cggsvd.f (3) - Linux Manuals


cggsvd.f -



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


 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 )


   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  )


   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) )




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


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


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


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


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


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


          K is INTEGER


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


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


          ALPHA is REAL array, dimension (N)


          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
            ALPHA(K+L+1:N) = 0
            BETA(K+L+1:N)  = 0


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


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


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


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


          RWORK is REAL array, dimension (2*N)


          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))
          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).


          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 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.



Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.


November 2011


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

Definition at line 335 of file cggsvd.f.


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