In this subsection, we will summarize all the available error bounds. Later subsections will provide further details. The reader may also refer to [11].

Bounds for individual eigenvalues and eigenvectors are provided by driver xGEEVX (subsection 2.2.4) or computational routine xTRSNA (subsection 2.3.5). Bounds for clusters of eigenvalues and their associated invariant subspace are provided by driver xGEESX (subsection 2.2.4) or computational routine xTRSEN (subsection 2.3.5).

We let be the `i-th` computed eigenvalue and
an `i-th` true eigenvalue.
Let be the
corresponding computed right eigenvector, and a true right
eigenvector (so ).
If is a subset of the
integers from 1 to `n`, we let denote the average of
the selected eigenvalues:
,
and similarly for . We also let
denote the subspace spanned by ; it is
called a right invariant subspace because if `v` is any vector in then
`Av` is also in . is the corresponding computed subspace.

The algorithms for the nonsymmetric eigenproblem are normwise backward stable:
they compute the exact eigenvalues, eigenvectors and invariant subspaces
of slightly perturbed matrices `A` + `E`, where .
Some of the bounds are stated in terms of and others in
terms of ; one may use to approximate
either quantity.
The code fragment in the previous subsection approximates
by , where
is returned by xGEEVX.

xGEEVX (or xTRSNA) returns two quantities for each
, pair: and .
xGEESX (or xTRSEN) returns two quantities for a selected subset
of eigenvalues: and .
(or ) is a reciprocal condition number for the
computed eigenvalue (or ),
and is referred to as `RCONDE` by xGEEVX (or xGEESX).
(or ) is a reciprocal condition number for
the right eigenvector (or ), and
is referred to as `RCONDV` by xGEEVX (or xGEESX).
The approximate error bounds for eigenvalues, averages of eigenvalues,
eigenvectors, and invariant subspaces
provided in Table 4.5 are
true for sufficiently small **||**`E`**||**, which is why they are called asymptotic.

**Table 4.5:** Asymptotic error bounds for the nonsymmetric eigenproblem

If the problem is ill-conditioned, the asymptotic bounds may only hold
for extremely small **||**`E`**||**. Therefore, in Table 4.6
we also provide global bounds
which are guaranteed to hold for all .

**Table 4.6:** Global error bounds for the nonsymmetric eigenproblem
assuming

We also have the following bound, which is true for all `E`:
all the lie in the union of `n` disks,
where the `i-th` disk is centered at and has
radius . If `k` of these disks overlap,
so that any two points inside the `k` disks can be connected
by a continuous curve lying entirely inside the `k` disks,
and if no larger set of `k` + 1 disks has this property,
then exactly `k` of the lie inside the
union of these `k` disks. Figure 4.1 illustrates
this for a 10-by-10 matrix, with 4 such overlapping unions
of disks, two containing 1 eigenvalue each, one containing 2
eigenvalues, and one containing 6 eigenvalues.

**Figure 4.1:** Bounding eigenvalues inside overlapping disks

Finally, the quantities `s` and `sep` tell use how we can best
(block) diagonalize a matrix `A` by a similarity,
, where each diagonal block
has a selected subset of the eigenvalues of `A`. Such a decomposition
may be useful in computing functions of matrices, for example.
The goal is to choose a `V` with a nearly minimum condition number
which performs this decomposition, since this generally minimizes the error
in the decomposition.
This may be done as follows. Let be
-by-. Then columns through
of `V` span the invariant
subspace of `A` corresponding
to the eigenvalues of ; these columns should be chosen to be any
orthonormal basis of this space (as computed by xGEESX, for example).
Let be the value corresponding to the
cluster of
eigenvalues of , as computed by xGEESX or xTRSEN. Then
, and no other choice of `V` can make
its condition number smaller than [17].
Thus choosing orthonormal
subblocks of `V` gets to within a factor `b` of its minimum
value.

In the case of a real symmetric (or complex Hermitian) matrix,
`s` = 1 and `sep` is the absolute gap, as defined in subsection 4.7.
The bounds in Table 4.5 then reduce to the
bounds in subsection 4.7.

Tue Nov 29 14:03:33 EST 1994