Introduction to Numerical Analysis I MATH/CMPSC 455 Introduction to Numerical Analysis I Numerical Integration

Numerical Integration Mathematical Problem: Example: Example:

By calculus, find that , then use Numerical Integration: replace by another function that approximates well and is easily integral, then we have

Newton-Cotes Formulas Idea: use polynomial interpolation to find the approximation function Step 1: Select nodes in [a,b] Step 2: Use Lagrange form of polynomial interpolation to find the approximation function Step 3:

Trapezoid rule Use two nodes: and

Simpson’s rule Use three nodes:

Example: Apply the Trapezoid Rule and Simpson’s Rule to approximate

Error of the trapezoid rule: The trapezoid rule is exact for all polynomial of degree less than or equal to 1.

Error of the Simpson’s rule: The Simpson’s rule is exact for all polynomial of degree less than or equal to 3.

The Composite Trapezoid Rule Why? ? The high order polynomial interpolations are unbounded! Step 1: Partition the interval into n subintervals by introducing points Step 2: Use the trapezoid rule on each subinterval Step 3: Sum over all subintervals

The Composite Simpson’s Rule

Error of Composite Rules Error of the composite trapezoid rule: Error of the composite Simpson’s rule:

Example: Apply the composite Trapezoid Rule and Simpson’s Rule ( 4 subintervals ) to approximate