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Arguments

A
(input/output) REAL or COMPLEX square array, shape $(:,:)$.
On entry, the matrix $A$.
On exit, the contents of A are destroyed.

w
(output) REAL or COMPLEX array, shape $(:)$ with $size$(w) $= size$(A,1).
The computed eigenvalues.
${\it w(:)}$ ::= WR(:), WI(:) $\mid$ W(:),
where
WR(:), WI(:) are of REAL type (for the real and imaginary parts) and W(:) is of COMPLEX type.
Note: If $A$ is real, then a complex-conjugate pair appear consecutively, with the eigenvalue having the positive imaginary part appearing first.

VL
Optional (output) REAL or COMPLEX square array, shape $(:,:)$ with $size$(VL,1) $= size$(A,1).
The left eigenvectors $u_j$ are stored in the columns of VL in the order of their eigenvalues. Each eigenvector is scaled so that the Euclidean norm is 1 and the largest component is real.
Note: If $A$ is real then complex eigenvectors, like their eigenvalues, occur in complex conjugate pairs. The real and imaginary parts of the first eigenvector of the pair are stored in VL$_{:,j}$ and VL$_{:,j+1}$, respectively. Thus a complex conjugate pair is given by

\begin{displaymath}u_j = {\bf VL}_{:,j}\,+\,i\, {\bf VL}_{:,j+1}\,,\;\;\;
u_{j+1} = {\bf VL}_{:,j}\,-\,i\, {\bf VL}_{:,j+1} \end{displaymath}

VR
Optional (output) REAL or COMPLEX square array, shape $(:,:)$ with $size$(VR,1) $= size$(A,1).
The right eigenvectors $v_j$ are stored in the columns of VR in the order of their eigenvalues. Each eigenvector is scaled so that the Euclidean norm is 1 and the largest component is real.
Note: If $A$ is real then complex eigenvectors, like their eigenvalues, occur in complex conjugate pairs. The real and imaginary parts of the first eigenvector of the pair are stored in VR$_{:,j}$ and VR$_{:,j+1}$, respectively. Thus a complex conjugate pair is given by

\begin{displaymath}v_j = {\bf VR}_{:,j}\,+\,i\, {\bf VR}_{:,j+1}\,,\;\;\;
v_{j+1} = {\bf VR}_{:,j}\,-\,i\, {\bf VR}_{:,j+1} \end{displaymath}

BALANC
Optional (input) CHARACTER(LEN=1).
Indicates whether the input matrix should be permuted and/or diagonally scaled.

\begin{optionarg}
\item[{$ =$\ 'N':}] Do not permute or scale;
\item[{$ =$\ 'P...
...ut do not permute;
\item[{$ =$\ 'B':}] Both permute and scale.
\end{optionarg}
Default value: 'N'.

ILO,IHI
Optional (output) INTEGER.
ILO and IHI are determined when $A$ is balanced. The balanced $A_{i,j} = 0$ if $i > j$ and $j = 1, \cdots ,{\bf ILO}-1$ or $i = {\bf IHI}+1,
\cdots ,size({\bf A},1)$.

SCALE
Optional (output) REAL array, shape $(:)$ with $size$(SCALE) $= size$(A,1).
Details of the permutations and scaling factors applied when balancing $A$. If $P_{j}$ is the index of the row and column interchanged with row and column $j$, and $D_j$ is the scaling factor applied to row and column $j$, then

\begin{displaymath}P_j = {\bf SCALE}_j, \;\;\; j = 1, \dots, {\bf ILO}-1 \; \mbox{ and } \; j= {\bf IHI}+1, \dots, n\end{displaymath}


\begin{displaymath}D_j = {\bf SCALE}_j, \;\;\; j = {\bf ILO}, \dots, {\bf IHI}.\end{displaymath}

ABNRM
Optional (output) REAL.
The $l_1$ norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).

RCONDE
Optional (output) REAL array, shape $(:)$ with $size$(RCONDE) $= size$(A,1).
RCONDE$_j$ is the reciprocal condition number of the $j^{th}$ eigenvalue.

RCONDV
Optional (output) REAL array, shape $(:)$, $size$(RCONDV) $= size$(A,1).
RCONDV$_j$ is the reciprocal condition number of the $j^{th}$ right eigenvector.

INFO
Optional (output) INTEGER.

\begin{infoarg}
\item[{$=$\ 0:}] successful exit.
\item[{$<$\ 0:}] if {\bf INF...
...and $i+1:n$\ of {\it w}
contain eigenvalues which have converged.
\end{infoarg}
If INFO is not present and an error occurs, then the program is terminated with an error message.
References: [1] and [17,9,20].
next up previous contents index
Next: Example (from Program LA_GEEVX_EXAMPLE) Up: Standard Nonsymmetric Eigenvalue Problems Previous: Purpose   Contents   Index
Susan Blackford 2001-08-19