DTGSJA computes the generalized singular value decomposition (GSVD)


 of two real upper triangular (or trapezoidal) matrices A and B.


 On entry, it is assumed that matrices A and B have the following


 forms, which may be obtained by the preprocessing subroutine DGGSVP


 from a general M-by-N matrix A and P-by-N matrix B:


              N-K-L  K    L


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


           L ( 0     0   A23 )


       M-K-L ( 0     0    0  )


            N-K-L  K    L


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


       M-K ( 0     0   A23 )


            N-K-L  K    L


    B =  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.


 On exit,


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


 where U, V and Q are orthogonal matrices.


 R is a nonsingular upper triangular matrix, and D1 and D2 are


 ``diagonal'' matrices, which are of the following structures:


 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 ) K


             L (  0    0   R22 ) L


 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.


 R = ( 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 computation of the orthogonal transformation matrices U, V or Q


 is optional.  These matrices may either be formed explicitly, or they


 may be postmultiplied into input matrices U1, V1, or Q1.

JOBU

          JOBU is CHARACTER*1


          = 'U':  U must contain an orthogonal matrix U1 on entry, and


                  the product U1*U is returned;


          = 'I':  U is initialized to the unit matrix, and the


                  orthogonal matrix U is returned;


          = 'N':  U is not computed.

JOBV

          JOBV is CHARACTER*1


          = 'V':  V must contain an orthogonal matrix V1 on entry, and


                  the product V1*V is returned;


          = 'I':  V is initialized to the unit matrix, and the


                  orthogonal matrix V is returned;


          = 'N':  V is not computed.

JOBQ

          JOBQ is CHARACTER*1


          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and


                  the product Q1*Q is returned;


          = 'I':  Q is initialized to the unit matrix, and the


                  orthogonal matrix Q is returned;


          = '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.

K

          K is INTEGER

L

          L is INTEGER


          K and L specify the subblocks in the input matrices A and B:


          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)


          of A and B, whose GSVD is going to be computed by DTGSJA.


          See Further Details.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)


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


          On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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 DOUBLE PRECISION array, dimension (LDB,N)


          On entry, the P-by-N matrix B.


          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains


          a part of R.  See Purpose for details.

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 convergence criteria for the Jacobi-


          Kogbetliantz iteration procedure. Generally, they are the


          same as used in the preprocessing step, say


              TOLA = max(M,N)*norm(A)*MAZHEPS,


              TOLB = max(P,N)*norm(B)*MAZHEPS.

ALPHA

          ALPHA is DOUBLE PRECISION array, dimension (N)

BETA

          BETA is DOUBLE PRECISION 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) = diag(C),


            BETA(K+1:K+L)  = diag(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.


          Furthermore, if K+L < N,


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


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

U

          U is DOUBLE PRECISION array, dimension (LDU,M)


          On entry, if JOBU = 'U', U must contain a matrix U1 (usually


          the orthogonal matrix returned by DGGSVP).


          On exit,


          if JOBU = 'I', U contains the orthogonal matrix U;


          if JOBU = 'U', U contains the product U1*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)


          On entry, if JOBV = 'V', V must contain a matrix V1 (usually


          the orthogonal matrix returned by DGGSVP).


          On exit,


          if JOBV = 'I', V contains the orthogonal matrix V;


          if JOBV = 'V', V contains the product V1*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)


          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually


          the orthogonal matrix returned by DGGSVP).


          On exit,


          if JOBQ = 'I', Q contains the orthogonal matrix Q;


          if JOBQ = 'Q', Q contains the product Q1*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 DOUBLE PRECISION array, dimension (2*N)

NCYCLE

          NCYCLE is INTEGER


          The number of cycles required for convergence.

INFO

          INFO is INTEGER


          = 0:  successful exit


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


          = 1:  the procedure does not converge after MAXIT cycles.

  Internal Parameters


  ===================


  MAXIT   INTEGER


          MAXIT specifies the total loops that the iterative procedure


          may take. If after MAXIT cycles, the routine fails to


          converge, we return INFO = 1..fi

Author:

Univ. of Tennessee 

Univ. of California Berkeley 

Univ. of Colorado Denver 

NAG Ltd. 

Date:

December 2016 

Further Details: 

  DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce


  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L


  matrix B13 to the form:


           U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,


  where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose


  of Z.  C1 and S1 are diagonal matrices satisfying


                C1**2 + S1**2 = I,


  and R1 is an L-by-L nonsingular upper triangular matrix.