NUMERICAL INTEGRATION. By : Dewi Rachmatin

Transcription

1 NUMERICAL INTEGRATION By : Dewi Rachmatin

2 The Trapezoidal Rule Theorem (Trapezoidal Rule) Consider y=f(x) over [x 0,x 1 ], where x 1 =x 0 +h. The trapezoidal rule is This is an numerical approximation to the integral of f(x) over [x 0,x 1 ] and we have the expression

3 The remainder term for the trapezoidal rule is where c lies somewhere between x0 and x1 have the equality

4 Composite Trapezoidal Rule An intuitive method of finding the area under a curve y = f(x) is by approximating that area with a series of trapezoids that lie above the intervals. When several trapezoids are used, we call it the composite trapezoidal rule.

5 Theorem (Composite Trapezoidal Rule) Consider y=f(x) over [a,b]. Suppose that the interval [a,b] is subdivided into m subintervals of equal width by using the equally spaced nodes x k =x 0 +kh for k=1,2,,m. The composite trapezoidal rule for m subintervals is

6 That is an numerical approximation to the integral of f(x) over [a,b] and we write

7 Algorithm Composite Trapezoidal Rule To approximate the integral by sampling f(x) at the m+1 equally spaced points x k =a+kh for k=0,1,,m, where. Notice that x 0 = a and x m = b.

8 Mathematica Subroutine (Trapezoidal Rule)

9 Example 1. Numerically approximate the integral by using the trapezoidal rule with m = 1, 2, 4, 8, and 16 subintervals. Example 2. Numerically approximate the integral in example 1 by using the trapezoidal rule with m = 50, 100, 200, 400 and 800 subintervals.

10 Simpson's Rule The numerical integration technique known as "Simpson's Rule" is credited to the mathematician Thomas Simpson ( ) of Leicestershire, England. His also worked in the areas of numerical interpolation and probability theory.

11 Theorem (Simpson's Rule) Consider y=f(x) over [x 0,x 2 ], where x 1 =x 0 +h and x 2 =x 0 +2h. Simpson's rule is This is an numerical approximation to the integral of over and we have the expression

12 The remainder term for Simpson's rule is where lies somewhere between x 0 and x 2 have the equality

13 Composite Simpson Rule Our next method of finding the area under a curve y=f(x) is by approximating that curve with a series of parabolic segments that lie above the intervals When several parabolas are used, we call it the composite Simpson rule.

14 Theorem (Composite Simpson's Rule) Consider y=f(x) over [a,b]. Suppose that the interval [a,b] is subdivided into 2m subintervals of equal width by using the equally spaced sample points x k =x 0 +kh for k=0,1,,m.

15 The composite Simpson's rule for subintervals is This is an numerical approximation to the integral of f(x) over [a,b] and we write

16 Algorithm Composite Simpson Rule To approximate the integral by sampling f(x) at the 2m+1 equally spaced sample points x k =a+kh for k=0,1,,2m, where. Notice that x 0 = a and x 2m = b.

17 Mathematica Subroutine (Simpson Rule)

18 Example 1. Numerically approximate the integral by using Simpson's rule with m = 1, 2, 4, and 8 subintervals. Example 2. Numerically approximate the integral in example 1 by using the Simpson's rule with m = 10, 20, 40, 80, and 160 subintervals.

19 Gauss-Legendre Quadrature We wish to find the area under the curve y=f(x) for -1 x 1. What method gives the best answer if only two function evaluations are to be made? We have already seen that the trapezoidal rule is a method for finding the area under the curve that uses two function evaluations at the endpoints (-1,f[-1]) and (1,f[1]). But if the graph of y = f(x) is concave, the error in approximation is the entire region that lies between the curve and the line segment joining the points.

20 If we are permitted to use the nodes x 1 and x 2 that lie inside the interval [-1,1], the line through the two points (x 1,f(x 1 )) and (x 2,f(x 2 )) crosses the curve, and the area under the line more closely approximates the area under the curve. This method is attributed to Johann Carl Friedrich Gauss ( ) and Adrien-Marie Legendre ( ).

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23 Theorem (Gauss- Legendre Quadrature) An approximation to the integral is obtained by sampling at f(x) the n unequally spaced abscissas where the corresponding weights are. The abscissa's and weights for Gauss- Legendre quadrature are often expressed in decimal form.

24 n=2 Rule where n=3 Rule where

25 Nilai-nilai w n, x n dan galat pemotongan untuk Kuadratur Gauss-Legendre 6 titik n Bobot w n Absis x n Galat Pemotongan 2 1, , , , , , , , , , , , , , , , , f(4) (c) f(6) (c) f(8) (c)

26 n Bobot w n Absis x n Galat Pemotongan 5 0, , , , , , , , , , , , , , , , , , , , , f(10) (c) f(12) (c)

27 The shifted Gauss- Legendre rule for [a,b] Theorem (The Gauss-Legendre Translation). Suppose that the abscissas and weights are given for the n-point Gauss-Legendre rule over [-1,1]. To apply the rule over the interval [a,b], use the change of variable Then the relationship is used to obtain the quadrature formula

28 Example 1. Use the Gauss-Legendre quadrature rules for n = 2, 3, 4 and 5 points to compute numerical approximations for. Example 2. Use the Gauss-Legendre quadrature rules for n = 2, 3, 4 and 5 points to compute numerical approximations for

29 Example 1. Use the shifted Gauss- Legendre rules for n = 3 points to compute approximations for the integrals