Some Experiments with Evaluation of Legendre Polynomials

Common practice is to recommend evaluation of polynomials by Horner’s rule. Here’s an example where it is fast but doesn’t work nearly as accurately as another fairly easy method. Can a method for Legendre polynomials be both fast and accurate? 1 1 Legendre Polynomials A substantial literature has grown up around the uses for orthonormal polynomials. Here we look at the example of Legendre polynomials (in particular, Legendre Functions of the First Kind) usually written as Pn, which we encountered most recently in looking at formulas for Gaussian quadrature. In this case we need to evaluate them at particular points, and must choose a sequence of operations for the computation. The polynomials can be defined in various algebraically equivalent ways, but with distinct numerical-roundoff behaviors. One popular method directly uses the well-known defining recurrence (for integer n ≥ 0): Pn(x) := (2n− 1)xPn−1 − (n− 1)Pn−2 n , P0 = 1, P1 = x Another method which might occur to someone with a computer algebra system handy and interested in minimizing the operation count is to expand this expression as a polynomial in x, extract the coefficients, and use Horner’s rule. Let us try as an example, P5(x) which is 63 x − 70 x + 15 x 8 . Using Horner’s Rule it can be expressed as x ( x ( 63 x − 70 ) + 15 ) 8 or by performing the indicated division by a constant: x ( x ( 7.875 x − 8.75 ) + 1.875 ) The recurrence, on the other hand, requires following a program. Here it is expressed in Macsyma: h(x):=block([p0,p1,p2,p3,p4], p0:1, p1:x, 1We have previously observed that Chebyshev polynomials (Tn)can be calculated using a recurrence that computes Tn+m from Tn and Tm. For this, see papers by Fateman and by Koepf [1, 3].