MA 780: Project #2, February 13.
Goal: Gaussian quadratures.
Problem 1 1. Find the expressions for the weights and nodes for GaussLobatto quadratures with n = 2, 3 and 4. Give those either by algebraic expressions or within a minimum of 15 accurate digits.
2. Read section 9.4 on composite Newton-Cotes formulas.
3. Consider the following integrals (examples 9.1, p.377, 9.2, p.381, 9.3,
p.382)
R 2π
−2π
π e−2π
,
• 0 xe−x cos(2x) dx = 3 (e −1)−10
25
•
R1
•
R5
0
x5/2 dx = 2/7,
1
−5 1+x2
dx = 2 arctan 5.
4. Compare the performance of Simpson, composite Simpson, GaussLegendre (seen in class), Gauss-Lobatto (derived above) for the above
three problems.
R1
Problem 2 We approximate the integral I = −1 f (x) dx by
In =
n
X
wk f (xk ).
(1)
k=0
1. Consider the code gauss.m below from N. Trefethen1 . Explain, line
by line, why this code corresponds to the Gauss-Legendre quadrature
seen in class.
2. The Clenshaw-Curtis quadratures are obtained by using Chebyshev
points as nodes, i.e., by using
xj = cos
jπ
,
n
j = 0, . . . , n
in (1). The weights are found by requiring that In = I for f ∈ Pn .
Find these weights and explain why the code clenshaw curtis.m
below corresponds to this type of quadrature.
3. Reproduce the results (figures) of Figure 2 from the paper cited under
point 1.
1 https://people.maths.ox.ac.uk/trefethen/publication/PDF/2008_127.pdf
function I = gauss(f,n)
beta = .5./sqrt(1−(2*(1:n)).ˆ(−2));
T = diag(beta,1) + diag(beta,−1);
[V,D] = eig(T);
x = diag(D); [x,i] = sort(x);
w = 2*V(1,i).ˆ2;
I = w*feval(f,x);
%
%
%
%
%
%
%
function I = clenshaw curtis(f,n)
x = cos(pi*(0:n)'/n);
fx = feval(f,x)/(2*n);
g = real(fft(fx([1:n+1 n:−1:2])));
a = [g(1); g(2:n)+g(2*n:−1:n+2); g(n+1)];
w = 0*a'; w(1:2:end) = 2./(1−(0:2:n).ˆ2);
I = w*a;
(n+1)−pt Gauss quadrature of f
3−term recurrence coeffs
Jacobi matrix
eigenvalue decomposition
nodes (= Legendre points)
weights
the integral% 3−term recurrence coeffs
%
%
%
%
%
%
%
(n+1)−pt C−C quadrature of f
Chebyshev points
f evaluated at these points
Fast Fourier Transform
Chebyshev coefficients
weight vector
the integral