Dickson polynomial

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:

$\left(\begin{array}{cc}-A_0& 0\\ 0& I\end{array}\right)\left(\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right)=\lambda \left(\begin{array}{cc}{A}_1& A_2\\ I& 0\end{array}\right)\left(\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right),$

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).
• Axel Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM Journal on Numerical Analysis 10 (4), 674-689 (1973).