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:

$A(\lambda) \mathbf{x} = 0 , \,$

where x is a vector (the nonlinear "eigenvector") and A is a matrix-valued function of the number $\lambda$ (the nonlinear "eigenvalue"). (More generally, $A(\lambda)$ 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 $\lambda$ (in some domain).

For example, an ordinary linear eigenproblem $B\mathbf{v} = \lambda \mathbf{v}$, where B is a square matrix, corresponds to $A(\lambda) = B - \lambda I$, 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:

$A(\lambda) \mathbf{x} = ( A_2 \lambda^2 + A_1 \lambda + A_0) \mathbf{x} = 0 , \,$

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 $\mathbf{y} = \lambda \mathbf{x}$. In terms of x and y, the quadratic eigenvalue problem becomes:

$\begin{pmatrix} -A_0 & 0 \\ 0 & I \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} = \lambda \begin{pmatrix} A_1 & A_2 \\ I & 0 \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} ,$

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).

References

• Françoise Tisseur and Karl Meerbergen, "The quadratic eigenvalue problem," SIAM Review 43 (2), 235-286 (2001).
• Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," Journal of Computational and Applied Mathematics 123, 35-65 (2000).
• Philippe Guillaume, "Nonlinear eigenproblems," SIAM J. Matrix. Anal. Appl. 20 (3), 575-595 (1999) (link).
• Axel Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM Journal on Numerical Analysis 10 (4), 674-689 (1973).