154,350 research outputs found
On the condition numbers of a multiple generalized eigenvalue
For standard eigenvalue problems, a closed-form expression for the condition
numbers of a multiple eigenvalue is known. In particular, they are uniformly 1
in the Hermitian case, and generally take different values in the non-Hermitian
case. We consider the generalized eigenvalue problem and identify the condition
numbers of a multiple eigenvalue. Our main result is that a multiple eigenvalue
generally has multiple condition numbers, even in the Hermitian definite case.
The condition numbers are characterized in terms of the singular values of the
outer product of the corresponding left and right eigenvectors
An integral method for solving nonlinear eigenvalue problems
We propose a numerical method for computing all eigenvalues (and the
corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that
lie within a given contour in the complex plane. The method uses complex
integrals of the resolvent operator, applied to at least column vectors,
where is the number of eigenvalues inside the contour. The theorem of
Keldysh is employed to show that the original nonlinear eigenvalue problem
reduces to a linear eigenvalue problem of dimension .
No initial approximations of eigenvalues and eigenvectors are needed. The
method is particularly suitable for moderately large eigenvalue problems where
is much smaller than the matrix dimension. We also give an extension of the
method to the case where is larger than the matrix dimension. The
quadrature errors caused by the trapezoid sum are discussed for the case of
analytic closed contours. Using well known techniques it is shown that the
error decays exponentially with an exponent given by the product of the number
of quadrature points and the minimal distance of the eigenvalues to the
contour
The nonconforming virtual element method for eigenvalue problems
We analyse the nonconforming Virtual Element Method (VEM) for the
approximation of elliptic eigenvalue problems. The nonconforming VEM allow to
treat in the same formulation the two- and three-dimensional case.We present
two possible formulations of the discrete problem, derived respectively by the
nonstabilized and stabilized approximation of the L^2-inner product, and we
study the convergence properties of the corresponding discrete eigenvalue
problem. The proposed schemes provide a correct approximation of the spectrum,
in particular we prove optimal-order error estimates for the eigenfunctions and
the usual double order of convergence of the eigenvalues. Finally we show a
large set of numerical tests supporting the theoretical results, including a
comparison with the conforming Virtual Element choice
The Bramble-Pasciak preconditioner for saddle point problems
The Bramble-Pasciak Conjugate Gradient method is a well known tool to solve linear systems in saddle point form. A drawback of this method in order to ensure applicability of Conjugate Gradients is the need for scaling the preconditioner which typically involves the solution of an eigenvalue problem. Here, we introduce a modified preconditioner and inner product which without scaling enable the use of a MINRES variant and can be used for the simplified Lanczos process. Furthermore, the modified preconditioner and inner product can be combined with the original Bramble-Pasciak setup to give new preconditioners and inner products. We undermine the new methods by showing numerical experiments for Stokes problems
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