Contents


NAME

     ctgsja - compute the generalized singular  value  decomposi-
     tion (GSVD) of two complex upper triangular (or trapezoidal)
     matrices A and B

SYNOPSIS

     SUBROUTINE CTGSJA(JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
           TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE,
           INFO)

     CHARACTER * 1 JOBU, JOBV, JOBQ
     COMPLEX A(LDA,*), B(LDB,*),  U(LDU,*),  V(LDV,*),  Q(LDQ,*),
     WORK(*)
     INTEGER M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
     REAL TOLA, TOLB
     REAL ALPHA(*), BETA(*)

     SUBROUTINE CTGSJA_64(JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
           TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE,
           INFO)

     CHARACTER * 1 JOBU, JOBV, JOBQ
     COMPLEX A(LDA,*), B(LDB,*),  U(LDU,*),  V(LDV,*),  Q(LDQ,*),
     WORK(*)
     INTEGER*8 M, P, N, K, L, LDA, LDB, LDU,  LDV,  LDQ,  NCYCLE,
     INFO
     REAL TOLA, TOLB
     REAL ALPHA(*), BETA(*)

  F95 INTERFACE
     SUBROUTINE TGSJA(JOBU, JOBV, JOBQ, [M], [P], [N], K, L, A, [LDA], B,
            [LDB], TOLA, TOLB, ALPHA, BETA, U, [LDU], V, [LDV], Q, [LDQ],
            [WORK], NCYCLE, [INFO])

     CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ
     COMPLEX, DIMENSION(:) :: WORK
     COMPLEX, DIMENSION(:,:) :: A, B, U, V, Q
     INTEGER :: M, P, N, K, L, LDA, LDB, LDU, LDV,  LDQ,  NCYCLE,
     INFO
     REAL :: TOLA, TOLB
     REAL, DIMENSION(:) :: ALPHA, BETA

     SUBROUTINE TGSJA_64(JOBU, JOBV, JOBQ, [M], [P], [N], K, L, A, [LDA],
            B, [LDB], TOLA, TOLB, ALPHA, BETA, U, [LDU], V, [LDV], Q, [LDQ],
            [WORK], NCYCLE, [INFO])

     CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ
     COMPLEX, DIMENSION(:) :: WORK
     COMPLEX, DIMENSION(:,:) :: A, B, U, V, Q
     INTEGER(8) :: M, P, N, K, L, LDA, LDB, LDU, LDV,  LDQ,  NCY-
     CLE, INFO
     REAL :: TOLA, TOLB
     REAL, DIMENSION(:) :: ALPHA, BETA

  C INTERFACE
     #include <sunperf.h>

     void ctgsja(char jobu, char jobv, char jobq, int m,  int  p,
               int  n, int k, int l, complex *a, int lda, complex
               *b, int ldb, float tola, float tolb, float *alpha,
               float  *beta, complex *u, int ldu, complex *v, int
               ldv, complex *q, int ldq, int *ncycle, int *info);

     void ctgsja_64(char jobu, char jobv, char jobq, long m, long
               p,  long  n, long k, long l, complex *a, long lda,
               complex *b, long  ldb,  float  tola,  float  tolb,
               float  *alpha,  float *beta, complex *u, long ldu,
               complex *v, long ldv, complex *q, long  ldq,  long
               *ncycle, long *info);

PURPOSE

     ctgsja 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  fol-
     lowing  forms,  which  may  be obtained by the preprocessing
     subroutine CGGSVP 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  non-
     singular 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'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ),

     where U, V and Q are unitary matrices, Z' denotes the conju-
     gate  transpose  of  Z,  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

               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.

     CTGSJA 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'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
     where U1, V1 and Q1 are unitary matrix, and Z' is the conju-
     gate transpose of Z.  C1 and S1 are diagonal matrices satis-
     fying
        C1**2 + S1**2 = I,
     and R1 is an L-by-L nonsingular upper triangular matrix.

ARGUMENTS

     JOBU (input)
               = '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 (input)
               = '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 (input)
               = '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 (input) The number of rows of the matrix A.  M >= 0.

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

     N (input) The number of columns of the matrices A and B.   N
               >= 0.
     K (input) 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 CTGSJA.  See  the  Further  Details
               section below.

     L (input) See the description of K.

     A (input/output)
               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 (input)
               The leading dimension  of  the  array  A.  LDA  >=
               max(1,M).

     B (input/output)
               On entry, the P-by-N matrix B.  On exit, if neces-
               sary, B(M-K+1:L,N+M-K-L+1:N) contains a part of R.
               See Purpose for details.

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

     TOLA (input)
               TOLA and TOLB are the convergence criteria for the
               Jacobi-  Kogbetliantz  iteration  procedure.  Gen-
               erally, they are the same as used in  the  prepro-
               cessing step, say TOLA = MAX(M,N)*norm(A)*MACHEPS,
               TOLB = MAX(P,N)*norm(B)*MACHEPS.

     TOLB (input)
               See the description of TOLA.

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

     BETA (output)
               See the description of ALPHA.

     U (input) On entry, if JOBU = 'U', U must contain  a  matrix
               U1   (usually   the  unitary  matrix  returned  by
               CGGSVP).  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 (input)
               The leading dimension  of  the  array  U.  LDU  >=
               max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.

     V (input) On entry, if JOBV = 'V', V must contain  a  matrix
               V1   (usually   the  unitary  matrix  returned  by
               CGGSVP).  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 (input)
               The leading dimension  of  the  array  V.  LDV  >=
               max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.

     Q (input) On entry, if JOBQ = 'Q', Q must contain  a  matrix
               Q1   (usually   the  unitary  matrix  returned  by
               CGGSVP).  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 (input)
               The leading dimension  of  the  array  Q.  LDQ  >=
               max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.

     WORK (workspace)
               dimension(2*N)

     NCYCLE (output)
               The number of cycles required for convergence.
     INFO (output)
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal value.
               = 1:  the procedure does not converge after  MAXIT
               cycles.