First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 5. APPROXIMATION THEORY 213
Out[13]: compsimpson (generic function with 1 method)
The integrals in Example 94 were computed using the composite Simpson rule with
n =20. For example, the second integral 〈e x ,T 1 〉 = ∫ π
0 ecos θ cos θdθ is computed as:
In [14]: compsimpson(x->exp(cos(x))*cos(x),0,pi,20)
Out[14]: 1.7754996892121808
Next we write two functions, polyChebCoeff(f::Function,n) and
polyCheb(x,n). The first function computes the coefficients 〈f,T j〉
of the least
〈f,T j 〉
α j
squares polynomial P n (x) = ∑ n
j=0
T j (x),j =0, 1, ..., n, for any f and n, where T j
are the Chebyshev polynomials. Note that the indices are shifted in the code so that
j starts at 1 not 0. The coefficients are stored in the global array A.
The integral 〈f,T j 〉 is transformed to the integral ∫ π
f(cos θ)cos(jθ)dθ, similar to
0
the derivation in Example 94, and then the transformed integral is computed using the
composite Simpson’s rule by polyChebCoeff.
In [15]: function polyChebCoeff(f::Function,n)
global A=Array{Float64}(undef,n+1)
A[1]=compsimpson(x->f(cos(x)),0,pi,20)/pi
for j in 2:n+1
A[j]=compsimpson(x->f(cos(x))*cos((j-1)*x),0,pi,20)*2/pi
end
end
Out[15]: polyChebCoeff (generic function with 1 method)
In [16]: function polyCheb(x,n)
sum=0.;
for j in 1:n+1
sum=sum+A[j]*cheb(x,(j-1))
end
return(sum)
end
Out[16]: polyCheb (generic function with 1 method)
α j