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

.in +1c
.ti -1c
.RI "subroutine \fBstgevc\fP (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info)"
.br
.RI "\fBSTGEVC\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine stgevc (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( lds, * ) s, integer lds, real, dimension( ldp, * ) p, integer ldp, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer info)"

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

.PP
.nf
!>
!> STGEVC computes some or all of the right and/or left eigenvectors of
!> a pair of real matrices (S,P), where S is a quasi-triangular matrix
!> and P is upper triangular\&.  Matrix pairs of this type are produced by
!> the generalized Schur factorization of a matrix pair (A,B):
!>
!>    A = Q*S*Z**T,  B = Q*P*Z**T
!>
!> as computed by SGGHRD + SHGEQZ\&.
!>
!> The right eigenvector x and the left eigenvector y of (S,P)
!> corresponding to an eigenvalue w are defined by:
!>
!>    S*x = w*P*x,  (y**H)*S = w*(y**H)*P,
!>
!> where y**H denotes the conjugate transpose of y\&.
!> The eigenvalues are not input to this routine, but are computed
!> directly from the diagonal blocks of S and P\&.
!>
!> This routine returns the matrices X and/or Y of right and left
!> eigenvectors of (S,P), or the products Z*X and/or Q*Y,
!> where Z and Q are input matrices\&.
!> If Q and Z are the orthogonal factors from the generalized Schur
!> factorization of a matrix pair (A,B), then Z*X and Q*Y
!> are the matrices of right and left eigenvectors of (A,B)\&.
!>
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP 
.PP
.nf
!>          SIDE is CHARACTER*1
!>          = 'R': compute right eigenvectors only;
!>          = 'L': compute left eigenvectors only;
!>          = 'B': compute both right and left eigenvectors\&.
!> 
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
!>          HOWMNY is CHARACTER*1
!>          = 'A': compute all right and/or left eigenvectors;
!>          = 'B': compute all right and/or left eigenvectors,
!>                 backtransformed by the matrices in VR and/or VL;
!>          = 'S': compute selected right and/or left eigenvectors,
!>                 specified by the logical array SELECT\&.
!> 
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
!>          SELECT is LOGICAL array, dimension (N)
!>          If HOWMNY='S', SELECT specifies the eigenvectors to be
!>          computed\&.  If w(j) is a real eigenvalue, the corresponding
!>          real eigenvector is computed if SELECT(j) is \&.TRUE\&.\&.
!>          If w(j) and w(j+1) are the real and imaginary parts of a
!>          complex eigenvalue, the corresponding complex eigenvector
!>          is computed if either SELECT(j) or SELECT(j+1) is \&.TRUE\&.,
!>          and on exit SELECT(j) is set to \&.TRUE\&. and SELECT(j+1) is
!>          set to \&.FALSE\&.\&.
!>          Not referenced if HOWMNY = 'A' or 'B'\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrices S and P\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIS\fP 
.PP
.nf
!>          S is REAL array, dimension (LDS,N)
!>          The upper quasi-triangular matrix S from a generalized Schur
!>          factorization, as computed by SHGEQZ\&.
!> 
.fi
.PP
.br
\fILDS\fP 
.PP
.nf
!>          LDS is INTEGER
!>          The leading dimension of array S\&.  LDS >= max(1,N)\&.
!> 
.fi
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.br
\fIP\fP 
.PP
.nf
!>          P is REAL array, dimension (LDP,N)
!>          The upper triangular matrix P from a generalized Schur
!>          factorization, as computed by SHGEQZ\&.
!>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
!>          of S must be in positive diagonal form\&.
!> 
.fi
.PP
.br
\fILDP\fP 
.PP
.nf
!>          LDP is INTEGER
!>          The leading dimension of array P\&.  LDP >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          VL is REAL array, dimension (LDVL,MM)
!>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
!>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
!>          of left Schur vectors returned by SHGEQZ)\&.
!>          On exit, if SIDE = 'L' or 'B', VL contains:
!>          if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
!>          if HOWMNY = 'B', the matrix Q*Y;
!>          if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
!>                      SELECT, stored consecutively in the columns of
!>                      VL, in the same order as their eigenvalues\&.
!>
!>          A complex eigenvector corresponding to a complex eigenvalue
!>          is stored in two consecutive columns, the first holding the
!>          real part, and the second the imaginary part\&.
!>
!>          Not referenced if SIDE = 'R'\&.
!> 
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
!>          LDVL is INTEGER
!>          The leading dimension of array VL\&.  LDVL >= 1, and if
!>          SIDE = 'L' or 'B', LDVL >= N\&.
!> 
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
!>          VR is REAL array, dimension (LDVR,MM)
!>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
!>          contain an N-by-N matrix Z (usually the orthogonal matrix Z
!>          of right Schur vectors returned by SHGEQZ)\&.
!>
!>          On exit, if SIDE = 'R' or 'B', VR contains:
!>          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
!>          if HOWMNY = 'B' or 'b', the matrix Z*X;
!>          if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
!>                      specified by SELECT, stored consecutively in the
!>                      columns of VR, in the same order as their
!>                      eigenvalues\&.
!>
!>          A complex eigenvector corresponding to a complex eigenvalue
!>          is stored in two consecutive columns, the first holding the
!>          real part and the second the imaginary part\&.
!>
!>          Not referenced if SIDE = 'L'\&.
!> 
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
!>          LDVR is INTEGER
!>          The leading dimension of the array VR\&.  LDVR >= 1, and if
!>          SIDE = 'R' or 'B', LDVR >= N\&.
!> 
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
!>          MM is INTEGER
!>          The number of columns in the arrays VL and/or VR\&. MM >= M\&.
!> 
.fi
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.br
\fIM\fP 
.PP
.nf
!>          M is INTEGER
!>          The number of columns in the arrays VL and/or VR actually
!>          used to store the eigenvectors\&.  If HOWMNY = 'A' or 'B', M
!>          is set to N\&.  Each selected real eigenvector occupies one
!>          column and each selected complex eigenvector occupies two
!>          columns\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (6*N)
!> 
.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\&.
!>          > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
!>                eigenvalue\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

