First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 5. APPROXIMATION THEORY 209
In [5]: using Base.MathConstants
The inner product, 〈 ∫
e x , 3 2 x2 − 2〉 1 1
=
−1 ex ( 3 2 x2 − 1 )dx is computed as
2
In [6]: gauss(x->((3/2)*x^2-1/2)*e^x)
Out[6]: 0.1431256282441218
Nowthatwehaveacodeleg(x,n) that generates the Legendre polynomials, we
can do the above computation without explicitly specifying the Legendre polynomial.
For example, since 3 2 x2 − 1 = L 2 2, we can apply the gauss function directly to L 2 (x)e x :
In [7]: gauss(x->leg(x,2)*e^x)
Out[7]: 0.14312562824412176
〈f,L j 〉
α j
The following function polyLegCoeff(f::Function,n) computes the coefficients
〈f,L j 〉
α j
L j (x),j =0, 1, ..., n, for any
of the least squares polynomial P n (x) = ∑ n
j=0
f and n, where L j are the Legendre polynomials. Note that the indices are shifted in
thecodesothatj starts at 1 not 0. The coefficients are stored in the global array A.
In [8]: function polyLegCoeff(f::Function,n)
global A=Array{Float64}(undef,n+1)
for j in 1:n+1
A[j]=gauss(x->leg(x,(j-1))*f(x))*(2(j-1)+1)/2
end
end
Out[8]: polyLegCoeff (generic function with 1 method)
Once the coefficients are computed, evaluating the polynomial can be done efficiently
by calling the array A. The next function polyLeg(x,n) evaluates the least
squares polynomial P n (x) = ∑ n 〈f,L j 〉
j=0 α j
L j (x),j =0, 1, ..., n, where the coefficients 〈f,L j〉
α j
are obtained from the array A.
In [9]: function polyLeg(x,n)
sum=0.;
for j in 1:n+1