Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function
This paper considers the computation of approximations of matrix functionals of form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>A</...
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MDPI AG
2023-01-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/2/105 |
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| author | Jihan Alahmadi Miroslav Pranić Lothar Reichel |
| author_facet | Jihan Alahmadi Miroslav Pranić Lothar Reichel |
| author_sort | Jihan Alahmadi |
| collection | DOAJ |
| description | This paper considers the computation of approximations of matrix functionals of form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mi>T</mi></msup><mi>f</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi mathvariant="bold-italic">v</mi></mrow></semantics></math></inline-formula>, where <i>A</i> is a large symmetric positive definite matrix, <i>v</i> is a vector, and <i>f</i> is a Stieltjes function. The functional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> is approximated by a rational Gauss quadrature rule with poles on the negative real axis (or part thereof) in the complex plane, and we focus on the allocation of the poles. Specifically, we propose that the poles, when considered positive point charges, be allocated to make the negative real axis (or part thereof) approximate an equipotential curve. This is easily achieved with the aid of conformal mapping. |
| first_indexed | 2024-03-11T09:09:40Z |
| format | Article |
| id | doaj.art-1af4a33ffeef4b5b9e2199b7d4a07d51 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T09:09:40Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-1af4a33ffeef4b5b9e2199b7d4a07d512023-11-16T19:05:18ZengMDPI AGAxioms2075-16802023-01-0112210510.3390/axioms12020105Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes FunctionJihan Alahmadi0Miroslav Pranić1Lothar Reichel2Department of Mathematics, Prince Sattam Bin Abdulaziz University, Al-Kharj 16273, Saudi ArabiaDepartment of Mathematics and Informatics, Faculty of Science M. Stojanovića 2, University of Banja Luka, 51000 Banja Luka, Bosnia and HerzegovinaDepartment of Mathematical Sciences, Kent State University, Kent, OH 44242, USAThis paper considers the computation of approximations of matrix functionals of form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mi>T</mi></msup><mi>f</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi mathvariant="bold-italic">v</mi></mrow></semantics></math></inline-formula>, where <i>A</i> is a large symmetric positive definite matrix, <i>v</i> is a vector, and <i>f</i> is a Stieltjes function. The functional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> is approximated by a rational Gauss quadrature rule with poles on the negative real axis (or part thereof) in the complex plane, and we focus on the allocation of the poles. Specifically, we propose that the poles, when considered positive point charges, be allocated to make the negative real axis (or part thereof) approximate an equipotential curve. This is easily achieved with the aid of conformal mapping.https://www.mdpi.com/2075-1680/12/2/105Stieltjes functionmatrix functionrational Gauss quadrature |
| spellingShingle | Jihan Alahmadi Miroslav Pranić Lothar Reichel Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function Axioms Stieltjes function matrix function rational Gauss quadrature |
| title | Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function |
| title_full | Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function |
| title_fullStr | Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function |
| title_full_unstemmed | Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function |
| title_short | Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function |
| title_sort | pole allocation for rational gauss quadrature rules for matrix functionals defined by a stieltjes function |
| topic | Stieltjes function matrix function rational Gauss quadrature |
| url | https://www.mdpi.com/2075-1680/12/2/105 |
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