00001 /* zggsvd.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 zggsvd_(char *jobu, char *jobv, char *jobq, integer *m, 00021 integer *n, integer *p, integer *k, integer *l, doublecomplex *a, 00022 integer *lda, doublecomplex *b, integer *ldb, doublereal *alpha, 00023 doublereal *beta, doublecomplex *u, integer *ldu, doublecomplex *v, 00024 integer *ldv, doublecomplex *q, integer *ldq, doublecomplex *work, 00025 doublereal *rwork, integer *iwork, 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 *); 00044 integer ncycle; 00045 extern /* Subroutine */ int xerbla_(char *, integer *); 00046 extern doublereal zlange_(char *, integer *, integer *, doublecomplex *, 00047 integer *, doublereal *); 00048 extern /* Subroutine */ int ztgsja_(char *, char *, char *, integer *, 00049 integer *, integer *, integer *, integer *, doublecomplex *, 00050 integer *, doublecomplex *, integer *, doublereal *, doublereal *, 00051 doublereal *, doublereal *, doublecomplex *, integer *, 00052 doublecomplex *, integer *, doublecomplex *, integer *, 00053 doublecomplex *, integer *, integer *), 00054 zggsvp_(char *, char *, char *, integer *, integer *, integer *, 00055 doublecomplex *, integer *, doublecomplex *, integer *, 00056 doublereal *, doublereal *, integer *, integer *, doublecomplex *, 00057 integer *, doublecomplex *, integer *, doublecomplex *, integer * 00058 , integer *, doublereal *, doublecomplex *, doublecomplex *, 00059 integer *); 00060 00061 00062 /* -- LAPACK driver routine (version 3.2) -- */ 00063 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00064 /* November 2006 */ 00065 00066 /* .. Scalar Arguments .. */ 00067 /* .. */ 00068 /* .. Array Arguments .. */ 00069 /* .. */ 00070 00071 /* Purpose */ 00072 /* ======= */ 00073 00074 /* ZGGSVD computes the generalized singular value decomposition (GSVD) */ 00075 /* of an M-by-N complex matrix A and P-by-N complex matrix B: */ 00076 00077 /* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) */ 00078 00079 /* where U, V and Q are unitary matrices, and Z' means the conjugate */ 00080 /* transpose of Z. Let K+L = the effective numerical rank of the */ 00081 /* matrix (A',B')', then R is a (K+L)-by-(K+L) nonsingular upper */ 00082 /* triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" */ 00083 /* matrices and of the following structures, respectively: */ 00084 00085 /* If M-K-L >= 0, */ 00086 00087 /* K L */ 00088 /* D1 = K ( I 0 ) */ 00089 /* L ( 0 C ) */ 00090 /* M-K-L ( 0 0 ) */ 00091 00092 /* K L */ 00093 /* D2 = L ( 0 S ) */ 00094 /* P-L ( 0 0 ) */ 00095 00096 /* N-K-L K L */ 00097 /* ( 0 R ) = K ( 0 R11 R12 ) */ 00098 /* L ( 0 0 R22 ) */ 00099 /* where */ 00100 00101 /* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), */ 00102 /* S = diag( BETA(K+1), ... , BETA(K+L) ), */ 00103 /* C**2 + S**2 = I. */ 00104 00105 /* R is stored in A(1:K+L,N-K-L+1:N) on exit. */ 00106 00107 /* If M-K-L < 0, */ 00108 00109 /* K M-K K+L-M */ 00110 /* D1 = K ( I 0 0 ) */ 00111 /* M-K ( 0 C 0 ) */ 00112 00113 /* K M-K K+L-M */ 00114 /* D2 = M-K ( 0 S 0 ) */ 00115 /* K+L-M ( 0 0 I ) */ 00116 /* P-L ( 0 0 0 ) */ 00117 00118 /* N-K-L K M-K K+L-M */ 00119 /* ( 0 R ) = K ( 0 R11 R12 R13 ) */ 00120 /* M-K ( 0 0 R22 R23 ) */ 00121 /* K+L-M ( 0 0 0 R33 ) */ 00122 00123 /* where */ 00124 00125 /* C = diag( ALPHA(K+1), ... , ALPHA(M) ), */ 00126 /* S = diag( BETA(K+1), ... , BETA(M) ), */ 00127 /* C**2 + S**2 = I. */ 00128 00129 /* (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored */ 00130 /* ( 0 R22 R23 ) */ 00131 /* in B(M-K+1:L,N+M-K-L+1:N) on exit. */ 00132 00133 /* The routine computes C, S, R, and optionally the unitary */ 00134 /* transformation matrices U, V and Q. */ 00135 00136 /* In particular, if B is an N-by-N nonsingular matrix, then the GSVD of */ 00137 /* A and B implicitly gives the SVD of A*inv(B): */ 00138 /* A*inv(B) = U*(D1*inv(D2))*V'. */ 00139 /* If ( A',B')' has orthnormal columns, then the GSVD of A and B is also */ 00140 /* equal to the CS decomposition of A and B. Furthermore, the GSVD can */ 00141 /* be used to derive the solution of the eigenvalue problem: */ 00142 /* A'*A x = lambda* B'*B x. */ 00143 /* In some literature, the GSVD of A and B is presented in the form */ 00144 /* U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) */ 00145 /* where U and V are orthogonal and X is nonsingular, and D1 and D2 are */ 00146 /* ``diagonal''. The former GSVD form can be converted to the latter */ 00147 /* form by taking the nonsingular matrix X as */ 00148 00149 /* X = Q*( I 0 ) */ 00150 /* ( 0 inv(R) ) */ 00151 00152 /* Arguments */ 00153 /* ========= */ 00154 00155 /* JOBU (input) CHARACTER*1 */ 00156 /* = 'U': Unitary matrix U is computed; */ 00157 /* = 'N': U is not computed. */ 00158 00159 /* JOBV (input) CHARACTER*1 */ 00160 /* = 'V': Unitary matrix V is computed; */ 00161 /* = 'N': V is not computed. */ 00162 00163 /* JOBQ (input) CHARACTER*1 */ 00164 /* = 'Q': Unitary matrix Q is computed; */ 00165 /* = 'N': Q is not computed. */ 00166 00167 /* M (input) INTEGER */ 00168 /* The number of rows of the matrix A. M >= 0. */ 00169 00170 /* N (input) INTEGER */ 00171 /* The number of columns of the matrices A and B. N >= 0. */ 00172 00173 /* P (input) INTEGER */ 00174 /* The number of rows of the matrix B. P >= 0. */ 00175 00176 /* K (output) INTEGER */ 00177 /* L (output) INTEGER */ 00178 /* On exit, K and L specify the dimension of the subblocks */ 00179 /* described in Purpose. */ 00180 /* K + L = effective numerical rank of (A',B')'. */ 00181 00182 /* A (input/output) COMPLEX*16 array, dimension (LDA,N) */ 00183 /* On entry, the M-by-N matrix A. */ 00184 /* On exit, A contains the triangular matrix R, or part of R. */ 00185 /* See Purpose for details. */ 00186 00187 /* LDA (input) INTEGER */ 00188 /* The leading dimension of the array A. LDA >= max(1,M). */ 00189 00190 /* B (input/output) COMPLEX*16 array, dimension (LDB,N) */ 00191 /* On entry, the P-by-N matrix B. */ 00192 /* On exit, B contains part of the triangular matrix R if */ 00193 /* M-K-L < 0. See Purpose for details. */ 00194 00195 /* LDB (input) INTEGER */ 00196 /* The leading dimension of the array B. LDB >= max(1,P). */ 00197 00198 /* ALPHA (output) DOUBLE PRECISION array, dimension (N) */ 00199 /* BETA (output) DOUBLE PRECISION array, dimension (N) */ 00200 /* On exit, ALPHA and BETA contain the generalized singular */ 00201 /* value pairs of A and B; */ 00202 /* ALPHA(1:K) = 1, */ 00203 /* BETA(1:K) = 0, */ 00204 /* and if M-K-L >= 0, */ 00205 /* ALPHA(K+1:K+L) = C, */ 00206 /* BETA(K+1:K+L) = S, */ 00207 /* or if M-K-L < 0, */ 00208 /* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 */ 00209 /* BETA(K+1:M) = S, BETA(M+1:K+L) = 1 */ 00210 /* and */ 00211 /* ALPHA(K+L+1:N) = 0 */ 00212 /* BETA(K+L+1:N) = 0 */ 00213 00214 /* U (output) COMPLEX*16 array, dimension (LDU,M) */ 00215 /* If JOBU = 'U', U contains the M-by-M unitary matrix U. */ 00216 /* If JOBU = 'N', U is not referenced. */ 00217 00218 /* LDU (input) INTEGER */ 00219 /* The leading dimension of the array U. LDU >= max(1,M) if */ 00220 /* JOBU = 'U'; LDU >= 1 otherwise. */ 00221 00222 /* V (output) COMPLEX*16 array, dimension (LDV,P) */ 00223 /* If JOBV = 'V', V contains the P-by-P unitary matrix V. */ 00224 /* If JOBV = 'N', V is not referenced. */ 00225 00226 /* LDV (input) INTEGER */ 00227 /* The leading dimension of the array V. LDV >= max(1,P) if */ 00228 /* JOBV = 'V'; LDV >= 1 otherwise. */ 00229 00230 /* Q (output) COMPLEX*16 array, dimension (LDQ,N) */ 00231 /* If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q. */ 00232 /* If JOBQ = 'N', Q is not referenced. */ 00233 00234 /* LDQ (input) INTEGER */ 00235 /* The leading dimension of the array Q. LDQ >= max(1,N) if */ 00236 /* JOBQ = 'Q'; LDQ >= 1 otherwise. */ 00237 00238 /* WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N) */ 00239 00240 /* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) */ 00241 00242 /* IWORK (workspace/output) INTEGER array, dimension (N) */ 00243 /* On exit, IWORK stores the sorting information. More */ 00244 /* precisely, the following loop will sort ALPHA */ 00245 /* for I = K+1, min(M,K+L) */ 00246 /* swap ALPHA(I) and ALPHA(IWORK(I)) */ 00247 /* endfor */ 00248 /* such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). */ 00249 00250 /* INFO (output) INTEGER */ 00251 /* = 0: successful exit. */ 00252 /* < 0: if INFO = -i, the i-th argument had an illegal value. */ 00253 /* > 0: if INFO = 1, the Jacobi-type procedure failed to */ 00254 /* converge. For further details, see subroutine ZTGSJA. */ 00255 00256 /* Internal Parameters */ 00257 /* =================== */ 00258 00259 /* TOLA DOUBLE PRECISION */ 00260 /* TOLB DOUBLE PRECISION */ 00261 /* TOLA and TOLB are the thresholds to determine the effective */ 00262 /* rank of (A',B')'. Generally, they are set to */ 00263 /* TOLA = MAX(M,N)*norm(A)*MAZHEPS, */ 00264 /* TOLB = MAX(P,N)*norm(B)*MAZHEPS. */ 00265 /* The size of TOLA and TOLB may affect the size of backward */ 00266 /* errors of the decomposition. */ 00267 00268 /* Further Details */ 00269 /* =============== */ 00270 00271 /* 2-96 Based on modifications by */ 00272 /* Ming Gu and Huan Ren, Computer Science Division, University of */ 00273 /* California at Berkeley, USA */ 00274 00275 /* ===================================================================== */ 00276 00277 /* .. Local Scalars .. */ 00278 /* .. */ 00279 /* .. External Functions .. */ 00280 /* .. */ 00281 /* .. External Subroutines .. */ 00282 /* .. */ 00283 /* .. Intrinsic Functions .. */ 00284 /* .. */ 00285 /* .. Executable Statements .. */ 00286 00287 /* Decode and test the input parameters */ 00288 00289 /* Parameter adjustments */ 00290 a_dim1 = *lda; 00291 a_offset = 1 + a_dim1; 00292 a -= a_offset; 00293 b_dim1 = *ldb; 00294 b_offset = 1 + b_dim1; 00295 b -= b_offset; 00296 --alpha; 00297 --beta; 00298 u_dim1 = *ldu; 00299 u_offset = 1 + u_dim1; 00300 u -= u_offset; 00301 v_dim1 = *ldv; 00302 v_offset = 1 + v_dim1; 00303 v -= v_offset; 00304 q_dim1 = *ldq; 00305 q_offset = 1 + q_dim1; 00306 q -= q_offset; 00307 --work; 00308 --rwork; 00309 --iwork; 00310 00311 /* Function Body */ 00312 wantu = lsame_(jobu, "U"); 00313 wantv = lsame_(jobv, "V"); 00314 wantq = lsame_(jobq, "Q"); 00315 00316 *info = 0; 00317 if (! (wantu || lsame_(jobu, "N"))) { 00318 *info = -1; 00319 } else if (! (wantv || lsame_(jobv, "N"))) { 00320 *info = -2; 00321 } else if (! (wantq || lsame_(jobq, "N"))) { 00322 *info = -3; 00323 } else if (*m < 0) { 00324 *info = -4; 00325 } else if (*n < 0) { 00326 *info = -5; 00327 } else if (*p < 0) { 00328 *info = -6; 00329 } else if (*lda < max(1,*m)) { 00330 *info = -10; 00331 } else if (*ldb < max(1,*p)) { 00332 *info = -12; 00333 } else if (*ldu < 1 || wantu && *ldu < *m) { 00334 *info = -16; 00335 } else if (*ldv < 1 || wantv && *ldv < *p) { 00336 *info = -18; 00337 } else if (*ldq < 1 || wantq && *ldq < *n) { 00338 *info = -20; 00339 } 00340 if (*info != 0) { 00341 i__1 = -(*info); 00342 xerbla_("ZGGSVD", &i__1); 00343 return 0; 00344 } 00345 00346 /* Compute the Frobenius norm of matrices A and B */ 00347 00348 anorm = zlange_("1", m, n, &a[a_offset], lda, &rwork[1]); 00349 bnorm = zlange_("1", p, n, &b[b_offset], ldb, &rwork[1]); 00350 00351 /* Get machine precision and set up threshold for determining */ 00352 /* the effective numerical rank of the matrices A and B. */ 00353 00354 ulp = dlamch_("Precision"); 00355 unfl = dlamch_("Safe Minimum"); 00356 tola = max(*m,*n) * max(anorm,unfl) * ulp; 00357 tolb = max(*p,*n) * max(bnorm,unfl) * ulp; 00358 00359 zggsvp_(jobu, jobv, jobq, m, p, n, &a[a_offset], lda, &b[b_offset], ldb, & 00360 tola, &tolb, k, l, &u[u_offset], ldu, &v[v_offset], ldv, &q[ 00361 q_offset], ldq, &iwork[1], &rwork[1], &work[1], &work[*n + 1], 00362 info); 00363 00364 /* Compute the GSVD of two upper "triangular" matrices */ 00365 00366 ztgsja_(jobu, jobv, jobq, m, p, n, k, l, &a[a_offset], lda, &b[b_offset], 00367 ldb, &tola, &tolb, &alpha[1], &beta[1], &u[u_offset], ldu, &v[ 00368 v_offset], ldv, &q[q_offset], ldq, &work[1], &ncycle, info); 00369 00370 /* Sort the singular values and store the pivot indices in IWORK */ 00371 /* Copy ALPHA to RWORK, then sort ALPHA in RWORK */ 00372 00373 dcopy_(n, &alpha[1], &c__1, &rwork[1], &c__1); 00374 /* Computing MIN */ 00375 i__1 = *l, i__2 = *m - *k; 00376 ibnd = min(i__1,i__2); 00377 i__1 = ibnd; 00378 for (i__ = 1; i__ <= i__1; ++i__) { 00379 00380 /* Scan for largest ALPHA(K+I) */ 00381 00382 isub = i__; 00383 smax = rwork[*k + i__]; 00384 i__2 = ibnd; 00385 for (j = i__ + 1; j <= i__2; ++j) { 00386 temp = rwork[*k + j]; 00387 if (temp > smax) { 00388 isub = j; 00389 smax = temp; 00390 } 00391 /* L10: */ 00392 } 00393 if (isub != i__) { 00394 rwork[*k + isub] = rwork[*k + i__]; 00395 rwork[*k + i__] = smax; 00396 iwork[*k + i__] = *k + isub; 00397 } else { 00398 iwork[*k + i__] = *k + i__; 00399 } 00400 /* L20: */ 00401 } 00402 00403 return 0; 00404 00405 /* End of ZGGSVD */ 00406 00407 } /* zggsvd_ */