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. 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 nonli...
Abstract. The Arnoldi method is currently a very popular algorithm to solve large-scale eigenvalue problems. The main goal of this paper is to generalize the Arnoldi method to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem (DEP).The DDE can equivalently be expressed with a linear infinite dimensional operator whose eigenvalues are the solutions to the DEP. We derive a new method by applying the Arnoldi method to the generalized eigenvalue problem (GEP) associated with a spectral discretization of the operator and by exploiting the structure. The result is a scheme where we expand a subspace not only in the traditional way done in the Arnoldi method. The subspace vectors are also expanded with one block of rows in each iteration. More importantly, the structure is such that if the Arnoldi method is started in an appropriate way, it has the (somewhat remarkable) property that it is in a sense independent of the number of discretization points. It is mathematically equivalent to an Arnoldi method with an infinite matrix, corresponding to the limit where we have an infinite number of discretization points.We also show an equivalence with the Arnoldi method in an operator setting. It turns out that with an appropriately defined operator over a space equipped with scalar product with respect to which Chebyshev polynomials are orthonormal, the vectors in the Arnoldi iteration can be interpreted as the coefficients in a Chebyshev expansion of a function. The presented method yields the same Hessenberg matrix as the Arnoldi method applied to the operator.
Abstract. Consider a symmetric matrix A(v) ∈ R n×n depending on a vector v ∈ R n and satisfying the property A(αv) = A(v) for any α ∈ R\{0}. We will here study the problem of finding (λ, v) ∈ R × R n \{0} such that (λ, v) is an eigenpair of the matrix A(v) and we propose a generalization of inverse iteration for eigenvalue problems with this type of eigenvector nonlinearity. The convergence of the proposed method is studied and several convergence properties are shown to be analogous to inverse iteration for standard eigenvalue problems, including local convergence properties. The algorithm is also shown to be equivalent to a particular discretization of an associated ordinary differential equation, if the shift is chosen in a particular way. The algorithm is adapted to a variant of the Schrödinger equation known as the Gross-Pitaevskii equation. We use numerical simulations to illustrate the convergence properties, as well as the efficiency of the algorithm and the adaption.
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