ECE 5340/6340 NUMERICAL INTEGRATION Page 1

a) TRAPEZOIDAL INTEGRATION

= Area under curve

Numerical Integration approximates the curve by a function and integrates this new approximate function.

Trapezoidal integration approximates the function by a line (1^st order polynomials).

Error

Straight lines are approximations of curved function f(x)

Calculate the area under each trapezoid

Area =

TRAPEZOIDAL RULE

Assuming equal-spaced points

ECE5340/6340 NUMERICAL INTEGRATION PAGE 2

SIMPSON’S INTEGRATION

Approximate by a set of 2^nd order LaGrange Polynomials

LaGrange Polynomial:

Third order derivatives cancel out for equal-spaced points.

SIMPSON’S RULE

Break region into even # of regions using odd # of points

Alternative form of algorithm:

To Calculate Expected Error for Realistic Simulation:

For Trapezoidal Integration, error is ON THE ORDER OF h³ f ''

For Simpson Integration, error is on the order of h⁵ f ⁽⁴⁾

BUT we don't know the derivatives f '' and f ⁽⁴⁾.

To approximate these

(a) Most accurate, most work: Calculate the derivatives of f(x) at the central location of each region (x= h/2, 3h/2, 5h/2, etc., and average these derivatives. The derivatives must be calculated numerically, because we assume we do not know f(x) (If we did, we could usually find a way to integrate it).

(b) Less accurate, less work: Calculate the derivatives at an average location (x = a + (b-a)/2)

(c) Even less accurate, even less work, most likely to be used in practice: If you have an idea what the order of the derivative is from the physical understanding of your application, use that. Otherwise, assume the derivative is on the order of 1.

For HW assignment #1, when you are asked to find the approximate error, use method b and compare with method c. IF you had only numerical data and could not calculate the derivative analytically, which of these methods would you be satisfied with?

Some tempting stumbling blocks:

What do you do with a knowledge of the error? How about subtracting it from your solution to get a better answer? No, don't do this. You only know the ORDER of the error, and it could be either positive or negative, and won't be exactly what you calculate. Use this ONLY to determine the decimal place or degree or accuracy of your solution.

2-D INTEGRATION (TRAPEZOIDAL METHOD)