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MIDPOINT quadrature 1. H. J. Stetter, "Stabilizing predictors 719 formulas for weakly unstable correctors," Math. Comp., v. 19, 1965, pp. 84-89. MR 31 #2833. 2. H. J. Stetter, "A study of strong and weak stability in discretization algorithms," /. Soc. Indust. Appl. Math. Ser. B Numer. Anal., v. 2, 1965, pp. 265-280. MR 32 #4859. 3. R. L. Crane & R. W. Klopfenstein, "A predictor-corrector algorithm with an increased range of absolute stability," /. Assoc Comput. Math., v. 12, 1965, pp. 227-241. MR 31 #6378. 4. F. T. Krogh, "Predictor-corrector methods of high order with improved stability charac- teristics," J. Assoc. Comput. Mach., v. 13, 1966, pp. 374-385. MR 33 #5127. 5. B. A. Fuchs & V. I. Levin, Fundions of a Complex Variable and Some of Their Applica- tions, Vol. II, Pergamon Press, New York, 1961; Addison-Wesley, Reading, Mass., 1962. MR 24 #A2654. 6. E. C. Titchmarsh, The Theory of Fundions, Oxford Univ. Press, Oxford, 1939. 7. Anthony Ralston, "Relative stability in the numerical solution of ordinary differential solution differential equations," SI AM Rev., v. 7, 1965, pp. 114-125. MR 31 #2831. 8. F. T. Krogh, "A test for instability in the numerical of ordinary equations," J. Assoc. Comput. Mach., v. 14, 1967, pp. 351-354. (This gives the stability diagram for the method proposed by Stetter in [1].) Midpoint Quadrature Formulas By Seymour Haber A family of quadrature formulas for the interval (0, 1) can be constructed in the following manner: For any positive integer n, we partition (0, 1) into subintervals Ii, I2, • ■■, In ili being the leftmost, I2 adjacent to it, etc.) of lengths ah a2, • ■-, an, respectively. Now let xk be the midpoint of Ik, for k = 1, ■• ■,n, and take (1) aifOxi) + • • • + a„f(Xn) as the approximation "midpoint" to flf(x)dx. The simplest of these rules is the "Euler's" or rule jJo fix)dx~fih). We will refer to the members of this family as "midpoint quadrature formulas" and determine their properties. We first find their "degrees of precision"—that is, for any formula, the highest integer p such that the formula is exact for all polynomials of degree p or lower. Theorem 1. The degree of precision of a midpoint quadrature formula is 1. Proof. The formula is exact for constants, since necessarily ai + a2 + • ■■ + an = 1. To check the exactness of the formula for f(x) = x, we first note that /r>\ (2) Ol xi = —,x2 , a2 = ai + — , ■■■,xn = ai+ r l l On ■■• + an-i + -5-. So for the integral J\ x dx, (1) gives us ffli(ai/2) + a2(ai + a2/2) + • • • + a„(ai + • • • + an-i + an/2) Received December 5, 1966. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use . 720 SEYMOUR HABER But this is just l/„ 2 2 (ai + a2 + • • • + an + 2aia2 + 2aias + • • • + 2an-ian) , or |(ai + • • • + an)2, which is \. Thus the degree of precision is at least one. To show it is no greater, we calculate error in integrating x2/2 by the rule : /12 n 2 i i » / \2 -7-dx — J^Oi-A = - —r 2 oiai + a2+ • • • + a,_i + +■J . Multiplying out and collecting terms in the last sum, we obtain: X OiX? = 7 ]Ç a»3+ ¿2 ata2 + 2 Z i ^ i where the indices of summation Ír*j aiajak , téj^k run from 1 to n. Now 1 = (ßi + • • • + an)3 = ^ a3 + 3 Z-, OiO2 + 6 i ir* i Z, a^jOk , ir^j^k so that 1 V> 2 1 / Ö — ¿_j OiXi = — (ai 3 , + , 3s. • • ■ + a» ) ■ It follows that /i 7-2 da; — 2 a,■-+2 = ¡-r (a/ + • • • + a«3) > 0 , , which proves the theorem. Theorem 2. The error of a midpoint quadrature formula, for an integrand continuous second derivative, is given by (4) with fgfix)dx - Ç aifixi) = ¿ (ax3+ • • • + aM3)/''(£) for some £ in (0, 1). Proof. By a general remainder in the form theorem (see, e.g., [1]) the error may be written (5) ff"(t)K(t)dt J 0 where K(t) = (1 ~ °2 - E <-<(*,- 0 . To derive (4) from (5) it is sufficient (0, 1) ; for then we may write to show that K(t) does not change sign in jlf"(t)K(t)dt=f"(£)jlK(t)dt, and, taking f(x) = x2/2, we see from (3) that License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 721 MIDPOINT QUADRATURE FORMULAS / K(t)dt = ^ (Cl* + ■• • + On') . We shall show that, in fact, K(t) ^ 0 for t G [0, 1]. For t between xk and xk+i, n 2K(t) = (1 - t)2 - 2 £ Oí(xí - t) i=k+l n = (1 - t)2 - 2 E n Oi(l - t) + 2 £ i=k+l a,-(l - Xi) . i=k+l Now, in fact n 2 X a>(l — ^») = iok+i + ak+2 + • • • + an)2. i=k+l To prove this by induction, 2akil we need only show that — xk) = a2 + 2ak(ak+i + ■■■ + an) , which follows directly from the fact that xk = 1 — an — On-i — • • ■ — ak+i — ak/2 Therefore, . in [xk, xk+i), 2K(t) = ((1 -t) - (ak+i + ■•■ +an))2^0; and it can similarly be shown that K is nonnegative in [0, Xi] and [x„,i]. It is easy to see that, given n, the coefficient (ai3 + • • • + aA)/2A in (4) is least when ai — a2 = ■ ■• = an = 1/n, so that for any n, the "best" midpoint quadrature rule is simply the repeated Euler's rule. National Bureau of Standards Washington, D. C. 20234 1. V. I. Krylov, Approximate Calculation of Integrals, Macmillan, MR 26 #2008. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use New York, 1962, p. 77.