ECE 5340/6340�� NUMERICAL INTEGRATION�� Page 1

a) TRAPEZOIDAL INTEGRATION

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�� =� Area under curve

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�� 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)

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�� h��

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�� Calculate the area under each trapezoid

�� Area =

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�� Area =�

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�� Area�� =�

�� TRAPEZOIDAL RULE

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�� Assuming equal-spaced points

ECE5340/6340�� NUMERICAL INTEGRATION�� PAGE 2

SIMPSON�S INTEGRATION

Approximate �by a set of 2^nd order LaGrange Polynomials

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LaGrange Polynomial:

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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)��

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