Nonlinear eigenproblem

A nonlinear eigenproblem is a generalization of an ordinary eigenproblem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form:

where x is a vector (the nonlinear "eigenvector") and A is a matrix-valued function of the number λ (the nonlinear "eigenvalue"). (More generally, A ( λ ) could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix.) A is usually required to be a holomorphic function of λ (in some domain).

For example, an ordinary linear eigenproblem B v = λ v , where B is a square matrix, corresponds to A ( λ ) = B − λ I , where I is the identity matrix.

One common case is where A is a polynomial matrix, which is called a polynomial eigenvalue problem. In particular, the specific case where the polynomial has degree two is called a quadratic eigenvalue problem, and can be written in the form:

in terms of the constant square matrices A_0,1,2. This can be converted into an ordinary linear generalized eigenproblem of twice the size by defining a new vector y = λ x . In terms of x and y, the quadratic eigenvalue problem becomes:

where I is the identity matrix. More generally, if A is a matrix polynomial of degree d, then one can convert the nonlinear eigenproblem into a linear (generalized) eigenproblem of d times the size.

Besides converting them to ordinary eigenproblems, which only works if A is polynomial, there are other methods of solving nonlinear eigenproblems based on the Jacobi-Davidson algorithm or based on Newton's method (related to inverse iteration).