Miroslav S. Pranić

• University of Banja Luka

This paper considers the computation of approximations of matrix functionals of form F(A):=vTf(A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. The functional F(A) is approximated by a rational Gauss quadrature rule with poles on the negative real axis (or part thereof) in the complex plane,...

This paper is concerned with computing approximations of matrix functionals of the form F(A):=vTf(A)v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(A):={{{\varvec{v}...

This paper is concerned with the approximation of matrix functionals of the form wTf(A)v, where \(A\in \mathbb {R}^{n\times n}\) is a large nonsymmetric matrix, \(\boldsymbol {w},\boldsymbol {v}\in \mathbb {R}^{n}\), and f is a function such that f(A) is well defined. We derive Gauss–Laurent quadrature rules for the approximation of these functiona...

Many functionals of a large symmetric matrix of interest in science and engineering can be expressed as a Stieltjes integral with a measure supported on the real axis. These functionals can be approximated by quadrature rules. Golub and Meurant proposed a technique for computing upper and lower error bounds for Stieltjes integrals with integrands w...

The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian)...

The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will describe such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian...

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel,...

Gauss quadrature can be naturally generalized in order to approximate quasi-definite linear functionals, where the interconnections with (formal) orthogonal polynomials, (complex) Jacobi matrices, and the Lanczos algorithm are analogous to those in the positive definite case. In this survey we review these relationships with giving references to th...

The rational Arnoldi process is a popular method for the computation of a few eigenvalues of a large non-Hermitian matrix A∈Cn×n and for the approximation of matrix functions. The method is particularly attractive when the rational functions that determine the process have only few distinct poles zj∈C, because then few factorizations of matrices of...

Gauss quadrature can be formulated as a method for approximating positive-definite linear functionals. Its mathematical context is extremely rich, with orthogonal polynomials, continued fractions and Padé approximation on one (functional analytic or approximation theory) side, and the method of moments,(real) Jacobi matrices, spectral decomposition...

Gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis. Laurie proposed an -point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated -point Gauss quadrature rule for all polynomials of degree up to . This rule is referr...

The existence of (standard) Gauss quadrature rules with respect to a nonnegative measure dμ with support on the real axis easily can be shown with the aid of orthogonal polynomials with respect to this measure. Efficient algorithms for computing the nodes and weights of an n-point Gauss rule use the n × n symmetric tridiagonal matrix determined by...

It has been shown that approximate extended Krylov subspaces can be computed, under certain assumptions, without any explicit inversion or system solves. Instead, the vectors spanning the extended Krylov space are retrieved in an implicit way, via unitary similarity transformations, from an enlarged Krylov subspace. In this paper this approach is g...

It is well known that members of families of polynomials, that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis, satisfy a three-term recursion relation. Analogous recursion formulas are available for orthogonal Laurent polynomials with a pole at the origin. This paper investigates recursion relat...

It is shown that extended Krylov subspaces-under some assumptions-can be computed approximately without any explicit inversion or system solves involved. Instead, the necessary computations are done in an implicit way using the information from an enlarged standard Krylov subspace. For both the classical and extended Krylov spaces, the matrices cap...

We continue with the study of the kernels Kn(z) in the remainder terms Rn(f) of the Gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at ∓1 and a sum of semi-axes ρ > 1. The weight function w of Bernstein–Szegő type here isw(t)≡wγ(-1/2)(t)=11-t2·1-4γ(1+γ)2t2-1,t∈(-1,1),γ∈(-1,0).Sufficient conditions are foun...

We consider quadrature rules with multiple nodes over a finite interval, taken to be [-1, 1] ò1-1 f(t) w(t) dt = ånv=1 å2si=0Ai,v f(i) (tv) +Rn,s(f),\int^{1}_{-1} f(t) w(t) dt = \sum \limits ^n_{v=1} \sum \limits ^{2s}_{i=0}\, {\rm A_i},v f^(i) \,(\tau v) \,+\, {\rm R}_n,s(f), (1) involving a positive weight function w, assumed integrable over...

We study the kernel of the remainder term of Gauss quadrature rules for analytic functions with respect to one class of Bernstein–Szegö weight functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures.

This paper is concerned with bounds on the remainder term of the Gauss-Turán quadrature formula,Rn, s (f) = underover(∫, - 1, 1) f (t) w (t) d t - underover(∑, ν = 1, n) underover(∑, i = 0, 2 s) λi, ν f(i) (τν), wherew (t) = wn, ℓ (t) = [Un - 1 (t) / n]2 ℓ (1 - t2)ℓ - 1 / 2 (ℓ ∈ N),Un - 1 denotes the (n - 1) th degree Chebyshev polynomial of the se...

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ϱ>1 for the Chebyshev weight functions of the first, second and third kind, and derive repr...

We study the kernel Kn, s(z) of the remainder term Rn, s(f) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L∞-error bounds of Gaus...

For analytic functions the remainder term of Gauss-Radau quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours with foci at the points ±1 and a sum of semi-axes [varrho]>1 for the Chebyshev weight function of the second kind. Starting from explicit expressions of the correspondi...

We study the kernels K_{n,s}(z) in the remainder terms R_{n,s}(f) of the Gauss-Turan quadrature formulae for analytic functions on elliptical contours with foci at pm 1 , when the weight omega is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel \vert...

We study the kernels K n,s (z) of the remainder term R n,s (f) of some Gauss-Turán-Kronrod quadrature rules for analytic functions when the weight function is the subclass of Gori-Micchelli weight functions. We investigate the location on the elliptic contours where the modulus of the kernel attains its maximum value, which leads to effective error...

We study the kernels K n,s (z) in the remainder terms R n,s (f) of Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at ±1, when the weight ω is Chebyshev weight function of the first and third kind. It is shown that the modulus of the kernel attains its maximum on the real axis ∀n ≥ n 0 , n 0 = n 0 (ρ, s) in t...