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cggsvd-f(3)

cggsvd.f –

Synopsis

Functions/Subroutines

subroutine cggsvd (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, RWORK, IWORK, INFO)
CGGSVD computes the singular value decomposition (SVD) for OTHER matrices

Function/Subroutine Documentation

subroutine cggsvd (characterJOBU, characterJOBV, characterJOBQ, integerM, integerN, integerP, integerK, integerL, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, real, dimension( * )ALPHA, real, dimension( * )BETA, complex, dimension( ldu, * )U, integerLDU, complex, dimension( ldv, * )V, integerLDV, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( * )WORK, real, dimension( * )RWORK, integer, dimension( * )IWORK, integerINFO)

CGGSVD computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

 CGGSVD computes the generalized singular value decomposition (GSVD)
 of an M-by-N complex matrix A and P-by-N complex matrix B:
 U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R )
 where U, V and Q are unitary matrices.
 Let K+L = the effective numerical rank of the
 matrix (A**H,B**H)**H, 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 unitary
 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**H.
 If ( A**H,B**H)**H has orthnormal 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**H*A x = lambda* B**H*B x.
 In some literature, the GSVD of A and B is presented in the form
 U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 )
 where U and V are orthogonal and X is nonsingular, and 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) )

 

Parameters:

JOBU

 JOBU is CHARACTER*1
 = 'U': Unitary matrix U is computed;
 = 'N': U is not computed.

JOBV

 JOBV is CHARACTER*1
 = 'V': Unitary matrix V is computed;
 = 'N': V is not computed.

JOBQ

 JOBQ is CHARACTER*1
 = 'Q': Unitary matrix Q is computed;
 = 'N': Q is not computed.

M

 M is INTEGER
 The number of rows of the matrix A. M >= 0.

N

 N is INTEGER
 The number of columns of the matrices A and B. N >= 0.

P

 P is INTEGER
 The number of rows of the matrix B. P >= 0.

K

 K is INTEGER

L

 L is INTEGER
 On exit, K and L specify the dimension of the subblocks
 described in Purpose.
 K + L = effective numerical rank of (A**H,B**H)**H.

A

 A is COMPLEX 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

 LDA is INTEGER
 The leading dimension of the array A. LDA >= max(1,M).

B

 B is COMPLEX array, dimension (LDB,N)
 On entry, the P-by-N matrix B.
 On exit, B contains part of the triangular matrix R if
 M-K-L < 0. See Purpose for details.

LDB

 LDB is INTEGER
 The leading dimension of the array B. LDB >= max(1,P).

ALPHA

 ALPHA is REAL array, dimension (N)

BETA

 BETA is REAL 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

 U is COMPLEX array, dimension (LDU,M)
 If JOBU = 'U', U contains the M-by-M unitary matrix U.
 If JOBU = 'N', U is not referenced.

LDU

 LDU is INTEGER
 The leading dimension of the array U. LDU >= max(1,M) if
 JOBU = 'U'; LDU >= 1 otherwise.

V

 V is COMPLEX array, dimension (LDV,P)
 If JOBV = 'V', V contains the P-by-P unitary matrix V.
 If JOBV = 'N', V is not referenced.

LDV

 LDV is INTEGER
 The leading dimension of the array V. LDV >= max(1,P) if
 JOBV = 'V'; LDV >= 1 otherwise.

Q

 Q is COMPLEX array, dimension (LDQ,N)
 If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
 If JOBQ = 'N', Q is not referenced.

LDQ

 LDQ is INTEGER
 The leading dimension of the array Q. LDQ >= max(1,N) if
 JOBQ = 'Q'; LDQ >= 1 otherwise.

WORK

 WORK is COMPLEX array, dimension (max(3*N,M,P)+N)

RWORK

 RWORK is REAL array, dimension (2*N)

IWORK

 IWORK is 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

 INFO is 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 CTGSJA.

 

Internal Parameters:

 TOLA REAL
 TOLB REAL
 TOLA and TOLB are the thresholds to determine the effective
 rank of (A**H,B**H)**H. Generally, they are set to
 TOLA = MAX(M,N)*norm(A)*MACHEPS,
 TOLB = MAX(P,N)*norm(B)*MACHEPS.
 The size of TOLA and TOLB may affect the size of backward
 errors of the decomposition.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 335 of file cggsvd.f.

Author

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