Convergence of the Method

Transcription

1 Gaussian Method - Integration Convergence of the Method Article Information Subject: This worksheet demonstrates the convergence of the Gaussian method of integration. Revised: 4 October 24 Authors: Nathan Collier, Autar Kaw, Loubna Guennoun Version: Mathcad 21 Introduction Gauss Quadrature Rule is another method of estimating an integral. The theory behind the point Gauss Quadrature Rule is to approximate the integral by taking the area under a stra line connecting any two points on the curve that are not predetermined as a and b, but as unknowns x 1 and x 2. For n-points rules, the general form to approximate the integral is b a fx ( ) ( ) ( ) ( ) d x= c fx + c 1 fx c n fx n where c i and x i are the weighting factors and function arguments used in Gauss Quadrature formulas, respectively. However, these factors and arguments are already defined to approximate any integral from -1 to 1. To be able to use them, the limits of the integral of t function f(x) need to be changed to [-1,1]. b a fx ( ) d x= 1 f 1 b a b + a x b a dx 2 NOTE: Weighting factors c and function arguments x used in Gauss Quadrature Rule have already been defined in the textbook for up to six points. The following procedure will illustrate the Gauss Quadrature Rule of integration. The user enter any function f(x), the lower and upper limit for the function, and the number of point the data section (up to six points). By entering this data, the program will calculate the exa value of the integral, followed by the results using the Gauss Quadrature Rule with n point The program will also display the true error, the absolute relative true percentage error, the approximate error, the absolute relative approximate percentage error, and the number of significant digits that are at least correct. 1 1/17/27

2 Inputs Integrand f(x) f( x) := 3 x 1 + e x Lower limit of the integral a a := Upper limit of the integral b b := 1 Maximum number of points, n. Note that n is allowed to be between 1 and 6.n := 6 2 1/17/27

3 Procedure for Gaussian Method First the weighting factors and functional arguements must be defined for up to 6 point Weighting Factors Function Arguments 1 point C 1, := 2. X 1, :=. 2 points C 1, 1 := 1. X 1, 1 := C 2, 1 := 1. X 2, 1 := points C 1, 2 := X 1, 2 := C 2, 2 := X 2, 2 :=. C 3, 2 := X 3, 2 := points C 1, 3 := X 1, 3 := C 2, 3 := X 2, 3 := C 3, 3 := X 3, 3 := C 4, 3 := X 4, 3 := points C 1, 4 := X 1, 4 := C 2, 4 := X 2, 4 := C 3, 4 := X 3, 4 :=. C 4, 4 := X 4, 4 := C 5, 4 := X 5, 4 := points C 1, 5 := X 1, 5 := C 2, 5 := X 2, 5 := C 3, 5 := X 3, 5 := C 4, 5 := X 4, 5 := C 5, 5 := X 5, 5 := C 6, 5 := X 6, 5 := /17/27

4 The integral given above has the limits of [a,b]. It needs to be converted into an integral w limits [-1,1] f new (x) is the new function that will be used for evaluating the integral using Gauss Quadrature rule f new ( x) := f b a x + 2 b + a 2 b a 2 The following procedure determines the approximate value of the integral. gauss( n) := n i = 1 ( C i, n 1 f new ( X i, n 1 )) range := 1, 2.. n 4 1/17/27

5 Exact Solution In this section, the program will evaluate the exact value for the integral of the function f( evaluated at the limits a and b. s exact := b a fx ( ) dx s exact = Figure 1: Entered function on given interval f(x) 5 1/17/27

6 6 1/17/27

7 The true error (E t ): E t ( n) := s exact gauss( n) The absolute relative true percentage error (ε t ): ε t ( n) := E t ( n) 1 s exact The approximate error (E a ): E a ( n) := gauss( n) gauss( n 1) if n > 1 "N/A" if n 1 The absolute relative approximate percentage error (ε a ): ε a ( n) := E a ( n) gauss( n) "N/A" if n 1 1 if n > 1 The least significant digits correct in your answer: Sig( n) := if n > 1 trunc 2 log otherwise ε a ( n).5 if ε a ( n) 5 otherwise 7 1/17/27

8 The following organizes the results in a table for display: Results := for i.. n 1 M M i, i+ 1 ( ) (, ) M i, 1 gauss M i, M i, 2 E t M i ( ) (, ) M i, 3 ε t M i, M i, 4 E a M i ( ) (, ) M i, 5 ε a M i, M i, 6 Sig M i Results = Number of Points Approximate Value True Error Relative True Error Approximate Error Least Relative Number of Approximate Significant Error Digits "N/A" "N/A" /17/27

9 Conclusions The following data and graphs show the approximate value of the integral, true error, absolute relative true percentage error, approximate error, absolute relative approximate percentage error, and least number of signficant digits as functions of number of points. Approximate Value Figure 2: Approximate value of the integral as a function of the number of points Approximate value of the integral 9 1/17/27

10 True Error Figure 3: True error as a function of the number of points True error Absolute Relative True Percentage Error Figure 4: Absolute relative true percentage error as a function of the number of points Absolute relative true percentage error 1 1/17/27

11 Approximate Error Figure 5: Approximate error as a function of the number of points Approximate error Absolute Relative Approximate Percentage Error Figure 6: Absolute relative approximate percentage error as a function of the number of points Absolute relative approximate percentage error 11 1/17/27

12 Least Number of Significant Digits Correct Figure 7: Least number of significant digits correct as a function of the number of points Least number of significant digits correct 12 1/17/27

13 13 1/17/27