Computing

To explain ** s_{i}**, ,
,
and
in Table 4.7 and Table 4.8,
we need to introduce a condition number for an individual eigenvalue,
block diagonalization of a matrix pair and
the separation of two matrix pairs.

Let
be a simple eigenvalue of **( A, B)** with
left and right
eigenvectors

We assume that the matrix pair **( A, B)** is in the generalized Schur form.
Consider a cluster of

An equivalence transformation that block-diagonalizes **( A, B)**
can be expressed as

Equation (4.14) is called a

and
for the eigenvalues of
**( A_{11}, B_{11})**
are defined as

The *separation* of two matrix pairs
**( A_{11}, B_{11})** and

Notice that
does not generally equal
(unless ** A_{ii}** and

In the error bounds of Tables 4.7 and 4.8,
and
denote
,
where
**( A_{11}, B_{11})** corresponds
to an individual eigenvalue pair
and a cluster of eigenvalues
defined by the subset ,
respectively. Similar notation is used
for
and
.
xGGESX reports estimates of
and
in

From a matrix representation of (4.14) it is possible to formulate an
exact expression of
as

and is the Kronecker product. A method based directly on forming

We instead compute an estimate of
as the reciprocal value of an
estimate of
,
where ** Z_{u}** is the matrix representation of the generalized Sylvester
operator. It is possible to estimate
by solving
generalized Sylvester equations
in triangular form.
We provide both Frobenius norm and
one norm
estimates [74].
The one norm estimate makes the condition estimation uniform with the
nonsymmetric eigenvalue problem. The Frobenius norm estimate
offers a low cost and equally reliable estimator.
The one norm estimate is a factor 3 to 10 times more
expensive [74]. From
the definition of
(4.17) we see that
estimates can be computed by using the algorithms for
estimating .

**Frobenius norm estimate**: From
the ** Z_{u}x = b** representation of the
generalized Sylvester equation (4.14) we get a
lower bound on
:

Methods for computing such

**One norm norm estimate**: From the relationship

The cost for computing this bound is roughly equal to the number of steps in the reverse communication times the cost for one generalized Sylvester solve.

The expert driver routines xGGEVX and xGGESX compute the Frobenius
norm estimate (4.20).
The routine xTGSNA also computes the Frobenius
norm estimate (4.20) of
and .
The routine xTGSEN optionally computes the Frobenius norm estimate
(4.20) or the one norm estimate (4.22).
The choice of estimate is controlled by the input parameter `IJOB`.