DGGSVP3 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


 DGGSVD3.

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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION

TOLB

          TOLB is DOUBLE PRECISION


          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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          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.

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.

August 2015

  The subroutine uses LAPACK subroutine DGEQP3 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.


  DGGSVP3 replaces the deprecated subroutine DGGSVP.