An Error Estimate for Stenger's Quadrature Formula

The basis of this paper is the quadrature formula where q = exp(2A), h being a chosen step length. This formula has been derived from the Trapezoidal Rule formula by F. Stenger. An explicit form of the error is given for the case where the integrand has a factor of the form (1 — x)a(\ + x)P, a,ß> -1. Application is made to the evaluation of Cauchy principal value integrals with endpoint singularities and an appropriate error form is derived. An alternative derivation is given for Stenger's quadrature formula for the finite interval [-1,1], with a more explicit form of the error in the case where the integrand has a factor of the form (\ -x)a(l +x)ß, a,ß>-\. Application is made to the evaluation of Cauchy principal value integrals with endpoint singularities and an appropriate error form is derived. An Alternative Derivation of Stenger's Formula With an Error Estimate. Stenger [2] derives the formula where q = exp(2«), « being a chosen step length. Taking a finite sum gives the modified form and, throughout this paper, we refer to (2) as Stenger's quadrature formula. Stenger [2] states that (1) is accurate even if/has singularities at the endpoints. For the form f(x)=(l-x)a(l+x)ßg(x), where a, ß >-I, and g(x) possesses differential coefficients of all orders for -1 ^ x < 1, Stenger later gave the error in (2) as ofexp^O+yy^A/1/2^'/2)], where y = min(a, ß). Received April 30, 1981. 1980 Mathematics Subject Classification. Primary 65D30. ©1982 American Mathematical Society 0025-5718/81/0000-1121/$02.50 539 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 540 S. BEIGHTON AND B. NOBLE We first give an alternative derivation of (2) which will lead to a more explicit form of the error. The substitution x = tanh u gives oí i+Mh). Next, consider the integral in expression (4): using the Euler-Maclaurin summation formula, r^^du = h I ^^-f{F(-M«) + JF(M«)} . v J-Mh cosh m r=-M cosh rh l (11) -^{F'(-Mh)-F'(Mh)} +0(h4), where F(u) =/(tanh w)/cosh2 u, so that (12) F(u)= ¿Mi«!2f^l tanh „. cosh u cosh u With the assumption above concerning the behavior of fix) near x = 1, (13) /(tanhM«) = eag,(l) + 0(ea+x), and /'(tanh Mh) = -V_,ag,(l) + 0(ea), License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use AN ERROR ESTIMATE FOR STENGER'S QUADRATURE FORMULA 541 so that (14) F'(Mh) = -4e°+1g,(l)(l +a) + 0(ea+1). Similarly, we obtain (15) F'(-Mh) = 4e'?+1g^(-l)(l + ß) + 0(eß+2). Finally, the sum of (11) may be transformed into Stenger's formula by taking q = exp(2/z), so that aJ 1 , (qJ + if tanh jh = *-, cosh2 jh = -^--1-, qJ +1 4qJ giving 06) h i ̂ %f = loèqî -2*1-/(1^1). r=-M cosh2 rh r=_M (1 + q'f \q + 1 / To simplify the error form, we will modify Stenger's formula by combining the term -h/2{F(-Mh) + F(Mh)} with the sum in (11). Thus, combining (9), (10), (14), (15) and (16) in (4), we obtain (17) /' f(x)dx = log q Í"-^S/(5tt)+£> y-> r=-M (l+qr) U + W where 2" has the usual meaning, namely that the first and last terms are halved, and + 0(e-2(r+2)«*) h2{0(e-*y+»Mh)} + 0(h4), with y = min(a, ß). Thus, for « small and Mh large enough to satisfy the approximation (8), we have the dominant part of the error term E for the modified form of Stenger's formula (17): (19) ^fgi(i)^2(a+,)MA + ^^-.(-iK2^1^Numerical Example. The errors incurred in using formula (17) when fix) = (1 — x)3/4 are given in Table 1 for a variety of values of M and « (h < 1). The values of the error expression (19) for the same values of M and « are given in Table 2. Below the dotted lines (Mh > 1) the two tables are very similar, whilst below the continuous line (Mh > 8) the error is less than { X 10~6. To clarify the relationship between « and M (to give minimum error) we give in Table 3 values of the actual errors for the arguments « and Mh. These results clearly indicate that the error depends on the product Mh rather than M or h separately. The pattern of these results was found to be substantially the same for the integral / (1 — x)a dx, J-\ with a = ± j, ± \ stnd -\. Naturally, the different values of a resulted in changes in the critical value of the product Mh. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 542 S. BEIGHTON AND B. NOBLE Evaluation of the Integral ft }(1 x)3/4 dx Using (17). Table 1 Actual errors incurred 1/4 1/8 1/32 1.674 1.433 J.004 0.429~ 0.062 hTable 2 Values of the approximate error expression (19) „1/2 1/4 1.8 1/16 1/32 3.860 2.841 .1,571 Ö.5f3 0.063 Mh 1/2 1 2 4 Table 3 Actual errors for values of the product Mh 1/2 1/4 1/8 1/16 1.032 0.440 0.067 0.0012 0 1.016 0.436 0.063 0.0012 0 1.006 0.431 0.062 0.0011 0 1.005 0.430 0.062 0.0011 0 1/32 1.004 0.429 0.062 0.0011 0 The zeros in the tables above represent numbers that are at most \ X 10 6 in modulus. An Application to Cauchy Principal Value Integrals With Endpoint Singularities. Consider (20) 7(x) = Pf + i(i-y)V+y)ßg(y) y dy, where the integral is defined in the Cauchy principal value sense, a and ß are greater than -1,-1 < x < 1, and g(y) possesses differential coefficients of all orders License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use AN ERROR ESTIMATE FOR STENGER'S QUADRATURE FORMULA 543 for -1 *£ y < 1. 'Subtracting out the singularity' gives (21) 7(x) =/ +V(x, y) dy + g(x)pf + ] {l ~y^\+^dy, j-\ j-x y~x where (22) H*,y) = i'-,jyx+y)[S(y)-S(*)l The first integral is evaluated by the modified Stenger formula (17), whilst the second may be directly evaluated using Erdélyi [ 1, p. 250]. To estimate the error we follow the procedure given earlier and observe that F(x, y) behaves like (1 — y)agx(y, x) near y — 1 and like (1 + y)ßg-x(y, x) near y = -1, where gi(y, x) = (1 +y)ß[g(y) g(x)]/ (y x) and g-X(y,x) = (\-y)*[g(y)-g(x)]/{y-x). We assume that gx(y, x) and g-X(y, x) may be expanded in Taylor series about y = 1 and y = -I, respectively, for given x. We may therefore develop an error term from (18) giving the dominant part as g(l)~g(x) £í(x) = Í___<r2(«+l)MA