Arguments

A

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

B

(input/output) REAL or COMPLEX square array, shape $(:,:)$ with $size({\bf B},1) = size({\bf A},1)$.
On entry, the matrix $B$.
On exit, the matrix $T$.

alpha

(output) REAL or COMPLEX array, shape $(:)$ with $size({\it alpha}) = size({\bf A},1)$.
The values of $\alpha$.
${\it alpha(:)}$ ::= ALPHAR(:), ALPHAI(:) $\mid$ ALPHA(:),
where
ALPHAR(:), ALPHAI(:) are of REAL type (for the real and imaginary parts) and ALPHA(:) is of COMPLEX type.

BETA

(output) REAL or COMPLEX array, shape $(:)$ with $size({\bf BETA}) = size({\bf A},1)$.
The values of $\beta$.
Note: The generalized eigenvalues of the pair $( A, B )$ are the scalars $\lambda_j = \alpha_j/\beta_j$. These quotients may easily over- or underflow, and $\beta_j$ may even be zero. Thus, the user should avoid computing them naively.
Note: If A and B are real then complex eigenvalues occur in complex conjugate pairs. Each pair is stored consecutively. Thus a complex conjugate pair is given by

$$\begin{array}{l} \lambda_j = ( {\bf ALPHAR}_j\,+\, i\, {\bf ... ...+1}\,+\, i\, {\bf ALPHAI}_{j+1})/{\bf BETA}_{j+1} \end{array}$$

where

$$\frac{{\bf ALPHAI}_j}{{\bf BETA}_j} = - \frac{{\bf ALPHAI}_{j+1}}{{\bf BETA}_{j+1}}$$

VSL

Optional (output) REAL or COMPLEX square array, shape $(:,:)$ with $size$(VSL,1) $= size$(A,1).
The left Schur vectors.

VSR

Optional (output) REAL or COMPLEX square array, shape $(:,:)$ with $size$(VSR,1) $= size$(A,1).
The right Schur vectors.

SELECT

Optional (input) LOGICAL FUNCTION

LOGICAL FUNCTION SELECT( alpha$_j$, BETA$_j$ )
type(wp), INTENT(IN) :: alpha$_j$, BETA$_j$
where
type ::= REAL $\mid$ COMPLEX
wp ::= KIND(1.0) $\mid$ KIND(1.0D0)
alpha$_j$ ::= ALPHAR$_j$, ALPHAI$_j$ $\mid$ ALPHA$_j$

Note: Select must be present if SDIM, RCONDE and RCONDF are desired.

SDIM

Optional (output) INTEGER.
The number of eigenvalues (after sorting) for which SELECT $=$ .TRUE. (If $A$ and $B$ are real, complex conjugate pairs for which SELECT $=$ .TRUE. for either eigenvalue count as 2).

RCONDE

Optional (output) REAL array, shape $(:)$ with $size$(RCONDE) $= 2$.
The reciprocal condition numbers for the average of the selected eigenvalues.

RCONDV

Optional (output) REAL array, shape $(:)$ with $size$(RCONDV) $= 2$.
The reciprocal condition numbers for the left and right deflating subspaces corresponding to the selected eigenvalues.

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 [17,9,20,21].