.TH "TESTING/EIG/dget52.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
TESTING/EIG/dget52.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBdget52\fP (left, n, a, lda, b, ldb, e, \fBlde\fP, alphar, alphai, beta, work, result)"
.br
.RI "\fBDGET52\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine dget52 (logical left, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( \fBlde\fP, * ) e, integer lde, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( * ) work, double precision, dimension( 2 ) result)"

.PP
\fBDGET52\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DGET52  does an eigenvector check for the generalized eigenvalue
 problem\&.

 The basic test for right eigenvectors is:

                           | b(j) A E(j) -  a(j) B E(j) |
         RESULT(1) = max   -------------------------------
                      j    n ulp max( |b(j) A|, |a(j) B| )

 using the 1-norm\&.  Here, a(j)/b(j) = w is the j-th generalized
 eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
 generalized eigenvalue of m A - B\&.

 For real eigenvalues, the test is straightforward\&.  For complex
 eigenvalues, E(j) and a(j) are complex, represented by
 Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp\&., so the test for that
 eigenvector becomes

                 max( |Wr|, |Wi| )
     --------------------------------------------
     n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )

 where

     Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)

     Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)

                         T   T  _
 For left eigenvectors, A , B , a, and b  are used\&.

 DGET52 also tests the normalization of E\&.  Each eigenvector is
 supposed to be normalized so that the maximum 'absolute value'
 of its elements is 1, where in this case, 'absolute value'
 of a complex value x is  |Re(x)| + |Im(x)| ; let us call this
 maximum 'absolute value' norm of a vector v  M(v)\&.
 if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
 vector\&.  The normalization test is:

         RESULT(2) =      max       | M(v(j)) - 1 | / ( n ulp )
                    eigenvectors v(j)
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fILEFT\fP 
.PP
.nf
          LEFT is LOGICAL
          =\&.TRUE\&.:  The eigenvectors in the columns of E are assumed
                    to be *left* eigenvectors\&.
          =\&.FALSE\&.: The eigenvectors in the columns of E are assumed
                    to be *right* eigenvectors\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The size of the matrices\&.  If it is zero, DGET52 does
          nothing\&.  It must be at least zero\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA, N)
          The matrix A\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of A\&.  It must be at least 1
          and at least N\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is DOUBLE PRECISION array, dimension (LDB, N)
          The matrix B\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of B\&.  It must be at least 1
          and at least N\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is DOUBLE PRECISION array, dimension (LDE, N)
          The matrix of eigenvectors\&.  It must be O( 1 )\&.  Complex
          eigenvalues and eigenvectors always come in pairs, the
          eigenvalue and its conjugate being stored in adjacent
          elements of ALPHAR, ALPHAI, and BETA\&.  Thus, if a(j)/b(j)
          and a(j+1)/b(j+1) are a complex conjugate pair of
          generalized eigenvalues, then E(,j) contains the real part
          of the eigenvector and E(,j+1) contains the imaginary part\&.
          Note that whether E(,j) is a real eigenvector or part of a
          complex one is specified by whether ALPHAI(j) is zero or not\&.
.fi
.PP
.br
\fILDE\fP 
.PP
.nf
          LDE is INTEGER
          The leading dimension of E\&.  It must be at least 1 and at
          least N\&.
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
          ALPHAR is DOUBLE PRECISION array, dimension (N)
          The real parts of the values a(j) as described above, which,
          along with b(j), define the generalized eigenvalues\&.
          Complex eigenvalues always come in complex conjugate pairs
          a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent
          elements in ALPHAR, ALPHAI, and BETA\&.  Thus, if the j-th
          and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1)
          is assumed to be equal to ALPHAR(j)/BETA(j)\&.
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          ALPHAI is DOUBLE PRECISION array, dimension (N)
          The imaginary parts of the values a(j) as described above,
          which, along with b(j), define the generalized eigenvalues\&.
          If ALPHAI(j)=0, then the eigenvalue is real, otherwise it
          is part of a complex conjugate pair\&.  Complex eigenvalues
          always come in complex conjugate pairs a(j)/b(j) and
          a(j+1)/b(j+1), which are stored in adjacent elements in
          ALPHAR, ALPHAI, and BETA\&.  Thus, if the j-th and (j+1)-st
          eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to
          be equal to  -ALPHAI(j)/BETA(j)\&.  Also, nonzero values in
          ALPHAI are assumed to always come in adjacent pairs\&.
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is DOUBLE PRECISION array, dimension (N)
          The values b(j) as described above, which, along with a(j),
          define the generalized eigenvalues\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (N**2+N)
.fi
.PP
.br
\fIRESULT\fP 
.PP
.nf
          RESULT is DOUBLE PRECISION array, dimension (2)
          The values computed by the test described above\&.  If A E or
          B E is likely to overflow, then RESULT(1:2) is set to
          10 / ulp\&.
.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
.PP

.PP
Definition at line \fB197\fP of file \fBdget52\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.