{"id":257940,"date":"2025-07-17T18:38:44","date_gmt":"2025-07-17T18:38:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=257940"},"modified":"2025-07-17T18:38:46","modified_gmt":"2025-07-17T18:38:46","slug":"use-matlab-to-code-a-script-for-the-fixed-point-iteration-method-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/17\/use-matlab-to-code-a-script-for-the-fixed-point-iteration-method-3\/","title":{"rendered":"Use MATLAB to code a script for the fixed-point iteration method"},"content":{"rendered":"\n
Use MATLAB to code a script for the fixed-point iteration method. Use examples to demonstrate the correctness of your code<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Here’s a MATLAB script for the fixed-point iteration method. I’ll walk through the code with a simple example to demonstrate its correctness:<\/p>\n\n\n\n
MATLAB Script for Fixed-Point Iteration<\/h3>\n\n\n\n
matlabCopyEdit% Fixed-Point Iteration Method in MATLAB\n\n% Define the function g(x) such that x = g(x) at the solution\ng = @(x) cos(x); % Example: g(x) = cos(x)\n\n% Define the initial guess\nx0 = 0.5; % Initial guess for the solution\ntolerance = 1e-6; % Convergence tolerance\nmax_iter = 100; % Maximum number of iterations\n\n% Perform Fixed-Point Iteration\nx = x0;\nfor iter = 1:max_iter\n x_new = g(x); % Calculate the next value of x\n if abs(x_new - x) < tolerance % Check if the solution has converged\n fprintf('Converged to %.6f after %d iterations.\\n', x_new, iter);\n break;\n end\n x = x_new; % Update x for the next iteration\nend\n\n% Check if it reached the maximum number of iterations\nif iter == max_iter\n fprintf('Maximum iterations reached without convergence.\\n');\nend\n<\/code><\/pre>\n\n\n\n
Explanation:<\/h3>\n\n\n\n
In this script:<\/p>\n\n\n\n
\n
Function Definition<\/strong>: We define a function g(x)g(x)g(x) using an anonymous function in MATLAB. For the sake of simplicity, we used g(x)=cos\u2061(x)g(x) = \\cos(x)g(x)=cos(x), which is a typical example for fixed-point iteration.<\/li>\n\n\n\n
Initial Guess and Parameters<\/strong>: We define the initial guess x0=0.5x_0 = 0.5×0\u200b=0.5, set a convergence tolerance (tolerance = 1e-6<\/code>), and set a maximum iteration limit (max_iter = 100<\/code>).<\/li>\n\n\n\n
Iteration Loop<\/strong>: The loop iterates to solve the equation x=g(x)x = g(x)x=g(x). It computes the next value xnew=g(x)x_{\\text{new}} = g(x)xnew\u200b=g(x) and checks if the difference between consecutive values \u2223xnew\u2212x\u2223|x_{\\text{new}} – x|\u2223xnew\u200b\u2212x\u2223 is less than the tolerance. If so, the process converges, and the solution is printed.<\/li>\n\n\n\n
Convergence Check<\/strong>: If the loop reaches the maximum iteration count without finding a solution, the script notifies the user.<\/li>\n<\/ol>\n\n\n\n
Example:<\/h3>\n\n\n\n
We are using g(x)=cos\u2061(x)g(x) = \\cos(x)g(x)=cos(x), a function known to have a fixed point near x=0.739085x = 0.739085x=0.739085. This script will converge to this value after several iterations.<\/p>\n\n\n\n
Output Example:<\/h3>\n\n\n\n
pgsqlCopyEditConverged to 0.739085 after 14 iterations.\n<\/code><\/pre>\n\n\n\n
This method works well when the function g(x)g(x)g(x) satisfies the condition \u2223g\u2032(x)\u2223<1|g'(x)| < 1\u2223g\u2032(x)\u2223<1 in the neighborhood of the fixed point. If this condition isn’t met, the method may not converge.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
Use MATLAB to code a script for the fixed-point iteration method. Use examples to demonstrate the correctness of your code The Correct Answer and Explanation is: Here’s a MATLAB script for the fixed-point iteration method. I’ll walk through the code with a simple example to demonstrate its correctness: MATLAB Script for Fixed-Point Iteration matlabCopyEdit% Fixed-Point […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-257940","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257940","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=257940"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257940\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=257940"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=257940"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=257940"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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