org.netlib.lapack
Class Dggsvd

java.lang.Object
  extended by org.netlib.lapack.Dggsvd

public class Dggsvd
extends java.lang.Object

Following is the description from the original
Fortran source.  For each array argument, the Java
version will include an integer offset parameter, so
the arguments may not match the description exactly.
Contact seymour@cs.utk.edu with any questions.

* .. * * 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. LDA >= 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 * * ===================================================================== * * .. Local Scalars ..


Constructor Summary
Dggsvd()
           
 
Method Summary
static void dggsvd(java.lang.String jobu, java.lang.String jobv, java.lang.String jobq, int m, int n, int p, intW k, intW l, double[] a, int _a_offset, int lda, double[] b, int _b_offset, int ldb, double[] alpha, int _alpha_offset, double[] beta, int _beta_offset, double[] u, int _u_offset, int ldu, double[] v, int _v_offset, int ldv, double[] q, int _q_offset, int ldq, double[] work, int _work_offset, int[] iwork, int _iwork_offset, intW info)
           
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

Dggsvd

public Dggsvd()
Method Detail

dggsvd

public static void dggsvd(java.lang.String jobu,
                          java.lang.String jobv,
                          java.lang.String jobq,
                          int m,
                          int n,
                          int p,
                          intW k,
                          intW l,
                          double[] a,
                          int _a_offset,
                          int lda,
                          double[] b,
                          int _b_offset,
                          int ldb,
                          double[] alpha,
                          int _alpha_offset,
                          double[] beta,
                          int _beta_offset,
                          double[] u,
                          int _u_offset,
                          int ldu,
                          double[] v,
                          int _v_offset,
                          int ldv,
                          double[] q,
                          int _q_offset,
                          int ldq,
                          double[] work,
                          int _work_offset,
                          int[] iwork,
                          int _iwork_offset,
                          intW info)