#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dggsvd_(char *jobu, char *jobv, char *jobq, integer *m, integer *n, integer *p, integer *k, integer *l, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *alpha, doublereal *beta, doublereal *u, integer *ldu, doublereal *v, integer *ldv, doublereal *q, integer *ldq, doublereal *work, integer *iwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= DGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) where U, V and Q are orthogonal matrices, and Z' is the transpose of Z. Let K+L = the effective numerical rank of the matrix (A',B')', then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the following structures, respectively: If M-K-L >= 0, K L D1 = K ( I 0 ) L ( 0 C ) M-K-L ( 0 0 ) K L D2 = L ( 0 S ) P-L ( 0 0 ) N-K-L K L ( 0 R ) = K ( 0 R11 R12 ) L ( 0 0 R22 ) where C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), S = diag( BETA(K+1), ... , BETA(K+L) ), C**2 + S**2 = I. R is stored in A(1:K+L,N-K-L+1:N) on exit. If M-K-L < 0, K M-K K+L-M D1 = K ( I 0 0 ) M-K ( 0 C 0 ) K M-K K+L-M D2 = M-K ( 0 S 0 ) K+L-M ( 0 0 I ) P-L ( 0 0 0 ) N-K-L K M-K K+L-M ( 0 R ) = K ( 0 R11 R12 R13 ) M-K ( 0 0 R22 R23 ) K+L-M ( 0 0 0 R33 ) where C = diag( ALPHA(K+1), ... , ALPHA(M) ), S = diag( BETA(K+1), ... , BETA(M) ), C**2 + S**2 = I. (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored ( 0 R22 R23 ) in B(M-K+1:L,N+M-K-L+1:N) on exit. The routine computes C, S, R, and optionally the orthogonal transformation matrices U, V and Q. In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B): A*inv(B) = U*(D1*inv(D2))*V'. If ( A',B')' has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem: A'*A x = lambda* B'*B x. In some literature, the GSVD of A and B is presented in the form U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) where U and V are orthogonal and X is nonsingular, D1 and D2 are ``diagonal''. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as X = Q*( I 0 ) ( 0 inv(R) ). Arguments ========= JOBU (input) CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed. JOBV (input) CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed. JOBQ (input) CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed. M (input) INTEGER The number of rows of the matrix A. M >= 0. N (input) INTEGER The number of columns of the matrices A and B. N >= 0. P (input) INTEGER The number of rows of the matrix B. P >= 0. K (output) INTEGER L (output) INTEGER On exit, K and L specify the dimension of the subblocks described in the Purpose section. K + L = effective numerical rank of (A',B')'. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A contains the triangular matrix R, or part of R. See Purpose for details. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). B (input/output) DOUBLE PRECISION array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, B contains the triangular matrix R if M-K-L < 0. See Purpose for details. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,P). ALPHA (output) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE PRECISION array, dimension (N) On exit, ALPHA and BETA contain the generalized singular value pairs of A and B; ALPHA(1:K) = 1, BETA(1:K) = 0, and if M-K-L >= 0, ALPHA(K+1:K+L) = C, BETA(K+1:K+L) = S, or if M-K-L < 0, ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 BETA(K+1:M) =S, BETA(M+1:K+L) =1 and ALPHA(K+L+1:N) = 0 BETA(K+L+1:N) = 0 U (output) DOUBLE PRECISION array, dimension (LDU,M) If JOBU = 'U', U contains the M-by-M orthogonal matrix U. If JOBU = 'N', U is not referenced. LDU (input) INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V (output) DOUBLE PRECISION array, dimension (LDV,P) If JOBV = 'V', V contains the P-by-P orthogonal matrix V. If JOBV = 'N', V is not referenced. LDV (input) INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q (output) DOUBLE PRECISION array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. If JOBQ = 'N', Q is not referenced. LDQ (input) INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P)+N) IWORK (workspace/output) INTEGER array, dimension (N) On exit, IWORK stores the sorting information. More precisely, the following loop will sort ALPHA for I = K+1, min(M,K+L) swap ALPHA(I) and ALPHA(IWORK(I)) endfor such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine DTGSJA. Internal Parameters =================== TOLA DOUBLE PRECISION TOLB DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective rank of (A',B')'. