Numerical Methods
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1 Numerical Methods Prof. M. K. Banda Botany Building: Prof. M. K. Banda (Tuks) WTW263 Semester II 1 / 27
2 Theme 1: Solving Nonlinear Equations Prof. M. K. Banda (Tuks) WTW263 Semester II 2 / 27
3 Fixed Point Iteration Learning Outcomes After completion of this unit you must: be able to write an equation f (x) = 0 in fixed point form g(x) = x; know that a fixed point of a function g is a solution of the equation g(x) = x; be able to use fixed point iteration for finding approximate solutions of equations; Prof. M. K. Banda (Tuks) WTW263 Semester II 3 / 27
4 Fixed Point Iteration Learning Outcomes be able to define and calculate the error, absolute error and the relative error of an estimate and be able to use it; be able to explain convergence/divergence of a sequence; know that if a sequence of fixed point iteration converges, then the limit of the sequence is a fixed point of the function; Prof. M. K. Banda (Tuks) WTW263 Semester II 4 / 27
5 Fixed Point Iteration Learning Outcomes be able to apply and prove the theorem establishing a condition for the uniqueness of a fixed point; be able to apply the theorems that claim convergence/divergence of a sequence generated by fixed point iterations; be able to illustrate graphically the convergence/divergence of sequences generated by fixed point iteration. Prof. M. K. Banda (Tuks) WTW263 Semester II 5 / 27
6 Fixed Point Iteration Bracketing Versus Open Methods Open methods differ from interval division (bracketing) methods, in that open methods require only a single starting value or two starting values that do not necessarily bracket a root. Prof. M. K. Banda (Tuks) WTW263 Semester II 6 / 27
7 Bracketing Vs Open Methods Fixed Point Iteration Figure: Source: A.Gilat, V. Subramaniam, Prof. M. K. Banda (Tuks) WTW263 Semester II 7 / 27
8 Assume a root has been isolated then consider f (x) = 0. Idea: Rearrange f (x) = 0 so that x is on the left-hand side of the equation: x = g(x) or x g(x) = 0. Prof. M. K. Banda (Tuks) WTW263 Semester II 8 / 27
9 If x is a solution of f (x) = 0, then x g(x) = 0. The point of intersection of the function y = x and y = g(x) is called a fixed point. g(x) is called the iteration function. Use g to predict a new approximation of root, x: x k = g(x k 1 ). Prof. M. K. Banda (Tuks) WTW263 Semester II 9 / 27
10 y Solving Nonlinear Equations Example: solve x = cos x or x cos x = 0. Recall: the solution lies in (0, π/2). x cos(x) x Prof. M. K. Banda (Tuks) WTW263 Semester II 10 / 27
11 Example: solve x = cos x or x cos x = 0. Take an initial guess: x 0 = 0. Use x k = g(x k 1 ) where g(x k 1 ) = cos(x k 1 ), k = 1, 2,.... x 1 = cos x 0 = cos 0 = 1; x 2 = cos x 1 = cos 1 = ; x 3 = cos x 2 = cos = ; Prof. M. K. Banda (Tuks) WTW263 Semester II 11 / 27
12 Example: solve x = cos x or x cos x = 0. Take an initial guess: x 0 = 0. x 1 = cos x 0 = cos 0 = 1; x 2 = cos x 1 = cos 1 = ; x 3 = cos x 2 = cos = ; Class Exercise: Find x 4 and x 5! Prof. M. K. Banda (Tuks) WTW263 Semester II 12 / 27
13 Example: solve x = cos x or x cos x = 0. k x k Prof. M. K. Banda (Tuks) WTW263 Semester II 13 / 27
14 Example: solve x = cos x or x cos x = 0. k x k Prof. M. K. Banda (Tuks) WTW263 Semester II 14 / 27
15 Example: solve x = cos x or x cos x = 0. k x k Note: x k+1 lies between x k 1 and x k. Prof. M. K. Banda (Tuks) WTW263 Semester II 15 / 27
16 Example 2 - Page 15 Solve f (x) = x 2 x 6 = 0 (Roots: x = 2 and x = 3.) Rewrite: x = x + 6 = g 1 (x) or x = x 2 6 = g 2 (x). Note: g 1 (x) will only approximate the positive root. Use x 0 = 3.1. Also imax = 10, err = 1e 06. Prof. M. K. Banda (Tuks) WTW263 Semester II 16 / 27
17 >> x = fpi(g1, x0, err, imax) iter. (xns) Sol g(xns) Tol x = Prof. M. K. Banda (Tuks) WTW263 Semester II 17 / 27
18 >> x = fpi(g2, x0, err, imax) iter (xns) Sol g(xns) Tol Solution was not obtained in 10 iterations. x = No answer Prof. M. K. Banda (Tuks) WTW263 Semester II 18 / 27
19 Example 2 - Page 15 Why did g 2 (x) fail to converge? Prof. M. K. Banda (Tuks) WTW263 Semester II 19 / 27
20 Convergence Assume x k approximates the root x. Expect x k x 0. Given E, an error tolerance, then require x k x E Note x k x is small if x k+1 x k is small. In practice, require x k+1 x k E. Note, the relative error can also be used. Prof. M. K. Banda (Tuks) WTW263 Semester II 20 / 27
21 Example Solve f (x) = e x x Re-write as x = g(x) by isolating x (example: x = e x ) Start with an initial guess: x 0 = 0 Prof. M. K. Banda (Tuks) WTW263 Semester II 21 / 27
22 Example Solving Nonlinear Equations k x k Err = x k+1 x k Class Exercise: Approximate the absolute errors. Prof. M. K. Banda (Tuks) WTW263 Semester II 22 / 27
23 Example Solving Nonlinear Equations k x k Err Continue until some tolerance is reached Prof. M. K. Banda (Tuks) WTW263 Semester II 23 / 27
24 Figure: Source: A.Gilat, V. Subramaniam, Prof. M. K. Banda (Tuks) WTW263 Semester II 24 / 27
25 Fixed Point Iteration Method Figure: Source: A.Gilat, V. Subramaniam, Prof. M. K. Banda (Tuks) WTW263 Semester II 25 / 27
26 Fixed Point Iteration Sometimes f (x) = 0 can not be re-written in the form x = g(x), it is advisable to use x + f (x) x = 0 Hence: x = x + f (x) = g(x) Prof. M. K. Banda (Tuks) WTW263 Semester II 26 / 27
27 Fixed Point Iteration Homework: Do Exercise 1.6. Prof. M. K. Banda (Tuks) WTW263 Semester II 27 / 27
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