In this paper, we have sought to perform a comprehensive review on the state-of-the-art research of eigenvalue and eigenvector derivatives and their important engineering applications. Major existing methods for different eigenvalue problems have been introduced and theoretically discussed, together with...
we use QMC methods to efficiently compute the expectations on each level; we exploit the smoothness in parameter space and reuse the eigenvector from a nearby QMC point to reduce the number of iterations of the eigensolver; and we utilize a two-grid discretization scheme to...
(1.31) and (1.84). An n×n matrix has n eigenvalues λi, some or all of which may coincide. If we insert, for example, an eigenvalue λ1 into (C.33), we get the equation (C.35)ax=λ1x which has a non-trivial solution for x. The solution is called the eigenvector ...
Eigenvalue l unchanged and eigenvector transformed to 1 V x - Same for left eigenvector 1 y U y - Prefer unitary , U V for numerical stability GENERALIZED SCHUR FORM Note that ( ) ( ) ( ) ( ) ( ) * det det det det U A B V U V A B l l - = - ...
If Mj are symmetric, i.e., M(λ) is symmetric, then A and B are symmetric. If M(λ)v=0, then [vT,λvT,…,λk−1vT]T is an eigenvector of A−λB. The proof is based on the following argument. A pair (λ,x) that fulfill M(λ)x=0 defines an eigenpair of the ...
We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div(A ∇ u)+Vu+f(u 2)u=λ u, ‖u‖L2=1. We focus in particular on the Fourier spectral approximation (for...
Peters and Wilkinson considered the closely related algorithm that consists of applying Newton's method, followed by a 2-norm normalization, to the nonlinear system of equations consisting of the eigenvalue-eigenvector equation and an equation requiring the eigenvector to have the square of its 2-...
There are problems for which only selected eigenvalues and associated eigenvectors are needed. If a real matrix has a simple eigenvalue of largest magnitude, the sequence xk=Axk–1 converges to the eigenvector corresponding to the largest eigenvalue, where x0 is a normalized initial approximation,...
A 'Nonzero Eigenvalue' is a scalar associated with an eigenvector of a square matrix that is not equal to zero. It represents the factor by which the eigenvector is scaled when multiplied by the matrix. AI generated definition based on: International Encyclopedia of the Social & Behavioral ...
Tensor eigenvalue problems have gained special attention in the realm of numerical multilinear algebra, and they have a wide range in practice; see [, , –]. For in- stance, we can use the smallest H-eigenvalues of tensors to determine their positive (semi)definiteness, ...