SGGSVP computes orthogonal matrices U, V and Q such that


                    N-K-L  K    L


  U**T*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;


                 L ( 0     0   A23 )


             M-K-L ( 0     0    0  )


                  N-K-L  K    L


         =     K ( 0    A12  A13 )  if M-K-L < 0;


             M-K ( 0     0   A23 )


                  N-K-L  K    L


  V**T*B*Q =   L ( 0     0   B13 )


             P-L ( 0     0    0  )


 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular


 upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,


 otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective


 numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T. 


 This decomposition is the preprocessing step for computing the


 Generalized Singular Value Decomposition (GSVD), see subroutine


 SGGSVD.

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.

P

          P is INTEGER


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

N

          N is INTEGER


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

A

          A is REAL array, dimension (LDA,N)


          On entry, the M-by-N matrix A.


          On exit, A contains the triangular (or trapezoidal) matrix


          described in the Purpose section.

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 described in


          the Purpose section.

LDB

          LDB is INTEGER


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

TOLA

          TOLA is REAL

TOLB

          TOLB is REAL


          TOLA and TOLB are the thresholds to determine the effective


          numerical rank of matrix B and a subblock of A. 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.

K

          K is INTEGER

L

          L is INTEGER


          On exit, K and L specify the dimension of the subblocks


          described in Purpose section.


          K + L = effective numerical rank of (A**T,B**T)**T.

U

          U is REAL array, dimension (LDU,M)


          If JOBU = 'U', U contains the 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 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 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.

IWORK

          IWORK is INTEGER array, dimension (N)

TAU

          TAU is REAL array, dimension (N)

WORK

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

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

The subroutine uses LAPACK subroutine SGEQPF for the QR factorization with column pivoting to detect the effective numerical rank of the a matrix. It may be replaced by a better rank determination strategy.