A linear eigenvalue algorithm for the nonlinear eigenvalue problem

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. Our first important result is a characterization of a general nonlinear eigenvalue problem (NEP) as a standard but infinite dimensional eigenvalue problem involving an integration operator denoted B. In this paper we present a new algorithm equivalent to the Arnoldi method for the operator B. Although the abstract construction is infinite dimensional, it turns out that we can carry out the iteration in an exact way (without approximation) by using only standard linear algebra operations involving matrices (not operators). This is achieved by working with coefficients in a basis of scalar functions, typically polynomials. Due to the fact that the constructed method has a complete equivalence with the standard Arnoldi method, it also inherits many of its attractive properties. Another somewhat unexpected consequence of the construction is that the matrix of basis vectors should be expanded not only in the way done in standard Arnoldi. We expand the matrix of basis vectors not only with a column to the right, but also a block row below. We also show that the method can be interpreted as the standard Arnoldi method if applied to the generalized eigenvalue problem resulting from the spectral discretization of the operator. With this equivalence we reach a recommendation on how the scalar product should be chosen for an important class of nonlinear eigenvalue problems. Noname manuscript No. (will be inserted by the editor) A linear eigenvalue algorithm for the nonlinear eigenvalue problem Elias Jarlebring · Wim Michiels · Karl Meerbergen Received: date / Accepted: date Abstract The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. Our first important result is a characterization of a general nonlinear eigenvalue problem (NEP) as a standard but infinite dimensional eigenvalue problem involving an integration operator denoted B. In this paper we present a new algorithm equivalent to the Arnoldi method for the operator B. Although the abstract construction is infinite dimensional, it turns out that we can carry out the iteration in an exact way (without approximation) by using only standard linear algebra operations involving matrices (not operators). This is achieved by working with coefficients in a basis of scalar functions, typically polynomials. Due to the fact that the constructed method has a complete equivalence with the standard Arnoldi method, it also inherits many of its attractive properties. Another somewhat unexpected consequence of the construction is that the matrix of basis vectors should be expanded not only in the way done in standard Arnoldi. We expand the matrix of basis vectors not only with a column to the right, but also a block row below. We also show that the method can be interpreted as the standard Arnoldi method if applied to the generalized eigenvalue problem resulting from the spectral discretization of the operator. With this equivalence we reach a recommendation on how the scalar product should be chosen for an important class of nonlinear eigenvalue problems.