First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 5. APPROXIMATION THEORY 207
Therefore
P 3 (x) = 3.977 + 3.55
π π x + 0.853
π (2x2 − 1) + 0.1393 (4x 3 − 3x)
π
=0.1774x 3 +0.5430x 2 +0.9970x +0.9944.
Julia code for orthogonal polynomials
Computing Legendre polynomials
Legendre polynomials satisfy the following recursion:
L n+1 (x) =
2n +1
n +1 xL n(x) −
n
n +1 L n−1(x)
for n =1, 2,..., with L 0 (x) =1,andL 1 (x) =x.
The Julia code implements this recursion, with a little modification: the index n +1is
shifted down to n, so the modified recursion is: L n (x) = 2n−1xL
n n−1(x) − n−1L
n n−2(x), for
n =2, 3,....
In [1]: using PyPlot
using LaTeXStrings
In [2]: function leg(x,n)
n==0 && return 1
n==1 && return x
((2*n-1)/n)*x*leg(x,n-1)-((n-1)/n)*leg(x,n-2)
end
Out[2]: leg (generic function with 1 method)
Here is a plot of the first five Legendre polynomials:
In [3]: xaxis=-1:1/100:1
legzero=map(x->leg(x,0),xaxis)
legone=map(x->leg(x,1),xaxis)
legtwo=map(x->leg(x,2),xaxis)
legthree=map(x->leg(x,3),xaxis)
legfour=map(x->leg(x,4),xaxis)
plot(xaxis,legzero,label=L"L_0(x)")