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 $size$(B,1) $= size$(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 $\alpha$.
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 $size$(BETA) $= size$(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

$$\frac{{\bf ALPHAI}_j}{{\bf BETA}_j} = - \frac{{\bf ALPHAI}_{j+1}}{{\bf BETA}_{j+1}}, \;\;i = \sqrt{-1}$$

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 is desired.

SDIM

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

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].