Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp

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1 Abscissas and Weight Coefficients for Lobatto Quadrature H. H. Michels Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp Stable URL: Mathematics of Computation is currently published by American Mathematical Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. Fri Jan 18 07:34:

2 Abscissas and Weight Coefficients for Lobatto Quadrature By H. H. Michels 1. Introduction. Recently, the numerical evaluation of certain collision illtegrals was studied using several different mechanical quadrature formulas, including Gaussian quadrature of high order [I, 21 and various n'ewton-cotes formulas. It was found that high accuracy could not easily be obtained, owing to the particular behavior of the integralid at the end points of integration, aiid it seemed likely that a "closed" type Gaussian formula of high order might be more eficient for this particular application. The existence of Gaussian-type quadrature formulas with one or more prescribed abscissas has beeii investigated by Lobatto [3] and Radau [I]. For the case where both ends of the integration interval are preassigned (Lobatto quadrature), the free abscissas aiid the corresponding weight coefficients have been evaluated by Radau [j]up to order 11. Nore recently abscissas and ~veights for Lobatto quadrature have beeii reported by Rabinowitz [GI for selected odd order up to 65. In some cases, however, an even-order quadrature formula may be desired aiid the results for such formulas of high order are reported in this communication. 2. Method of Computation. We are concerned with the Lobatto quadrature formulas of order n normalized by a change of variables to the interval (- 1, I) Formula (I) is exact for all polynoi~lials f(x) of degree S 2n - 3, whereas Gaussian quadrature rules are exact for degree 62n - 1. However, if the function f(x) is zero at both ends of the integration interval, only i. - 2 ordinates are involved in the calculation and a higher effective degree of precision is obtainable thail if an open Gaussian type formula is used. The free abscissas xi (12 = 2, 3,..., n - 1) are the zeros of the first derivative of the Legendre polynomial of order n - I The corresponding le eight coefficients Hi, can be found froin the expression TI-hereP?I-I(~b) is the normalized Legendre polynoinial of order n - 1. The ~veights corresponding to the fixed abscissas at n: = il are found to be Received Septe~~lber 18,

3 X first approximation to the zeros of the derivative of the Legendre polyiloinial P,'(x) can be obtained in several ways. It can be sho~~n [7]that Since the zeros of P,,'(z) are the same as those of the associated Legendre polynomial P,' (.xi through the relation (6) p,,'(:c) = (2-I)~'~P,~ (e) equation (5) can be used to relate the zeros of P,;(X) to the successive zeros of the Bessel function Jl(z). A better approxinlation to the zeros of P,'(.E) can be obtained by making use of the inequalities derived by Szego [8]for the zeros of the generalized Jacobi polynoillial ~,'",~'(x). An examination of the upper and lower bounds of the zeros of Pi,-,(x) showed that two or three decimal places could be established using the relatioil where jl,a are the successire zeros of the Bessel functioil Jl(r).These initial approxiillations to the roots were iinproved using Xen-ton-Raphson iteration The Legendre polynomials and their derivatives were coillputed using the recursion formulas The eight coefficients were coinputed directly using equations (3) and (4) 3. Results. Abscissas and weights for Lobatto quadrature are presented in Table I for order n = 3(l)16, 24, 32, 40, 48, 64, 80, 96. All computatioiis were perforilled on an IBAI 7090 digital computer using extended precision routines. The tolerance for iteration 011 the roots mas set at 1 X 10-'! Several hand calculations of the roots and weight coefficients were performed. Coinplete agreement

5 TABLEI-Continucil Abscissas Weights : : G(i fi8(i1884g i iSA ',) : (i (i (i2399!)2e K : : = v O3Oq qJ Ci :32(jl s * G47Oti $ : t % , = 1,; *724592(55 038: ~297ti2988~ (i;i;il!) OS R <571I t5 0,42O(i: : t :35339< :382; ~ ISv{ <51 // = :3:33:3 3: : : ~ : (iS : : ; lt54O ; ~~ : iz = , t5cj3c50, t O77Ovj t , e566:3; tll : ::22(?182.j q5, : : :3212e ~ K60:1942: :37228v (i0 71 = 3" ? e t ? ? C8L t :3:3: , e5e :3 362X :344e71 0.0<? i80s5e5869t5.? ,> , : q5305?~ : < : S :39GO ,i73578:2:307

6 Abscissas Weights

7 Abscissas Weights

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9 244 H. H. MICHELS with the machine results was found in all cases to 21 decimals. In addition, the following relation was used as a check on the accuracy of the results Equation (12) was satisfied to within 2 units in the 21st decimal place for all cases reported here. In the table only the positive abscissas are reported since all abscissas and weights are symmetric with z-a = -z,, and H-I, = HA. 4. Acknowledgements. This work was sponsored in part by the Air Force Office of Scientific Research of the Office of Aerosp&ce Research, and in part by the United Aircraft Corporation Research Laboratories. The author wishes to thank Miss Anne Putnam for her assistance with the calculations. Research Laboratories United Aircraft Corporation East Hartford 8, Connecticut 1. P. DAVIS & P. RABINOWITZ, "Abscissas and Weights for Gaussian Quadratures of High Order," htbsj. of Research, v. 56, 1956, p P. DAVIS & P. RABINOWITZ, "Additional Abscissas and Weights for Gaussian Quadrature of High Order: values for n = 64, 80, and 96," NBS J. of Research, v. 60, 1958, p R. LOBATTO, Lessen over de Integraal-Rekening, The Hague, 1852, p M. R. RADAU, "Etude sur les formules d'ap roximation qui servant 3, calculer la valeur numcrique d'une integrale dhfinie." J. de Math., 6),v. 6, 1880, p. %3. 5. Ibid, p P. RABINOWITZ, "Abscissas and Weights for Lobatto Quadrature of High Order," Math. Comp., v. 14, 1960, p E. T. WHITAKER & G. N. WATSOS,Modern Analysis, 4th Edition, Cambridge University Press, H. SZEG~, Orthogonal Polynomials, American Math. Soc. Publ., v. 23, American Math. Soc., N. Y., 1939.

10 LINKED CITATIONS - Page 1 of 1 - You have printed the following article: Abscissas and Weight Coefficients for Lobatto Quadrature H. H. Michels Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp Stable URL: This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. [Bibliography] 6 Abscissas and Weights for Lobatto Quadrature of High Order Philip Rabinowitz Mathematics of Computation, Vol. 14, No. 69. (Jan., 1960), pp Stable URL: NOTE: The reference numbering from the original has been maintained in this citation list.

The American Mathematical Monthly, Vol. 100, No. 8. (Oct., 1993), pp

A Visual Explanation of Jensen's Inequality Tristan Needham The American Mathematical Monthly, Vol. 100, No. 8. (Oct., 1993), pp. 768-771. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28199310%29100%3a8%3c768%3aaveoji%3e2.0.co%3b2-8