Verified backward error analysis of eigenpairs.
For a square complex (or real) matrix A, this function computes a vector of eigenvalues lambda and a matrix of eigenvectors X in the usual Octave way
[X, L] = eig (A); lambda = diag (L);
and additionally a vector ep with the following property: for each
i there exists a matrix, say A[i], verified to satisfy
max (max (abs (A - A[i]))) <= ep(i) such that
(lambda(i), X(:, i)) is verified to be an exact eigenpair
of A[i]. If A, lambda(i), and X(:, i) are
real then A[i] can be taken real, otherwise it is complex in
general. The maximal value of ep(i) is usually very small (of order
1e-013 to 1e-016), which shows that Octave computes eigenvalues and
eigenvectors with great accuracy.
Based on the inequality (3.13) in J. Rohn, A Handbook of Results on Interval Linear Problems, posted at http://www.cs.cas.cz/~rohn, which also holds for complex eigenpairs (unpublished).
This work was supported by the Czech Republic National Research Program “Information Society”, project 1ET400300415.
See also: eig.
Package: interval