version 1.3, 2010/08/06 15:28:53
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version 1.15, 2016/08/27 15:34:48
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*> \brief <b> ZGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b> |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download ZGGSVD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggsvd.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggsvd.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggsvd.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, |
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* LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, |
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* RWORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBQ, JOBU, JOBV |
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* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IWORK( * ) |
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* DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * ) |
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* COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), |
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* $ U( LDU, * ), V( LDV, * ), WORK( * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> This routine is deprecated and has been replaced by routine ZGGSVD3. |
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*> |
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*> ZGGSVD computes the generalized singular value decomposition (GSVD) |
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*> of an M-by-N complex matrix A and P-by-N complex matrix B: |
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*> |
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*> U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R ) |
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*> |
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*> where U, V and Q are unitary matrices. |
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*> Let K+L = the effective numerical rank of the |
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*> matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper |
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*> triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" |
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*> matrices and of the following structures, respectively: |
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*> |
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*> If M-K-L >= 0, |
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*> |
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*> K L |
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*> D1 = K ( I 0 ) |
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*> L ( 0 C ) |
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*> M-K-L ( 0 0 ) |
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*> |
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*> K L |
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*> D2 = L ( 0 S ) |
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*> P-L ( 0 0 ) |
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*> |
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*> N-K-L K L |
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*> ( 0 R ) = K ( 0 R11 R12 ) |
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*> L ( 0 0 R22 ) |
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*> where |
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*> |
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*> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), |
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*> S = diag( BETA(K+1), ... , BETA(K+L) ), |
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*> C**2 + S**2 = I. |
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*> |
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*> R is stored in A(1:K+L,N-K-L+1:N) on exit. |
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*> |
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*> If M-K-L < 0, |
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*> |
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*> K M-K K+L-M |
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*> D1 = K ( I 0 0 ) |
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*> M-K ( 0 C 0 ) |
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*> |
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*> K M-K K+L-M |
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*> D2 = M-K ( 0 S 0 ) |
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*> K+L-M ( 0 0 I ) |
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*> P-L ( 0 0 0 ) |
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*> |
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*> N-K-L K M-K K+L-M |
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*> ( 0 R ) = K ( 0 R11 R12 R13 ) |
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*> M-K ( 0 0 R22 R23 ) |
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*> K+L-M ( 0 0 0 R33 ) |
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*> |
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*> where |
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*> |
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*> C = diag( ALPHA(K+1), ... , ALPHA(M) ), |
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*> S = diag( BETA(K+1), ... , BETA(M) ), |
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*> C**2 + S**2 = I. |
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*> |
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*> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored |
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*> ( 0 R22 R23 ) |
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*> in B(M-K+1:L,N+M-K-L+1:N) on exit. |
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*> |
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*> The routine computes C, S, R, and optionally the unitary |
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*> transformation matrices U, V and Q. |
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*> |
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*> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of |
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*> A and B implicitly gives the SVD of A*inv(B): |
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*> A*inv(B) = U*(D1*inv(D2))*V**H. |
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*> If ( A**H,B**H)**H has orthnormal columns, then the GSVD of A and B is also |
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*> equal to the CS decomposition of A and B. Furthermore, the GSVD can |
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*> be used to derive the solution of the eigenvalue problem: |
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*> A**H*A x = lambda* B**H*B x. |
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*> In some literature, the GSVD of A and B is presented in the form |
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*> U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 ) |
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*> where U and V are orthogonal and X is nonsingular, and D1 and D2 are |
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*> ``diagonal''. The former GSVD form can be converted to the latter |
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*> form by taking the nonsingular matrix X as |
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*> |
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*> X = Q*( I 0 ) |
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*> ( 0 inv(R) ) |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] JOBU |
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*> \verbatim |
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*> JOBU is CHARACTER*1 |
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*> = 'U': Unitary matrix U is computed; |
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*> = 'N': U is not computed. |
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*> \endverbatim |
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*> |
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*> \param[in] JOBV |
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*> \verbatim |
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*> JOBV is CHARACTER*1 |
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*> = 'V': Unitary matrix V is computed; |
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*> = 'N': V is not computed. |
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*> \endverbatim |
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*> |
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*> \param[in] JOBQ |
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*> \verbatim |
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*> JOBQ is CHARACTER*1 |
