Gauss-Kronrod integration rules for Cauchy principal value integrals

Rabinowitz, Philip

doi:10.1090/S0025-5718-1983-0701624-2

Gauss-Kronrod integration rules for Cauchy principal value integrals
HTML articles powered by AMS MathViewer

by Philip Rabinowitz PDF

Math. Comp. 41 (1983), 63-78 Request permission

Corrigendum: Math. Comp. 50 (1988), 655.
Corrigendum: Math. Comp. 50 (1988), 655-657.
Correction: Math. Comp. 45 (1985), 277.

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 $n + 2$ 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.

References

Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0448814
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, DOI 10.1090/S0025-5718-1979-0514825-2
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
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, DOI 10.1007/BF01596155
Giovanni Monegato, Stieltjes polynomials and related quadrature rules, SIAM Rev. 24 (1982), no. 2, 137–158. MR 652464, DOI 10.1137/1024039

QUADPACK, A Quadrature Subroutine Package

Angew. Informatik

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
Philip Rabinowitz, The exact degree of precision of generalized Gauss-Kronrod integration rules, Math. Comp. 35 (1980), no. 152, 1275–1283. MR 583504, DOI 10.1090/S0025-5718-1980-0583504-6
G. Szegö, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann. 110 (1935), no. 1, 501–513 (German). MR 1512952, DOI 10.1007/BF01448041
Gábor Szegő, Orthogonal polynomials, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1967. MR 0310533