Nonlinear Eigenproblems

This chapter collects several examples of nonlinear eigenvalue problems for which effective algorithms exist. There is no single approach for all nonlinearities, so each example is treated differently.

The simplest kind of nonlinear eigenproblem is the
*quadratic eigenvalue problem* (QEP)

Here is called an eigenvalue, and is a (right) eigenvector. It is called

Higher degree polynomial eigenproblems can be similarly treated. See §9.3.

Finally, §9.4 considers nonlinear eigenproblems
which can be expressed as maximizing a scalar function
over the set of by orthonormal matrices . The simplest
case is maximizing , where is Hermitian and ; the
answer is the largest eigenvalue of . If
,
i.e., the *trace* or sum of diagonal entries of , then
the answer is the sum of the largest eigenvalues of the Hermitian
matrix .
For these problems more effective algorithms are available than the
conjugate-gradient-based optimization scheme presented here, but its
advantage is that it generalizes to far more functions .
We give two more examples.
First, if
are by real symmetric matrices that should
have
a common set of eigenvectors but have been corrupted by noise so that
this is no longer so, then we seek an orthogonal that
minimizes the sums of norms of the offdiagonal entries of all the
.
Second, in quantum mechanical calculations using the
local density approximation, one wishes to maximize
, where is a complicated nonlinear term
representing the energies of electron-electron interactions. The optimization
approach presented here handles quite general functions .