Introduction to Numerical Analysis I

MATH/CMPSC 455

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 FORMULASIdea: 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 RULEUse two nodes: and

SIMPSON’S RULEUse three nodes:

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

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