The Legendre polynomial of degree 3 is

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Special functions and orthogonal polynomials · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

P_3(x)=\tfrac12(5x^3-3x)

, and the three-point Clenshaw-Curtis nodes on

[-1,1]

are

x_k=\cos(k\pi/2)

for

k=0,1,2

. ((a)) Find the three roots of

P_3

(one of them is

x=0

) and take them as the nodes of a three-node Gauss-Legendre rule. Using the symmetry

x_1=-x_3

x_2=0

, write its weights as

w_1=w_3=a

and

w_2=b

, and find

a

and

b

by demanding exactness for

f(x)=1

and

f(x)=x^2

. ((b)) Check that the same

a,b

also make the rule exact for

f(x)=x^4

, and state the degree of exactness this three-node rule reaches. ((c)) Evaluate the three Clenshaw-Curtis nodes

x_k=\cos(k\pi/2)

, and find their weights

p

(at

x=\pm1

) and

q

(at

x=0

) from exactness for

f(x)=1

and

f(x)=x^2

. Identify the classical rule this reproduces. ((d)) Test the Clenshaw-Curtis rule of part (c) on

f(x)=x^4

and state its degree of exactness. Compare, at the same cost of three function evaluations, the degrees of exactness reached by the Gauss and Clenshaw-Curtis rules.

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