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*> = 'Q': Unitary matrix Q is computed; |
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*> = 'N': Q is not computed. |
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*> \endverbatim |
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*> |
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*> \param[in] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of rows of the matrix A. M >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The number of columns of the matrices A and B. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] P |
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*> \verbatim |
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*> P is INTEGER |
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*> The number of rows of the matrix B. P >= 0. |
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*> \endverbatim |
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*> |
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*> \param[out] K |
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*> \verbatim |
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*> K is INTEGER |
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*> \endverbatim |
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*> |
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*> \param[out] L |
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*> \verbatim |
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*> L is INTEGER |
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*> |
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*> On exit, K and L specify the dimension of the subblocks |
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*> described in Purpose. |
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*> K + L = effective numerical rank of (A**H,B**H)**H. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is COMPLEX*16 array, dimension (LDA,N) |
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*> On entry, the M-by-N matrix A. |
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*> On exit, A contains the triangular matrix R, or part of R. |
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*> See Purpose for details. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,M). |
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*> \endverbatim |
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*> |
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*> \param[in,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension (LDB,N) |
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*> On entry, the P-by-N matrix B. |
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*> On exit, B contains part of the triangular matrix R if |
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*> M-K-L < 0. See Purpose for details. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,P). |
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*> \endverbatim |
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*> |
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*> \param[out] ALPHA |
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*> \verbatim |
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*> ALPHA is DOUBLE PRECISION array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] BETA |
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*> \verbatim |
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*> BETA is DOUBLE PRECISION array, dimension (N) |
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*> |
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*> On exit, ALPHA and BETA contain the generalized singular |
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*> value pairs of A and B; |
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*> ALPHA(1:K) = 1, |
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*> BETA(1:K) = 0, |
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*> and if M-K-L >= 0, |
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*> ALPHA(K+1:K+L) = C, |
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*> BETA(K+1:K+L) = S, |
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*> or if M-K-L < 0, |
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*> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 |
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*> BETA(K+1:M) =S, BETA(M+1:K+L) =1 |
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*> and |
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*> ALPHA(K+L+1:N) = 0 |
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*> BETA(K+L+1:N) = 0 |
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*> \endverbatim |
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*> |
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*> \param[out] U |
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*> \verbatim |
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*> U is COMPLEX*16 array, dimension (LDU,M) |
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*> If JOBU = 'U', U contains the M-by-M unitary matrix U. |
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*> If JOBU = 'N', U is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDU |
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*> \verbatim |
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*> LDU is INTEGER |
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*> The leading dimension of the array U. LDU >= max(1,M) if |
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*> JOBU = 'U'; LDU >= 1 otherwise. |
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*> \endverbatim |
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*> |
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*> \param[out] V |
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*> \verbatim |
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*> V is COMPLEX*16 array, dimension (LDV,P) |
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*> If JOBV = 'V', V contains the P-by-P unitary matrix V. |
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*> If JOBV = 'N', V is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDV |
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*> \verbatim |
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*> LDV is INTEGER |
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*> The leading dimension of the array V. LDV >= max(1,P) if |
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*> JOBV = 'V'; LDV >= 1 otherwise. |
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*> \endverbatim |
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*> |
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*> \param[out] Q |
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*> \verbatim |
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*> Q is COMPLEX*16 array, dimension (LDQ,N) |
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*> If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q. |
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*> If JOBQ = 'N', Q is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDQ |
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*> \verbatim |
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*> LDQ is INTEGER |
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*> The leading dimension of the array Q. LDQ >= max(1,N) if |
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*> JOBQ = 'Q'; LDQ >= 1 otherwise. |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is COMPLEX*16 array, dimension (max(3*N,M,P)+N) |
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*> \endverbatim |
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*> |
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*> \param[out] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, dimension (2*N) |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (N) |
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*> On exit, IWORK stores the sorting information. More |
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*> precisely, the following loop will sort ALPHA |
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*> for I = K+1, min(M,K+L) |
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*> swap ALPHA(I) and ALPHA(IWORK(I)) |
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*> endfor |
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*> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit. |
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*> < 0: if INFO = -i, the i-th argument had an illegal value. |
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*> > 0: if INFO = 1, the Jacobi-type procedure failed to |
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*> converge. For further details, see subroutine ZTGSJA. |
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*> \endverbatim |
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* |
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*> \par Internal Parameters: |
