Papers by Miroslav Pranic
Numerische Mathematik
This paper is concerned with computing approximations of matrix functionals of the form F(A) := v... more This paper is concerned with computing approximations of matrix functionals of the form F(A) := v T f (A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. We approximate F(A) with the aid of rational Gauss quadrature rules. Associated rational Gauss-Radau and rational anti-Gauss rules are developed. Pairs of rational Gauss and rational Gauss-Radau quadrature rules, or pairs of rational Gauss and rational anti-Gauss quadrature rules, can be used to determine upper and lower bounds, or approximate upper and lower bounds, for F(A). The application of rational Gauss rules, instead of standard Gauss rules, is beneficial when the function f has singularities close to the spectrum of A.
Journal of Computational and Applied Mathematics, 2021
Many functionals of a large symmetric matrix of interest in science and engineering can be expres... more 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 whose derivatives do not change sign on the convex hull of the support of the measure. This technique is based on evaluating pairs of a Gauss quadrature rule and a suitably chosen Gauss-Radau or Gauss-Lobatto quadrature rule. However, when derivatives of the integrand change sign on the convex hull of the support of the measure, this technique is not guaranteed to give upper and lower error bounds for the functional. We describe an extension of the technique by Golub and Meurant that yields upper and lower error bounds for the functional in situations when only some derivatives of the integrand do not change sign on the convex hull of the support of the measure. This extension is based on the use of pairs of Gauss, and suitable generalized Gauss-Radau or Gauss-Lobatto rules. New methods to evaluate generalized Gauss-Radau and Gauss-Lobatto rules also are described.
Numerical Algorithms, 2021
Applicable Analysis and Discrete Mathematics, 2019
The paper deals with new contributions to the theory of the Gauss quadrature formulas with multip... more 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, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286].
Maximum of the modulus of kernels in Gauss-Tur�n quadratures
Math Comput, 2008
We study the kernels K_{n,s}(z) in the remainder terms R_{n,s}(f) of the Gauss-Turan quadrature f... more 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 K_{n,s}(z)\vert attains its maximum on the real axis (positive real semi-axis) for each ngeq n_0, n_0Dn_0(rho,s) . It was stated as a conjecture in [Mathematics of Computation 72 (2003), 1855-1872]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes n in the corresponding Gauss-Turan quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each ngeq n_0, n_0Dn_0(rho,s) . Numerical examples are included. Retrieve articles in all Journals with MSC (1991): [41]41A55, [42]65D30, [43]65D32
Bounds of the error of Gauss�Tur�n-type quadratures, II
Appl Numer Math, 2010
This paper is concerned with bounds on the remainder term of the Gauss–Turán quadrature formula,
Journal of Computational and Applied Mathematics, 2015
Gauss quadrature is a popular approach to approximate the value of a desired integral determined ... more 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 (n + 1)-point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n-point Gauss quadrature rule for all polynomials of degree up to 2n + 1. This rule is referred to as an anti-Gauss rule. It is useful for the estimation of the error in the approximation of the desired integral furnished by the n-point Gauss rule. This paper describes a modification of the (n + 1)-point anti-Gauss rule, that has n + k nodes and gives an error of the same magnitude and of opposite sign as the associated n-point Gauss quadrature rule for all polynomials of degree up to 2n+2k -1 for some k > 1. We refer to this rule as a generalized anti-Gauss rule. An application to error estimation of matrix functionals is presented.
SIAM Journal on Numerical Analysis, 2014
We present three types of reccurence relations for orthogonal rational functions with arbitrary r... more We present three types of reccurence relations for orthogonal rational functions with arbitrary real and/or complex conjugate poles and their application to rational Gauss quadrature formulae. A matrix which has the same role for computation of rational Gauss quadrature as Jacobi matrix does for classical Gauss quadrature is now septadiagonal matrix (or pentadiagonal if we do not have complex poles) with 5 × 5 and/or 3 × 3 blocks along a diagonal.
Numerische Mathematik, 2012
It is well known that members of families of polynomials, that are orthogonal with respect to an ... more 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 relations for orthogonal rational functions with arbitrary prescribed real or complex conjugate poles. The number of terms in the recursion relation is shown to be related to the structure of the orthogonal rational functions.
Error estimates for Gauss-Turan quadratures and their Kronrod extensions
IMA Journal of Numerical Analysis, 2008
Page 1. IMA Journal of Numerical Analysis (2009) 29, 486507 doi:10.1093/imanum/ drm040 Advance A... more Page 1. IMA Journal of Numerical Analysis (2009) 29, 486507 doi:10.1093/imanum/ drm040 Advance Access publication on May 16, 2008 Error estimates for GaussTurán quadratures and their Kronrod extensions GRADIMIR ...
Numerical Linear Algebra with Applications, 2016
SummaryThe rational Arnoldi process is a popular method for the computation of a few eigenvalues ... more SummaryThe rational Arnoldi process is a popular method for the computation of a few eigenvalues of a large non‐Hermitian matrix 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 , because then few factorizations of matrices of the form A − zjI have to be computed. We discuss recursion relations for orthogonal bases of rational Krylov subspaces determined by rational functions with few distinct poles. These recursion formulas yield a new implementation of the rational Arnoldi process. Applications of the rational Arnoldi process to the approximation of matrix functions as well as to the computation of eigenvalues and pseudospectra of A are described. The new implementation is compared to several available implementations.
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Papers by Miroslav Pranic