A Punched Card Technique to Obtain Coefficients of Orthogonal Polynomials

Abstract

IN PAST YEARS many methods have been developed for fitting polynomials. Although the use of orthogonal polynomials for fitting continuous functions probably originated with Legendre, Tchebycheff2 seems to be the first to use orthogonal polynomials for fitting discrete observations with and without equal intervals. Orthogonal polynomials have been the subject of many mathematical dissertations and an entire volume3 of references and cross references on this subject has been compiled by a committee, of which J. Shohat is chairman. fitting of polynomials by the use of orthogonal functions has been mainly by two methods: (1) summation; and, (2) multiplication of the variates by actual values of the orthogonal polynomials as given in a table. Pearson4 and Isserlis,i dealt with the fitting of non-equidistant and unequally weighted data. R. A. Fisher developed methods of calculation of Tchebycheff polynomials by successive summation up to the 5th degree,6 and in his classic work on Yield of Wheat at Rothamsted7 refers to work by Esscher.8 F. E. Allan published a paper, The General Form of Orthogonal Polynomials for Simple Series.9 C. Jordan'0 develops the theory in full, and crediting the method to Tchetverikoff in a paper published in Russian in 1926, demonstrates the building up of general expressions for the orthogonal polynomials in factorial form originally used by G. F. Hardy in Graduation of British Offices Tables, 1863-93. Jordan also suggests the numerical method of building up the polynomials by summation and furnishes tables for this purpose. J. Shohat gave the mathematical approach in Stieltje's Integrals in Mathematical Statistics. A. C. Aitken12 gives a relatively simple demonstration of the theory and includes a series of tables up