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

.PP
.nf
!>
!>  Allocation of workspace:
!>  ---------- -- ---------
!>
!>     WORK( j ) = 1-norm of j-th column of A, above the diagonal
!>     WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
!>     WORK( 2*N+1:3*N ) = real part of eigenvector
!>     WORK( 3*N+1:4*N ) = imaginary part of eigenvector
!>     WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
!>     WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
!>
!>  Rowwise vs\&. columnwise solution methods:
!>  ------- --  ---------- -------- -------
!>
!>  Finding a generalized eigenvector consists basically of solving the
!>  singular triangular system
!>
!>   (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)
!>
!>  Consider finding the i-th right eigenvector (assume all eigenvalues
!>  are real)\&. The equation to be solved is:
!>       n                   i
!>  0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,\&. \&. \&.,1
!>      k=j                 k=j
!>
!>  where  C = (A - w B)  (The components v(i+1:n) are 0\&.)
!>
!>  The  method is:
!>
!>  (1)  v(i) := 1
!>  for j = i-1,\&. \&. \&.,1:
!>                          i
!>      (2) compute  s = - sum C(j,k) v(k)   and
!>                        k=j+1
!>
!>      (3) v(j) := s / C(j,j)
!>
!>  Step 2 is sometimes called the  step, since it is an
!>  inner product between the j-th row and the portion of the eigenvector
!>  that has been computed so far\&.
!>
!>  The  method consists basically in doing the sums
!>  for all the rows in parallel\&.  As each v(j) is computed, the
!>  contribution of v(j) times the j-th column of C is added to the
!>  partial sums\&.  Since FORTRAN arrays are stored columnwise, this has
!>  the advantage that at each step, the elements of C that are accessed
!>  are adjacent to one another, whereas with the rowwise method, the
!>  elements accessed at a step are spaced LDS (and LDP) words apart\&.
!>
!>  When finding left eigenvectors, the matrix in question is the
!>  transpose of the one in storage, so the rowwise method then
!>  actually accesses columns of A and B at each step, and so is the
!>  preferred method\&.
!> 
.fi
.PP
 
.RE
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Definition at line \fB293\fP of file \fBstgevc\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.