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:

${\displaystyle {\begin{pmatrix}A_{0}&A_{1}\\0&I\end{pmatrix}}{\begin{pmatrix}\mathbf {x} \\\mathbf {y} \end{pmatrix}}=\lambda {\begin{pmatrix}0&-A_{2}\\I&0\end{pmatrix}}{\begin{pmatrix}\mathbf {x} \\\mathbf {y} \end{pmatrix}},}$

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).