LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
ztgsja.f File Reference

Functions/Subroutines

subroutine ztgsja (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)
 ZTGSJA More...
 

Function/Subroutine Documentation

subroutine ztgsja ( character  JOBU,
character  JOBV,
character  JOBQ,
integer  M,
integer  P,
integer  N,
integer  K,
integer  L,
complex*16, dimension( lda, * )  A,
integer  LDA,
complex*16, dimension( ldb, * )  B,
integer  LDB,
double precision  TOLA,
double precision  TOLB,
double precision, dimension( * )  ALPHA,
double precision, dimension( * )  BETA,
complex*16, dimension( ldu, * )  U,
integer  LDU,
complex*16, dimension( ldv, * )  V,
integer  LDV,
complex*16, dimension( ldq, * )  Q,
integer  LDQ,
complex*16, dimension( * )  WORK,
integer  NCYCLE,
integer  INFO 
)

ZTGSJA

Download ZTGSJA + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 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.
Parameters
[in]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.
[in]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.
[in]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.
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.
[in]P
          P is INTEGER
          The number of rows of the matrix B.  P >= 0.
[in]N
          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.
[in]K
          K is INTEGER
[in]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.
[in,out]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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).
[in,out]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.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).
[in]TOLA
          TOLA is DOUBLE PRECISION
[in]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.
[out]ALPHA
          ALPHA is DOUBLE PRECISION array, dimension (N)
[out]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.
[in,out]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.
[in]LDU
          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.
[in,out]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.
[in]LDV
          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.
[in,out]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.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.
[out]WORK
          WORK is COMPLEX*16 array, dimension (2*N)
[out]NCYCLE
          NCYCLE is INTEGER
          The number of cycles required for convergence.
[out]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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
November 2011
Further Details:
  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.