Generation and Use of Orthogonal Polynomials for Data-Fitting with a Digital Computer

Methods in Data Reduction: 1. Another Look at Least Squares

“…Since the formulation of this method is well treated in the original reference (Forsythe 1957) we give here only a brief summary of the results. We start by noting that if we are given a set of / + 1 polynomials, P^x), i -0, • • •, I, which are completely arbitrary except that the leading term of each is just x 1 , then by suitably choosing the coefficients, S¿, the original polynomial, equation (1), can in all generality be written y = 2 SjP^x) .…”

“…On the other hand, with socalled mini-or micro-computers, with relatively slow multiply and divide cycles, one pays a substantial time penalty, as well as a loss of storage if one is required to perform double precision calculations. We outline here a variation of the usual leastsquares algorithm which has been available for some time in applied mathematics and statistics circles (i.e., Forsythe 1957;Cooper 1968), and which is part of standard statistical packages available at most large computer installations (BMDP 1975;IMSL 1976). It is treated only incompletely or for the special case of equal increments in the x coordinate in the standard texts (i.e., Hildebrand 1956;Bevington 1969;Carnahan, Luther, and Wilkes 1969;Martin 1971) and, in particular, it does not seem to be generally known among astronomers (although the application is hinted at in a recent paper by Groth (1975)).…”

Calculating the coefficients of certain linear predictors

“…Taking expected values on both sides of equation ( 1 ), one can write If these equations are all multiplied by Po(j) and then summed, and this process is repeated but with Pi(j), ■ ■ ■ , Ph(j), one obtains the equations (by using (4) and (7) Values for a¡ were obtained by using the following set of polynomials, which are orthogonal over any set of points symmetric about the origin [2] : (15) ft-^-rT*' i = i,2,...,k-i.…”

Discrete Least Squares Approximation by Trigonometric Polynomials

“…We refer to this procedure as the Stieltjes procedure for Szegö polynomials because it is analogous to the Stieltjes procedure that is often used to compute the recurrence coefficients and values of orthogonal polynomials that satisfy a three-term recurrence relation (see Gautschi [7,8] and Forsythe [6]). …”