ZTGSJA computes the generalized singular value decomposition (GSVD)


 of two complex 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 ZGGSVP


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


 where U, V and Q are unitary 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 unitary 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 a unitary matrix U1 on entry, and


                  the product U1*U is returned;


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


                  unitary matrix U is returned;


          = 'N':  U is not computed.

JOBV

          JOBV is CHARACTER*1


          = 'V':  V must contain a unitary matrix V1 on entry, and


                  the product V1*V is returned;


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


                  unitary matrix V is returned;


          = 'N':  V is not computed.

JOBQ

          JOBQ is CHARACTER*1


          = 'Q':  Q must contain a unitary matrix Q1 on entry, and


                  the product Q1*Q is returned;


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


                  unitary 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 ZTGSJA.


          See Further Details.

A

          A is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDU,M)


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


          the unitary matrix returned by ZGGSVP).


          On exit,


          if JOBU = 'I', U contains the unitary 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 COMPLEX*16 array, dimension (LDV,P)


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


          the unitary matrix returned by ZGGSVP).


          On exit,


          if JOBV = 'I', V contains the unitary 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 COMPLEX*16 array, dimension (LDQ,N)


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


          the unitary matrix returned by ZGGSVP).


          On exit,


          if JOBQ = 'I', Q contains the unitary 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 COMPLEX*16 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.

  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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  ZTGSJA 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**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,


  where U1, V1 and Q1 are unitary matrix.


  C1 and S1 are diagonal matrices satisfying


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


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