Numerical Analysis

Math 370  MWF 2:30-3:25
Spring 1997  201 Fowler

(04.02.97) \underline{Numerical Integration} \underline{Warm Up} Let’s try and approximate the area of a unit circle using these methods of numerical quadrature you already know. We will do this by estimating the area under the curve f(x) = \sqrt{1 − x^{2}} from 0 to 1, depicted below:

How close did you get to the exact value of the area of the circle? What order do you think the quadrature methods you used are? i.e. O(h)? O(h^{2})?

Something else?

2-point Quadrature Rule Let’s assume that we only have 2 data points (x_{0}, f(x_{0})) and (x_{1}, f(x_{1})). We can use a first degree Lagrange polynomial to interpolate f(x), integrate it and thus derive a quadrature rule we call Error Term for Trapezoid Rule

Simpson’s Rule A similar method can be used to derive the formula for Simpson’s Rule. Assume that we have 3 data points (x_{0}, f_{0}), (x_{1}, f_{1}) and (x_{2}, f_{2}). Write down the formula one would have to evaluate to derive the 3-point quadrature rule more commonly known as Simpson’s Rule:

Do we believe this will get us the answer? What’s wrong with it?

Let’s try something else... First, find the 4th degree Taylor expansion of f(x) about the point x_{1} and write it down here:

∫_{x_{0}}^{x_{2}} f(x) dx

Definition of Precision A numerical quadrature method is said to have precision (or accuracy) n (where n is a positive integer) if E(P_{k}) = 0 for all polynomials P_{k}(x) of degree k ≤ n but E(P_{n+1}) ≠ 0

QUESTION: What degree of precision does Simpson’s Rule have?

QUESTION: What degree of precision does the Trapezoidal Rule have?