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition. Further Details =============== 2-96 Based on modifications by Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA ===================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1, u_offset, v_dim1, v_offset, i__1, i__2; /* Local variables */ static integer i__, j; static doublereal ulp; static integer ibnd; static doublereal tola; static integer isub; static doublereal tolb, unfl, temp, smax; extern logical lsame_(char *, char *); static doublereal anorm, bnorm; extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *); static logical wantq, wantu, wantv; extern doublereal dlamch_(char *), dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dtgsja_(char *, char *, char *, integer *, integer *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *); static integer ncycle; extern /* Subroutine */ int xerbla_(char *, integer *), dggsvp_( char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, doublereal *, doublereal *, integer *); a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --alpha; --beta; u_dim1 = *ldu; u_offset = 1 + u_dim1; u -= u_offset; v_dim1 = *ldv; v_offset = 1 + v_dim1; v -= v_offset; q_dim1 = *ldq; q_offset = 1 + q_dim1; q -= q_offset; --work; --iwork; /* Function Body */ wantu = lsame_(jobu, "U"); wantv = lsame_(jobv, "V"); wantq = lsame_(jobq, "Q"); *info = 0; if (! (wantu || lsame_(jobu, "N"))) { *info = -1; } else if (! (wantv || lsame_(jobv, "N"))) { *info = -2; } else if (! (wantq || lsame_(jobq, "N"))) { *info = -3; } else if (*m < 0) { *info = -4; } else if (*n < 0) { *info = -5; } else if (*p < 0) { *info = -6; } else if (*lda < max(1,*m)) { *info = -10; } else if (*ldb < max(1,*p)) { *info = -12; } else if (*ldu < 1 || wantu && *ldu < *m) { *info = -16; } else if (*ldv < 1 || wantv && *ldv < *p) { *info = -18; } else if (*ldq < 1 || wantq && *ldq < *n) { *info = -20; } if (*info != 0) { i__1 = -(*info); xerbla_("DGGSVD", &i__1); return 0; } /* Compute the Frobenius norm of matrices A and B */ anorm = dlange_("1", m, n, &a[a_offset], lda, &work[1]); bnorm = dlange_("1", p, n, &b[b_offset], ldb, &work[1]); /* Get machine precision and set up threshold for determining the effective numerical rank of the matrices A and B. */ ulp = dlamch_("Precision"); unfl = dlamch_("Safe Minimum"); tola = max(*m,*n) * max(anorm,unfl) * ulp; tolb = max(*p,*n) * max(bnorm,unfl) * ulp; /* Preprocessing */ dggsvp_(jobu, jobv, jobq, m, p, n, &a[a_offset], lda, &b[b_offset], ldb, & tola, &tolb, k, l, &u[u_offset], ldu, &v[v_offset], ldv, &q[ q_offset], ldq, &iwork[1], &work[1], &work[*n + 1], info); /* Compute the GSVD of two upper "triangular" matrices */ dtgsja_(jobu, jobv, jobq, m, p, n, k, l, &a[a_offset], lda, &b[b_offset], ldb, &tola, &tolb, &alpha[1], &beta[1], &u[u_offset], ldu, &v[ v_offset], ldv, &q[q_offset], ldq, &work[1], &ncycle, info); /* Sort the singular values and store the pivot indices in IWORK Copy ALPHA to WORK, then sort ALPHA in WORK */ dcopy_(n, &alpha[1], &c__1, &work[1], &c__1); /* Computing MIN */ i__1 = *l, i__2 = *m - *k; ibnd = min(i__1,i__2); i__1 = ibnd; for (i__ = 1; i__ <= i__1; ++i__) { /* Scan for largest ALPHA(K+I) */ isub = i__; smax = work[*k + i__]; i__2 = ibnd; for (j = i__ + 1; j <= i__2; ++j) { temp = work[*k + j]; if (temp > smax) { isub = j; smax = temp; } /* L10: */ } if (isub != i__) { work[*k + isub] = work[*k + i__]; work[*k + i__] = smax; iwork[*k + i__] = *k + isub; } else { iwork[*k + i__] = *k + i__; } /* L20: */ } return 0; /* End of DGGSVD */ } /* dggsvd_ */