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

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

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

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

BALANC

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


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

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

$$D_j = {\bf SCALE}_j, \;\;\; j = {\bf ILO}, \dots, {\bf IHI}.$$

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.


If INFO is not present and an error occurs, then the program is terminated with an error message.

References: [1] and [17,9,20].