2. The best polynomial approximation

  In what follows we will refer to the interval . There is not lack of generality, for projecting any closed interval [a,b] onto the interval [-1,1] is a trivial problem.

When using a Lagrange polynomial P_n(x) of degree n for interpolating a continuous function f(x) we get
 
The function E_n(x) is the Lagrange remainder given by

where Q(x) is the polynomial of degree n+1

It is well known that he maximum absolute value of Q(x) is minimum when the abscissas are chosen to be the zeros of the Chebyshev polynomial T_n+1(x) of degree n+1. Nevertheless, this polynomial could be not the best approximation to the function f(x). One will succeed with the best approximation when it is achieved in Eq. (1 (click here)) the minimum value of |E_n| over the entire interval. Therefore, we will say that the polynomial P_n^*(x) is the best approximation of degree n to the function f(x) when

An important property characterizes the best approximation: at least at n+2 critical points of the interval [-1,1] the error function E(x)=P_n^*(x)-f(x) assumes extremes values that are equal in magnitude and alternating in sign (see, for instance, Vallée-Poussin 1989). From this property, if a set of critical points is known it is clear that P_n^*(x) is easily determined from the n+2 linear equations

in the unknowns and the n+1 coefficients of P_n^*(x). Consequently the problem of computing the best approximation to a function f(x) can be reduced to determine the set of n+2 points where

The critical points are not known a priori, but starting from a set of references points close enough to the critical one we can proceed by iterations to compute the polynomial coefficients. Murnaghan & Wrench (1959) assume the convergence of the iterations and find the best approximation by changing the complete set of abscissas at each step. Without need of this assumption, Stiefel (1958) demonstrated that, when proceeding by interchange in the transition from one set of reference points to the next, the iteration takes place in a finite number of steps.

Basically the procedure consists of the following steps:

1. select a trial set of reference points
2. compute the set of coefficients of the polynomial approximation
3. check that it is the best approximation
4. if not, change the set of reference points and come back to step 2.

Assuming that we can approximate the astronomical ephemerides by truncated series of Chebyshev polynomials, the error in their best approximation is in most cases comparable to the first of the neglected terms. Consequently the zeros of the derivative of the Chebyshev polynomial of degree n+1 can be selected at step 1 as a first approach to the reference points (Deprit et al. 1945).

The procedure is different when approximating a function given by a finite set of m points f(x_i), . Since we want to compute a polynomial of degree n which fits the data, we face up to an undetermined system: m points and n coefficients (n<m). The direct approach to the problem is to solve the inconsistent system

of n+2 linear equations where the solution c_j (the coefficients of the polynomial) minimizes . Then, the computation of the best approximation consists in determining the set of coefficients c_j which minimizes
 

The problem can be attacked in two ways. The first one begins with an initial estimation of the coefficients and uses an algorithm that tries to diminish the error function |P_n(x)-f(x)| at each step. The other, consists in finding the set of reference points x_k that maximize |P_n(x_k)-f(x_k)|. Stiefel (1960) demonstrated that both methods are equivalent to the simplex method of linear programming.

In our work we construct the approximating polynomials by using the Stiefel exchange-method (Stiefel 1958). A set of n+2 points is selected as reference points from the set of m. The starter reference points are those with closest abscissas to the zeros of the derivative with respect to x of the Chebyshev polynomial T_n+1, plus both ends . The exchange-method is an iterative one that, by changing the reference points, computes those coefficients c_j such that for i=1 to m they minimize Eq. (2 (click here)). Uniform approximation is ensured when Remez's rippling occurs (Remez 1957), i.e. the errors at reference points are equal in absolute value but opposite in sign. When this situation occurs the polynomial obtained is optimal.

The Stiefel exchange-method has been implemented by Barrodale (1975) and Schmitt (1971). Deprit & Picard (1979) used the Barrodale algorithm with encouraging results. In our work we first used the Barrodale algorithm but the Schmitt one showed better results.