Solving singular integral equations using Gaussian quadrature and overdetermined system

S. Kim

doi:10.1016/S0898-1221(98)00073-X

Solving singular integral equations using Gaussian quadrature and overdetermined system

S. Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

Gauss-Chebyshev quadrature and collocation at the zeros of the Chebyshev polynomial of the first kind T_n(x), and second kind U_n(x) leads to an overdetermined system of linear algebraic equations. The size of the coefficient matrix for the overdetermined system depends on the degrees of Chebyshev polynomials used. We show that we can get more accurate solution using T_4n+4(x), than other T_n(x). The regularization method using Generalized Singular Value Decomposition is described and compared to Gauss-Newton method for solving the overdetermined system of equations. Computational tests show that GSVD with an appropriate choice of regularization parameter gives better solution in solving singular integral equations.

Original language	English
Pages (from-to)	63-71
Number of pages	9
Journal	Computers and Mathematics with Applications
Volume	35
Issue number	10
DOIs	https://doi.org/10.1016/S0898-1221(98)00073-X
State	Published - May 1998

Bibliographical note

Funding Information:
Research supported by Korean Ministry of Education, BSRI-97-1430 and Ewha Women's University, 1996. The author wishes to thank R. P. Tewarson for his hospitality during the author's visit to SUNY Stony Brook and R. P. Srivastav for his helpful comments.

Keywords

Gauss-Chebyshev quadrature
Generalized singular
Overdetermined systems
Tikhonov regularization
Value decomposition

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10.1016/S0898-1221(98)00073-X

Cite this

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title = "Solving singular integral equations using Gaussian quadrature and overdetermined system",

abstract = "Gauss-Chebyshev quadrature and collocation at the zeros of the Chebyshev polynomial of the first kind Tn(x), and second kind Un(x) leads to an overdetermined system of linear algebraic equations. The size of the coefficient matrix for the overdetermined system depends on the degrees of Chebyshev polynomials used. We show that we can get more accurate solution using T4n+4(x), than other Tn(x). The regularization method using Generalized Singular Value Decomposition is described and compared to Gauss-Newton method for solving the overdetermined system of equations. Computational tests show that GSVD with an appropriate choice of regularization parameter gives better solution in solving singular integral equations.",

keywords = "Gauss-Chebyshev quadrature, Generalized singular, Overdetermined systems, Tikhonov regularization, Value decomposition",

author = "S. Kim",

note = "Funding Information: Research supported by Korean Ministry of Education, BSRI-97-1430 and Ewha Women's University, 1996. The author wishes to thank R. P. Tewarson for his hospitality during the author's visit to SUNY Stony Brook and R. P. Srivastav for his helpful comments.",

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N1 - Funding Information: Research supported by Korean Ministry of Education, BSRI-97-1430 and Ewha Women's University, 1996. The author wishes to thank R. P. Tewarson for his hospitality during the author's visit to SUNY Stony Brook and R. P. Srivastav for his helpful comments.

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N2 - Gauss-Chebyshev quadrature and collocation at the zeros of the Chebyshev polynomial of the first kind Tn(x), and second kind Un(x) leads to an overdetermined system of linear algebraic equations. The size of the coefficient matrix for the overdetermined system depends on the degrees of Chebyshev polynomials used. We show that we can get more accurate solution using T4n+4(x), than other Tn(x). The regularization method using Generalized Singular Value Decomposition is described and compared to Gauss-Newton method for solving the overdetermined system of equations. Computational tests show that GSVD with an appropriate choice of regularization parameter gives better solution in solving singular integral equations.

AB - Gauss-Chebyshev quadrature and collocation at the zeros of the Chebyshev polynomial of the first kind Tn(x), and second kind Un(x) leads to an overdetermined system of linear algebraic equations. The size of the coefficient matrix for the overdetermined system depends on the degrees of Chebyshev polynomials used. We show that we can get more accurate solution using T4n+4(x), than other Tn(x). The regularization method using Generalized Singular Value Decomposition is described and compared to Gauss-Newton method for solving the overdetermined system of equations. Computational tests show that GSVD with an appropriate choice of regularization parameter gives better solution in solving singular integral equations.

KW - Gauss-Chebyshev quadrature

KW - Generalized singular

KW - Overdetermined systems

KW - Tikhonov regularization

KW - Value decomposition

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Solving singular integral equations using Gaussian quadrature and overdetermined system

Abstract

Bibliographical note

Keywords

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