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 and A is a matrix-valued function of the number . A is usually required to be a holomorphic function of .
For example, an ordinary linear eigenproblem, where B is a square matrix, corresponds to, 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 A0,1,2. This can be converted into an ordinary linear generalized eigenproblem of twice the size by defining a new vector. 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 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.
