.TH "SRC/stgsja.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/stgsja.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBstgsja\fP (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)"
.br
.RI "\fBSTGSJA\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

.PP
\fBSTGSJA\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIJOBV\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIJOBQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          M is INTEGER
!>          The number of rows of the matrix A\&.  M >= 0\&.
!> 
.fi
.PP
.br
\fIP\fP 
.PP
.nf
!>          P is INTEGER
!>          The number of rows of the matrix B\&.  P >= 0\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The number of columns of the matrices A and B\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIK\fP 
.PP
.nf
!>          K is INTEGER
!> 
.fi
.PP
.br
\fIL\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
!>          LDA is INTEGER
!>          The leading dimension of the array A\&. LDA >= max(1,M)\&.
!> 
.fi
.PP
.br
\fIB\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
!>          LDB is INTEGER
!>          The leading dimension of the array B\&. LDB >= max(1,P)\&.
!> 
.fi
.PP
.br
\fITOLA\fP 
.PP
.nf
!>          TOLA is REAL
!> 
.fi
.PP
.br
\fITOLB\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
!>          ALPHA is REAL array, dimension (N)
!> 
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIU\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
!>          LDU is INTEGER
!>          The leading dimension of the array U\&. LDU >= max(1,M) if
!>          JOBU = 'U'; LDU >= 1 otherwise\&.
!> 
.fi
.PP
.br
\fIV\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
!>          LDV is INTEGER
!>          The leading dimension of the array V\&. LDV >= max(1,P) if
!>          JOBV = 'V'; LDV >= 1 otherwise\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>          The leading dimension of the array Q\&. LDQ >= max(1,N) if
!>          JOBQ = 'Q'; LDQ >= 1 otherwise\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (2*N)
!> 
.fi
.PP
.br
\fINCYCLE\fP 
.PP
.nf
!>          NCYCLE is INTEGER
!>          The number of cycles required for convergence\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.RE
.PP
.PP
.nf
!>  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
.PP
 
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  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\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB375\fP of file \fBstgsja\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.