at Berkeley, USA */ 00272 00273 /* ===================================================================== */ 00274 00275 /* .. Local Scalars .. */ 00276 /* .. */ 00277 /* .. External Functions .. */ 00278 /* .. */ 00279 /* .. External Subroutines .. */ 00280 /* .. */ 00281 /* .. Intrinsic Functions .. */ 00282 /* .. */ 00283 /* .. Executable Statements .. */ 00284 00285 /* Test the input parameters */ 00286 00287 /* Parameter adjustments */ 00288 a_dim1 = *lda; 00289 a_offset = 1 + a_dim1; 00290 a -= a_offset; 00291 b_dim1 = *ldb; 00292 b_offset = 1 + b_dim1; 00293 b -= b_offset; 00294 --alpha; 00295 --beta; 00296 u_dim1 = *ldu; 00297 u_offset = 1 + u_dim1; 00298 u -= u_offset; 00299 v_dim1 = *ldv; 00300 v_offset = 1 + v_dim1; 00301 v -= v_offset; 00302 q_dim1 = *ldq; 00303 q_offset = 1 + q_dim1; 00304 q -= q_offset; 00305 --work; 00306 --iwork; 00307 00308 /* Function Body */ 00309 wantu = lsame_(jobu, "U"); 00310 wantv = lsame_(jobv, "V"); 00311 wantq = lsame_(jobq, "Q"); 00312 00313 *info = 0; 00314 if (! (wantu || lsame_(jobu, "N"))) { 00315 *info = -1; 00316 } else if (! (wantv || lsame_(jobv, "N"))) { 00317 *info = -2; 00318 } else if (! (wantq || lsame_(jobq, "N"))) { 00319 *info = -3; 00320 } else if (*m < 0) { 00321 *info = -4; 00322 } else if (*n < 0) { 00323 *info = -5; 00324 } else if (*p < 0) { 00325 *info = -6; 00326 } else if (*lda < max(1,*m)) { 00327 *info = -10; 00328 } else if (*ldb < max(1,*p)) { 00329 *info = -12; 00330 } else if (*ldu < 1 || wantu && *ldu < *m) { 00331 *info = -16; 00332 } else if (*ldv < 1 || wantv && *ldv < *p) { 00333 *info = -18; 00334 } else if (*ldq < 1 || wantq && *ldq < *n) { 00335 *info = -20; 00336 } 00337 if (*info != 0) { 00338 i__1 = -(*info); 00339 xerbla_("DGGSVD", &i__1); 00340 return 0; 00341 } 00342 00343 /* Compute the Frobenius norm of matrices A and B */ 00344 00345 anorm = dlange_("1", m, n, &a[a_offset], lda, &work[1]); 00346 bnorm = dlange_("1", p, n, &b[b_offset], ldb, &work[1]); 00347 00348 /* Get machine precision and set up threshold for determining */ 00349 /* the effective numerical rank of the matrices A and B. */ 00350 00351 ulp = dlamch_("Precision"); 00352 unfl = dlamch_("Safe Minimum"); 00353 tola = max(*m,*n) * max(anorm,unfl) * ulp; 00354 tolb = max(*p,*n) * max(bnorm,unfl) * ulp; 00355 00356 /* Preprocessing */ 00357 00358 dggsvp_(jobu, jobv, jobq, m, p, n, &a[a_offset], lda, &b[b_offset], ldb, & 00359 tola, &tolb, k, l, &u[u_offset], ldu, &v[v_offset], ldv, &q[ 00360 q_offset], ldq, &iwork[1], &work[1], &work[*n + 1], info); 00361 00362 /* Compute the GSVD of two upper "triangular" matrices */ 00363 00364 dtgsja_(jobu, jobv, jobq, m, p, n, k, l, &a[a_offset], lda, &b[b_offset], 00365 ldb, &tola, &tolb, &alpha[1], &beta[1], &u[u_offset], ldu, &v[ 00366 v_offset], ldv, &q[q_offset], ldq, &work[1], &ncycle, info); 00367 00368 /* Sort the singular values and store the pivot indices in IWORK */ 00369 /* Copy ALPHA to WORK, then sort ALPHA in WORK */ 00370 00371 dcopy_(n, &alpha[1], &c__1, &work[1], &c__1); 00372 /* Computing MIN */ 00373 i__1 = *l, i__2 = *m - *k; 00374 ibnd = min(i__1,i__2); 00375 i__1 = ibnd; 00376 for (i__ = 1; i__ <= i__1; ++i__) { 00377 00378 /* Scan for largest ALPHA(K+I) */ 00379 00380 isub = i__; 00381 smax = work[*k + i__]; 00382 i__2 = ibnd; 00383 for (j = i__ + 1; j <= i__2; ++j) { 00384 temp = work[*k + j]; 00385 if (temp > smax) { 00386 isub = j; 00387 smax = temp; 00388 } 00389 /* L10: */ 00390 } 00391 if (isub != i__) { 00392 work[*k + isub] = work[*k + i__]; 00393 work[*k + i__] = smax; 00394 iwork[*k + i__] = *k + isub; 00395 } else { 00396 iwork[*k + i__] = *k + i__; 00397 } 00398 /* L20: */ 00399 } 00400 00401 return 0; 00402 00403 /* End of DGGSVD */ 00404 00405 } /* dggsvd_ */