A linear eigenvalue algorithm for the nonlinear eigenvalue problem

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. 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...

Determining bound states in a semiconductor device with contacts using a nonlinear eigenvalue solver

“…In the recent literature we find several subspace based nonlinear eigensolvers [17][18][19][20]. This type of methods have the advantage that they are able to compute several eigenpairs at once.…”

A successive quadratic approximations method for nonlinear eigenvalue problems

“…Nonlinear eigenvalue problem arises naturally in a variety of science and engineering applications, such as the dynamic analysis of structures, the optimization of the acoustic emissions of high speed trains, and the solution of optimal control problems (see [5,6,11,12,14,16,20,26,28,29] and therein). The eigenvalue results, particularly including the special matrices area, are used in developing numerical methods to solve the ordinary and partial differential equations (see [3,4,18,19]).…”

Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems

“…Most linearizations used in the literature can be written in a similar form, i.e., L(λ) = A−λB, where the parts below the first block rows of A and B have the following Kronecker structure: M ⊗ I n and N ⊗ I n , respectively. Note that the pencil (A, B) also covers the dynamically growing linearization pencils used in [10,11,21,9]. The construction of the polynomial or rational approximation of A(λ) can be obtained using results on approximation theory or can be constructed dynamically during the solution process [21,9].…”

“…For methods with dynamically growing linearizations, such as the infinite Arnoldi method [11] and the Newton rational Krylov method [21], the structured starting vector (5.3) corresponds to take f := e 1 , with e 1 the first unit vector. Also in cases where it is inappropriate to choose f as an unit vector, i.e., in Lagrange basis, the structured starting vector (5.3) is advantageous, since we only need to store one vector of dimension n instead of d vectors of dimension n.…”

“…For the general nonlinear case, i.e., nonpolynomial eigenvalue problem, A(λ) is first approximated by a matrix polynomial [6,11,21] or rational matrix polynomial [9,17] before a convenient linearization is applied. Most linearizations used in the literature can be written in a similar form, i.e., L(λ) = A−λB, where the parts below the first block rows of A and B have the following Kronecker structure: M ⊗ I n and N ⊗ I n , respectively.…”