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Standard and Rational Gauss Quadrature Rules for the Approximation of Matrix Functionals

Alahmadi, Jihan
http://rave.ohiolink.edu/etdc/view?acc_num=kent1633624009163651

Abstract Details

2021, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
In this thesis we develop efficient numerical methods for the approximation of matrix functionals of the form F(A):=w^Tf(A)v, where A is a large symmetric or nonsymmetric matrix, w,v are given vectors, and f is a function. Golub and Meurant describe a technique for computing upper and lower bounds for matrix functionals F(A) based on the connection between the Lanczos process, orthogonal polynomials, and Gauss-type quadrature rules. Their technique considers the expression F(A) as a Stieltjes integral. If the derivatives of the integrand f do not change sign on the convex hull of the support of the measure, then Gauss-type quadrature rules can be applied to compute upper and lower bounds for F(A). However, when A is symmetric and derivatives of the integrand f change sign in the convex hull of the spectrum of A, or when the matrix A is nonsymmetric, then this approach is not guaranteed to yield upper and lower bounds. We consider several extensions of the technique by Golub and Meurant for approximating matrix functions of the form F(A). Let A be a large symmetric matrix. Our first extension is based on the use of pairs of Gauss, and suitable generalized Gauss--Radau or generalized Gauss--Lobatto rules that yield upper and lower bounds for F(A) when some of the derivatives of f change sign on the convex hull of the support of the measure. We also describe new methods to evaluate these quadrature rules. Our other extensions are concerned with the situation when the function f cannot be approximated accurately by a polynomial of small to moderate degree. Then Gauss-type rules may yield poor approximations of the functional F(A). This situation occurs, for instance, when the function f has one or several singularities close to the support of the measure. This difficulty can be remedied by using rational Gauss rules. We discuss two approaches related to this case. First, we develop a technique to approximate matrix functionals of the form F(A) with A a large nonsymmetric matrix when the function f has a singularity at the origin. We derive Gauss--Laurent quadrature rules that yield significantly more accurate approximations of F(A) than Gauss-type rules with the same number of nodes. Also, we develop associated anti-Gauss--Laurent quadrature rules. Pairs of Gauss–Laurent and anti-Gauss–Laurent rules can be applied to compute upper and lower bounds for F(A) under suitable conditions. Second, we consider matrix functionals of the form F(A), where A is a large symmetric positive definite matrix, and f is a Stieltjes function. In this case, upper and lower bounds can be computed using pairs of rational Gauss and Gauss--Radau rules or, under suitable conditions, pairs of rational Gauss and anti-Gauss rules that are determined by prescribed poles.
Lothar Reichel (Advisor)
Miroslav Pranic' (Advisor)
Jing Li (Committee Member)
Jun Li (Committee Member)
Mikhail Nesterenko (Committee Member)
Arvind Bansal (Committee Member)
112 p.
Applied Mathematics
Lanczos process, Matrix functionals, Gauss quadrature rules, Orthogonal polynomials, Rational functions

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Citations

  • Alahmadi, J. (2021). Standard and Rational Gauss Quadrature Rules for the Approximation of Matrix Functionals [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1633624009163651

    APA Style (7th edition)

  • Alahmadi, Jihan. Standard and Rational Gauss Quadrature Rules for the Approximation of Matrix Functionals. 2021. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1633624009163651.

    MLA Style (8th edition)

  • Alahmadi, Jihan. "Standard and Rational Gauss Quadrature Rules for the Approximation of Matrix Functionals." Doctoral dissertation, Kent State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=kent1633624009163651

    Chicago Manual of Style (17th edition)

kent1633624009163651
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This open access ETD is published by Kent State University and OhioLINK.
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