[SOLVED] MATH 151A Assignment 4

Description

1. Let f(x) be a function defined on the interval [−1,1], and f ∈ C⁴[−1,1] .
   • Let h(x) be the Lagrange interpolation polynomial of f(x) at the nodes x = −1,0, Write down the expression of h(x).
   • Write down the error term E(x) := f(x) − h(x) in terms of the derivatives of f(x). (Recall the theorem about the error between the interpolation formula h and the exact function f.)
   • Compute the integral

exactly in terms of the values of f(x) at points x = −1,0,1.

• If we approximate the integral by    , is it true that the above

approximation is exact if f is a polynomial of degree less than or equal to 2 ? Why ?

• Write down an error bound of this approximation rule suggested in (d) directly basedon the result in (b).

2. A function f has the values shown as below:

• Use Simpson’s Rule and only the function values at x = 0,2,4 to approximate the

integral.

• Use composite Simpson’s Rule and the functions values at x = 0,1,2,3,4 to approx-

imate the same integral .

3. (Programming problem) Consider the integral:

1

• Write a program to use the composite trapezoidal to approximate the above integralby dividing [0,π] to N equal spaces.
• Write a program to use the composite Simpson’s approximate the above integral bydividing [0,π] to N equal spaces.

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