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

2. Numerical Integration
Mathematical Problem:
Example:
Example:

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

4. 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:

5. Trapezoid rule
Use two nodes: and

6. Simpson’s rule
Use three nodes:

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

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

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

10. 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

11. The Composite Simpson’s Rule

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

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