Statistics for Spatial Data.

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

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Linear And Nonlinear Programming

We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.

Some modified matrix eigenvalue problems.

Potential theory on locally compact abelian groups

When using the Arnoldi method for approximating $f(A){\mathbf b}$, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which has been universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numer...

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Efficient and stable Arnoldi restarts for matrix functions based on quadrature

To approximate $f(A){b}$---the action of a matrix function on a vector---by a Krylov subspace method, restarts may become mandatory due to storage requirements for the Arnoldi basis or due to the growing computational complexity of evaluating $f$ on a Hessenberg matrix of growing size. A number of restarting methods have been proposed in the literature in recent years and there has been substantial algorithmic advancement concerning their stability and computational efficiency. However, the question under which circumstances convergence of these methods can be guaranteed has remained largely unanswered. In this paper we consider the class of Stieltjes functions and a related class, which contain important functions like the (inverse) square root and the matrix logarithm. For these classes of functions we present new theoretical results which prove convergence for Hermitian positive definite matrices $A$ and arbitrary restart lengths. We also propose a modification of the Arnoldi approximation which guaran...

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Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices

We consider the task of updating a matrix function $f(A)$ when the matrix $A\in\mathbb{C}^{n \times n}$ is subject to a low-rank modification. In other words, we aim at approximating $f(A+D)-f(A)$ for a matrix $D$ of rank $k \ll n$. The approach proposed in this paper attains efficiency by projecting onto tensorized Krylov subspaces produced by matrix-vector multiplications with $A$ and $A^*$. We prove the approximations obtained from $m$ steps of the proposed methods are exact if $f$ is a polynomial of degree at most $m$ and use this as a basis for proving a variety of convergence results, in particular for the matrix exponential and for Markov functions. We illustrate the performance of our method by considering various examples from network analysis, where our approach can be used to cheaply update centrality and communicability measures.

Low-Rank Updates of Matrix Functions

This thesis deals with Krylov subspace methods for computing the action of a matrix function on a vector. The main result of the thesis is a new integral representation for the error in Arnoldi's method, which is valid for large classes of functions, including holomorphic functions represented via the Cauchy integral formula and Stieltjes functions. 
It is then shown how this error representation can be used to improve upon the standard Arnoldi method: First, a restarted Arnoldi method which evaluates the error function via numerical quadrature is developed and shown to be an improvement over existing methods by numerical experiments on various standard model problems. Afterwards, the method is analyzed theoretically and convergence for Stieltjes functions of Hermitian positive definite matrices is proven (independent of the restart length). 
The second main use of the error representation is the efficient computation of error estimators in Arnoldi's method. An algorithm which computes retrospective error estimates essentially for free (with cost independent of the matrix size) is presented, and it is shown that it is possible to compute guaranteed lower and upper bounds for the error norm for Stieltjes functions of Hermitian positive definite matrices.
 
As an extension of the developed results, it is briefly discussed how most of the results of the thesis can be generalized to extended Krylov subspace methods, with emphasis on the efficient computation of error estimators.

http://elpub.bib.uni-wuppertal.de/servlets/DerivateServlet/Derivate-5346/dc1534.pdf

Restarting and error estimation in polynomial and extended Krylov subspace methods for the approximation of matrix functions

We present an extended Krylov subspace analogue of the two-sided Lanczos method, i.e., a method which, given a nonsingular matrix A and vectors b, c with $\left \langle {{\mathbf {b}},{\mathbf {c}}}\right \rangle \neq 0$
 , constructs bi-orthonormal bases of the extended Krylov subspaces ${\mathcal {E}}_{m}(A,{\mathbf {b}})$
 and ${\mathcal {E}}_{m}(A^{T}\!,{\mathbf {c}})$
 via short recurrences. We investigate the connection of the proposed method to rational moment matching for bilinear forms c
 T
 f(A)b, similar to known results connecting the two-sided Lanczos method to moment matching. Numerical experiments demonstrate the quality of the resulting approximations and the numerical behavior of the new extended Krylov subspace method.

Marcel Schweitzer

Papers

Efficient and stable Arnoldi restarts for matrix functions based on quadrature

Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices

Low-Rank Updates of Matrix Functions

Restarting and error estimation in polynomial and extended Krylov subspace methods for the approximation of matrix functions

A two-sided short-recurrence extended Krylov subspace method for nonsymmetric matrices and its relation to rational moment matching