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NM4_integration_s02 - CGN 3421 Computer Methods Gurley...

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CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 4 - Numerical Integration / Differentiation page 83 of 88 Numerical Methods Lecture 4 - Numerical Integration / Differentiation Numerical Integration - approximating the area under a function We will consider 3 methods: Trapezoidal rule, Simpson’s rule, Gauss Quadrature The basic concept is to break the area up into smaller pieces with simple shapes fitted to approximate the func- tion over short lengths. The area under these simple shapes are then added up to approximate the total. Trapezoidal rule - piece-wise linear approximation to curve f(x) • chop area into N smaller pieces (sample of two pieces on the right) • connect areas with straight lines, creating trapezoids • find the area under each trapezoid A=1/2*dx*(f(x)+f(x+dx)) • add up the trapezoids Example, the figure at the bottom of the page chops a function into 4 trapezoids over the range [a, b]. The trapeziodal rule adds up the area of each of the 4 trape- zoids to represent the area under the fucntion f(x). dx = (b-a)/N (a, b are limits, N is number of pieces we divide the area into) A1= 1/2 * dx * (f(x1) + f(x2)) A2 = 1/2* dx * ( f(x2) + f(x3)) A3 = 1/2* dx * ( f(x3) + f(x4)) A4 = 1/2* dx * ( f(x4) + f(x5)) _____________________________________________ Area = 1/2 * dx * (f(x1)+ 2 *f(x2)+ 2 *f(x3)+ 2 *f(x4)+f(x5)) The generalized equation expressing trapezoidal rule is then ==> Error: The error (in blue) between the area of the trapezoid and the curve it represents builds up Error reduction: We can reduce error if we can shorten the straight lines used to estimate curves. Increasing the number (increase N) of trapezoids used over the same range [a,b] will reduce error in the area estimate. A1 A2 x1 x2 x3 dx f(x) A1=1/2*dx*(f(x1)+f(x2)) !"#$ ! " -- %& ’ & ! ( ) " ’ & ( ( ) ( " # ) ’ & ) ! $ ( ) $ $ fx x a b Numerically estimate area under fx within [a, b] A1 A2 A3 A4 dx x1 x2 x3 x4 x5 Error
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