cggsvd3()

subroutine cggsvd3	(	character	jobu,
		character	jobv,
		character	jobq,
		integer	m,
		integer	n,
		integer	p,
		integer	k,
		integer	l,
		complex, dimension( lda, * )	a,
		integer	lda,
		complex, dimension( ldb, * )	b,
		integer	ldb,
		real, dimension( * )	alpha,
		real, dimension( * )	beta,
		complex, dimension( ldu, * )	u,
		integer	ldu,
		complex, dimension( ldv, * )	v,
		integer	ldv,
		complex, dimension( ldq, * )	q,
		integer	ldq,
		complex, dimension( * )	work,
		integer	lwork,
		real, dimension( * )	rwork,
		integer, dimension( * )	iwork,
		integer	info )

CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

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Purpose:

!>
!> CGGSVD3 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) 
!> 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 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**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

[in]	JOBU	!> JOBU is CHARACTER*1 !> = 'U': Unitary matrix U is computed; !> = 'N': U is not computed. !>
[in]	JOBV	!> JOBV is CHARACTER*1 !> = 'V': Unitary matrix V is computed; !> = 'N': V is not computed. !>
[in]	JOBQ	!> JOBQ is CHARACTER*1 !> = 'Q': Unitary matrix Q is computed; !> = 'N': Q is not computed. !>
[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrices A and B. N >= 0. !>
[in]	P	!> P is INTEGER !> The number of rows of the matrix B. P >= 0. !>
[out]	K	!> K is INTEGER !>
[out]	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. !>
[in,out]	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. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[in,out]	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. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,P). !>
[out]	ALPHA	!> ALPHA is REAL array, dimension (N) !>
[out]	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 !>
[out]	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. !>
[in]	LDU	!> LDU is INTEGER !> The leading dimension of the array U. LDU >= max(1,M) if !> JOBU = 'U'; LDU >= 1 otherwise. !>
[out]	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. !>
[in]	LDV	!> LDV is INTEGER !> The leading dimension of the array V. LDV >= max(1,P) if !> JOBV = 'V'; LDV >= 1 otherwise. !>
[out]	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. !>
[in]	LDQ	!> LDQ is INTEGER !> The leading dimension of the array Q. LDQ >= max(1,N) if !> JOBQ = 'Q'; LDQ >= 1 otherwise. !>
[out]	WORK	!> WORK is COMPLEX array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= 1. !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the optimal size of the WORK array, returns !> this value as the first entry of the WORK array, and no error !> message related to LWORK is issued by XERBLA. !>
[out]	RWORK	!> RWORK is REAL array, dimension (2*N) !>
[out]	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). !>
[out]	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.

Contributors:

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

Further Details:

CGGSVD3 replaces the deprecated subroutine CGGSVD.