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* ========================= |
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*> |
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*> \verbatim |
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*> TOLA DOUBLE PRECISION |
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*> TOLB DOUBLE PRECISION |
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*> TOLA and TOLB are the thresholds to determine the effective |
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*> rank of (A**H,B**H)**H. Generally, they are set to |
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*> TOLA = MAX(M,N)*norm(A)*MAZHEPS, |
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*> TOLB = MAX(P,N)*norm(B)*MAZHEPS. |
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*> The size of TOLA and TOLB may affect the size of backward |
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*> errors of the decomposition. |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \date November 2011 |
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* |
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*> \ingroup complex16OTHERsing |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Ming Gu and Huan Ren, Computer Science Division, University of |
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*> California at Berkeley, USA |
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*> |
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* ===================================================================== |
SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, |
SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, |
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, |
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, |
$ RWORK, IWORK, INFO ) |
$ RWORK, IWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine (version 3.4.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBQ, JOBU, JOBV |
CHARACTER JOBQ, JOBU, JOBV |
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$ U( LDU, * ), V( LDV, * ), WORK( * ) |
$ U( LDU, * ), V( LDV, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZGGSVD computes the generalized singular value decomposition (GSVD) |
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* of an M-by-N complex matrix A and P-by-N complex matrix B: |
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* |
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* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) |
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* |
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* where U, V and Q are unitary matrices, and Z' means the conjugate |
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* transpose of Z. Let K+L = the effective numerical rank of the |
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* matrix (A',B')', then R is a (K+L)-by-(K+L) nonsingular upper |
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* triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" |
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* matrices and of the following structures, respectively: |
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* |
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* If M-K-L >= 0, |
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* |
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* K L |
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* D1 = K ( I 0 ) |
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* L ( 0 C ) |
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* M-K-L ( 0 0 ) |
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* |
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* K L |
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* D2 = L ( 0 S ) |
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* P-L ( 0 0 ) |
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* |
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* N-K-L K L |
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* ( 0 R ) = K ( 0 R11 R12 ) |
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* L ( 0 0 R22 ) |
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* where |
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* |
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* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), |
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* S = diag( BETA(K+1), ... , BETA(K+L) ), |
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* C**2 + S**2 = I. |
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* |
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* R is stored in A(1:K+L,N-K-L+1:N) on exit. |
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* |
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* If M-K-L < 0, |
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* |
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* K M-K K+L-M |
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* D1 = K ( I 0 0 ) |
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* M-K ( 0 C 0 ) |
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* |
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* K M-K K+L-M |
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* D2 = M-K ( 0 S 0 ) |
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* K+L-M ( 0 0 I ) |
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* P-L ( 0 0 0 ) |
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* |
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* N-K-L K M-K K+L-M |
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* ( 0 R ) = K ( 0 R11 R12 R13 ) |
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* M-K ( 0 0 R22 R23 ) |
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* K+L-M ( 0 0 0 R33 ) |
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* |
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* where |
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* |
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* C = diag( ALPHA(K+1), ... , ALPHA(M) ), |
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* S = diag( BETA(K+1), ... , BETA(M) ), |
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* C**2 + S**2 = I. |
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* |
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* (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored |
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* ( 0 R22 R23 ) |
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* in B(M-K+1:L,N+M-K-L+1:N) on exit. |
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* |
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* The routine computes C, S, R, and optionally the unitary |
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* transformation matrices U, V and Q. |
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* |
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* In particular, if B is an N-by-N nonsingular matrix, then the GSVD of |
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* A and B implicitly gives the SVD of A*inv(B): |
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* A*inv(B) = U*(D1*inv(D2))*V'. |
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* If ( A',B')' has orthnormal columns, then the GSVD of A and B is also |
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* equal to the CS decomposition of A and B. Furthermore, the GSVD can |
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* be used to derive the solution of the eigenvalue problem: |
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* A'*A x = lambda* B'*B x. |
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* In some literature, the GSVD of A and B is presented in the form |
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* U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) |
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* where U and V are orthogonal and X is nonsingular, and D1 and D2 are |
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* ``diagonal''. The former GSVD form can be converted to the latter |
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* form by taking the nonsingular matrix X as |
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* |
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* X = Q*( I 0 ) |
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* ( 0 inv(R) ) |
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* |
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* Arguments |
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* ========= |
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* |
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* JOBU (input) CHARACTER*1 |
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* = 'U': Unitary matrix U is computed; |
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* = 'N': U is not computed. |
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* |
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* JOBV (input) CHARACTER*1 |
