SGGSVD computes the generalized singular value decomposition (GSVD)


 of an M-by-N real matrix A and P-by-N real matrix B:


       U**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )


 where U, V and Q are orthogonal matrices.


 Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,


 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**T.


 If ( A**T,B**T)**T  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**T*A x = lambda* B**T*B x.


 In some literature, the GSVD of A and B is presented in the form


                  U**T*A*X = ( 0 D1 ),   V**T*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) ).

JOBU

          JOBU is CHARACTER*1


          = 'U':  Orthogonal matrix U is computed;


          = 'N':  U is not computed.

JOBV

          JOBV is CHARACTER*1


          = 'V':  Orthogonal matrix V is computed;


          = 'N':  V is not computed.

JOBQ

          JOBQ is CHARACTER*1


          = 'Q':  Orthogonal 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**T,B**T)**T.

A

          A is REAL 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 REAL 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

          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 REAL array, dimension (LDU,M)


          If JOBU = 'U', U contains the M-by-M orthogonal 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 REAL array, dimension (LDV,P)


          If JOBV = 'V', V contains the P-by-P orthogonal 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 REAL array, dimension (LDQ,N)


          If JOBQ = 'Q', Q contains the N-by-N orthogonal 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 REAL array,


                      dimension (max(3*N,M,P)+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 STGSJA.

  TOLA    REAL


  TOLB    REAL


          TOLA and TOLB are the thresholds to determine the effective


          rank of (A**T,B**T)**T. 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