One way to study the sensitivity of the eigenvalues to perturbations
on 
is to study
the spectrum of 
for various perturbations 
.
We therefore introduce pseudoeigenvalues and pseudospectra of matrices.
The complex number z is defined as a normwise 
-pseudoeigenvalue
of A if z is an eigenvalue of 
with 
such
that 
.
Consequently, 
is an \
-pseudoeigenvalue of 
is the normwise 
-pseudospectrum of A.
It can be shown that
[13].
The level curve 
is the border of the 
-pseudospectrum of A and corresponds to the
set of eigenvalues corresponding to a backward error equal to 
(see [1]). So 
is the set of
all approximate eigenvalues of A associated with a backward error less than
or equal to 
.
The spectral portrait, introduced by Godunov [10], consists
in the representation of
in the complex plane. Its computation requires:
for each point z of the discretization.
be the singular values of the matrix
sorted by decreasing order.
We are required to estimate 
for each point z.
is to use the Singular Value Decomposition method. This method is very reliable but has the drawback to be
too expensive in terms of computational time and storage requirement.
.
is
.
To compute 
which is the smallest eigenvalue (in modulus) of
this matrix, we use a code (see [11]) implementing a restarted
Lanczos method in combination with a shift and invert around 0.
The block structure of this matrix
has been taken into account in order to optimize both the computational
time and the storage requirements.
The code is tuned for sparse matrices stored in Compressed Sparse Row
format and uses routines from SPARSKIT and the Harwell subroutine library
( ME48 for the sparse linear solver).
The computational time can become prohibitive on sequential computers
for large problems (i.e. large matrices 
or fine discretizations).
Parallel computing is the only alternative in such a case to overcome this
problem.