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* = 'V': Unitary matrix V is computed; |
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* = 'N': V is not computed. |
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* |
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* JOBQ (input) CHARACTER*1 |
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* = 'Q': Unitary matrix Q is computed; |
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* = 'N': Q is not computed. |
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* |
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* M (input) INTEGER |
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* The number of rows of the matrix A. M >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns of the matrices A and B. N >= 0. |
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* |
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* P (input) INTEGER |
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* The number of rows of the matrix B. P >= 0. |
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* |
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* K (output) INTEGER |
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* L (output) INTEGER |
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* On exit, K and L specify the dimension of the subblocks |
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* described in Purpose. |
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* K + L = effective numerical rank of (A',B')'. |
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* |
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* A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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* On entry, the M-by-N matrix A. |
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* On exit, A contains the triangular matrix R, or part of R. |
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* See Purpose for details. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,M). |
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* |
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* B (input/output) COMPLEX*16 array, dimension (LDB,N) |
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* On entry, the P-by-N matrix B. |
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* On exit, B contains part of the triangular matrix R if |
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* M-K-L < 0. See Purpose for details. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= max(1,P). |
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* |
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* ALPHA (output) DOUBLE PRECISION array, dimension (N) |
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* BETA (output) DOUBLE PRECISION array, dimension (N) |
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* On exit, ALPHA and BETA contain the generalized singular |
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* value pairs of A and B; |
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* ALPHA(1:K) = 1, |
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* BETA(1:K) = 0, |
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* and if M-K-L >= 0, |
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* ALPHA(K+1:K+L) = C, |
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* BETA(K+1:K+L) = S, |
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* or if M-K-L < 0, |
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* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 |
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* BETA(K+1:M) = S, BETA(M+1:K+L) = 1 |
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* and |
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* ALPHA(K+L+1:N) = 0 |
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* BETA(K+L+1:N) = 0 |
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* |
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* U (output) COMPLEX*16 array, dimension (LDU,M) |
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* If JOBU = 'U', U contains the M-by-M unitary matrix U. |
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* If JOBU = 'N', U is not referenced. |
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* |
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* LDU (input) INTEGER |
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* The leading dimension of the array U. LDU >= max(1,M) if |
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* JOBU = 'U'; LDU >= 1 otherwise. |
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* |
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* V (output) COMPLEX*16 array, dimension (LDV,P) |
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* If JOBV = 'V', V contains the P-by-P unitary matrix V. |
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* If JOBV = 'N', V is not referenced. |
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* |
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* LDV (input) INTEGER |
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* The leading dimension of the array V. LDV >= max(1,P) if |
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* JOBV = 'V'; LDV >= 1 otherwise. |
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* |
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* Q (output) COMPLEX*16 array, dimension (LDQ,N) |
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* If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q. |
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* If JOBQ = 'N', Q is not referenced. |
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* |
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* LDQ (input) INTEGER |
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* The leading dimension of the array Q. LDQ >= max(1,N) if |
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* JOBQ = 'Q'; LDQ >= 1 otherwise. |
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* |
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* WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N) |
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* |
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* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) |
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* |
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* IWORK (workspace/output) INTEGER array, dimension (N) |
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* On exit, IWORK stores the sorting information. More |
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* precisely, the following loop will sort ALPHA |
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* for I = K+1, min(M,K+L) |
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* swap ALPHA(I) and ALPHA(IWORK(I)) |
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* endfor |
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* such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit. |
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* < 0: if INFO = -i, the i-th argument had an illegal value. |
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* > 0: if INFO = 1, the Jacobi-type procedure failed to |
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* converge. For further details, see subroutine ZTGSJA. |
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* |
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* Internal Parameters |
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* =================== |
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* |
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* TOLA DOUBLE PRECISION |
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* TOLB DOUBLE PRECISION |
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* TOLA and TOLB are the thresholds to determine the effective |
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* rank of (A',B')'. Generally, they are set to |
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* TOLA = MAX(M,N)*norm(A)*MAZHEPS, |
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* TOLB = MAX(P,N)*norm(B)*MAZHEPS. |
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* The size of TOLA and TOLB may affect the size of backward |
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* errors of the decomposition. |
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* |
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* Further Details |
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* =============== |
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* |
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* 2-96 Based on modifications by |
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* Ming Gu and Huan Ren, Computer Science Division, University of |
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* California at Berkeley, USA |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Local Scalars .. |
* .. Local Scalars .. |