Legendre Polynomials

A global polynomial is defined over the entire region of space

This polynomial is of degree m (highest power is x^m) and order m+1 (m+1 parameters {c_j}). If we are given a set of m+1 points

then Lagrange's formula gives a polynomial of degree m that goes through the m+1 points:

Note that each coefficient of y_j is a polynomial of degree m that vanishes at the points {x_j} (except for one value of j) and takes the value of 1.0 at that point, i.e.

If the function f(x) is known, the error in the approximation is [Abramowitz and Stegun, 1964]

The evaluation of P_m(x) at a point other than at the defining points can be made with Neville's algorithm [Presset al. 1986]. Let P₁ be the value at x of the unique function passing through the point (x₁,y₁); i.e. P₁=y₁. Let P₁₂ be the value at x of the unique polynomial passing through the points x₁ and x₂. Likewise, P_ijk...r is the unique polynomial passing through the points x_i, x_j, x_k, ...x_r. Then use the table

These entries are defined using

For example, consider P₁₂₃₄. The terms on the right-hand side involve P₁₂₃ and P₂₃₄. The "parents," P₁₂₃ and P₂₃₄, already agree at points 2 and 3. Here i=1, m=3; thus, the parents agree at x_i+1, ..., x_i+m-1 already. The formula makes P_{i(i+1)...(i+m)} agree with the function at the additional points x_i+m and x_i. Thus, P_{i(i+1)...(i+m)} agrees with the function at all the points {x_i, x_i+1, ...x_i+m}.

Another form of the polynomials is obtained by defining them so that they are orthogonal. We require P_m(x) to be orthogonal to P_k(x) for all k=0,...,m-1.

The orthogonality includes a non-negative weight function, W(x)0 for all axb. This procedure specifies the set of polynomials to within multiplicative constants, which can be set by either requiring the leading coefficient to be one or by requiring the norm to be one.

The polynomial P_m(x) has m roots in the closed interval a to b.

The polynomial

minimizes

when

Note that each c_j is independent of m, the number of terms retained in the series. The minimum value of I is

Such functions are useful for continuous data, i.e. when f(x) is known for all x.

Typical polynomials are given in Table I. Chebyshev polynomials are used in spectral methods (link). The last few entries are widely used in the orthogonal collocation method within chemical engineering.

Table I Orthogonal Polynomials

[Courant and Hilbert, 1953], [Press, et al.,1986]

The last entry is defined by

where a=1, 2, or 3 is for planar, cylindrical, or spherical geometry. These functions are useful if it can be proven that the solution is an even function of x.