Gauss-Kronrod integration rules for Cauchy principal value integrals

Author:
Philip Rabinowitz

Journal:
Math. Comp. **41** (1983), 63-78

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701624-2

Corrigendum:
Math. Comp. **50** (1988), 655.

Corrigendum:
Math. Comp. **50** (1988), 655-657.

Correction:
Math. Comp. **45** (1985), 277.

MathSciNet review:
701624

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Abstract | References | Similar Articles | Additional Information

Abstract: Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using points which requires only values of the integrand. However, this new rule does not exist for all *n*, and when it does, several significant figures are lost in its use.

**[1]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR**0448814****[2]**David Elliott and D. F. Paget,*Gauss type quadrature rules for Cauchy principal value integrals*, Math. Comp.**33**(1979), no. 145, 301–309. MR**514825**, https://doi.org/10.1090/S0025-5718-1979-0514825-2**[3]**Walter Gautschi,*A survey of Gauss-Christoffel quadrature formulae*, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR**661060****[4]**Giovanni Monegato,*On polynomials orthogonal with respect to particular variable-signed weight functions*, Z. Angew. Math. Phys.**31**(1980), no. 5, 549–555 (English, with Italian summary). MR**599514**, https://doi.org/10.1007/BF01596155**[5]**Giovanni Monegato,*Stieltjes polynomials and related quadrature rules*, SIAM Rev.**24**(1982), no. 2, 137–158. MR**652464**, https://doi.org/10.1137/1024039**[6]**R. Piessens, E. de Doncker, C. Uberhuber & D. Kahaner,*QUADPACK, A Quadrature Subroutine Package*. (To appear.)**[7]**R. Piessens, M. van Roy-Branders & I. Mertens, "The automatic evaluation of Cauchy principal value integrals,"*Angew. Informatik*, v. 18, 1976, pp. 31-35.**[8]**P. Rabinowitz,*The numerical evaluation of Cauchy principal value integrals*, Symposium on numerical mathematics (Durban, 1978) Univ. Natal, Durban, 1978, pp. 53–82. MR**728242****[9]**Philip Rabinowitz,*The exact degree of precision of generalized Gauss-Kronrod integration rules*, Math. Comp.**35**(1980), no. 152, 1275–1283. MR**583504**, https://doi.org/10.1090/S0025-5718-1980-0583504-6**[10]**G. Szegö,*Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören*, Math. Ann.**110**(1935), no. 1, 501–513 (German). MR**1512952**, https://doi.org/10.1007/BF01448041**[11]**Gábor Szegő,*Orthogonal polynomials*, 3rd ed., American Mathematical Society, Providence, R.I., 1967. American Mathematical Society Colloquium Publications, Vol. 23. MR**0310533**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701624-2

Keywords:
Cauchy principal value integral,
Kronrod rule,
Gauss integration rule,
Lobatto integration rule,
Gegenbauer polynomials,
Szegö polynomials

Article copyright:
© Copyright 1983
American Mathematical Society