Loading...
Searching...
No Matches

## stgsja()

subroutine stgsja | ( | character | jobu,
|  | character | jobv,
|  | character | jobq,
|  | integer | m,
|  | integer | p,
|  | integer | n,
|  | integer | k,
|  | integer | l,
|  | real, dimension( lda, * ) | a,
|  | integer | lda,
|  | real, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | real | tola,
|  | real | tolb,
|  | real, dimension( * ) | alpha,
|  | real, dimension( * ) | beta,
|  | real, dimension( ldu, * ) | u,
|  | integer | ldu,
|  | real, dimension( ldv, * ) | v,
|  | integer | ldv,
|  | real, dimension( ldq, * ) | q,
|  | integer | ldq,
|  | real, dimension( * ) | work,
|  | integer | ncycle,
|  | integer | info )

STGSJA

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

Purpose:

      !>
      !> STGSJA 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 SGGSVP
      !> 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.
      !>

Parameters

  [in]

  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.
      !>

  [in]

  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.
      !>

  [in]

  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.
      !>

  [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 STGSJA.
      !>          See Further Details.
      !>

  [in,out]

  A

      !>          A is REAL 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 REAL 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 REAL
      !>

  [in]

  TOLB

      !>          TOLB is REAL
      !>
      !>          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)*MACHEPS,
      !>              TOLB = max(P,N)*norm(B)*MACHEPS.
      !>

  [out]

  ALPHA

      !>          ALPHA is REAL array, dimension (N)
      !>

  [out]

  BETA

      !>          BETA is REAL 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 REAL array, dimension (LDU,M)
      !>          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
      !>          the orthogonal matrix returned by SGGSVP).
      !>          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.
      !>

  [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 REAL array, dimension (LDV,P)
      !>          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
      !>          the orthogonal matrix returned by SGGSVP).
      !>          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.
      !>

  [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 REAL array, dimension (LDQ,N)
      !>          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
      !>          the orthogonal matrix returned by SGGSVP).
      !>          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.
      !>

  [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 REAL 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.

Further Details:

      !>
      !>  STGSJA 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.
      !>