{"id":257928,"date":"2025-07-17T18:29:47","date_gmt":"2025-07-17T18:29:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=257928"},"modified":"2025-07-17T18:29:50","modified_gmt":"2025-07-17T18:29:50","slug":"use-matlab-to-code-a-script-for-the-fixed-point-iteration-method","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\/","title":{"rendered":"Use MATLAB to code a script for the fixed-point iteration method"},"content":{"rendered":"\n<p>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<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The fixed-point iteration method is an iterative numerical technique used to approximate the solution of an equation of the form:x=g(x)x = g(x)x=g(x)<\/p>\n\n\n\n<p>Where g(x)g(x)g(x) is a function derived from the original equation f(x)=0f(x) = 0f(x)=0. The method starts with an initial guess x0x_0x0\u200b and iteratively computes xn+1=g(xn)x_{n+1} = g(x_n)xn+1\u200b=g(xn\u200b) until the value converges to the solution.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">MATLAB Script for Fixed-Point Iteration<\/h3>\n\n\n\n<p>Here&#8217;s a MATLAB script to implement the fixed-point iteration method:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">matlabCopyEdit<code>% Fixed-Point Iteration Method\n% The function g(x) should be derived from the equation f(x) = 0\n\n% Define the function g(x)\ng = @(x) cos(x);  % Example: g(x) = cos(x)\n\n% Define the tolerance and maximum number of iterations\ntolerance = 1e-6;\nmax_iter = 100;\n\n% Initial guess\nx0 = 0.5;\n\n% Fixed-Point Iteration\nx = x0;  % Starting value\niter = 0;  % Iteration counter\n\nfprintf('Iteration\\tValue of x\\n');\nfprintf('%d\\t%.6f\\n', iter, x);\n\nwhile iter &lt; max_iter\n    x_new = g(x);  % Compute the next approximation\n    iter = iter + 1;  % Increment the iteration counter\n    \n    % Display the iteration and current value\n    fprintf('%d\\t%.6f\\n', iter, x_new);\n    \n    % Check for convergence\n    if abs(x_new - x) &lt; tolerance\n        fprintf('Convergence achieved after %d iterations\\n', iter);\n        break;\n    end\n    \n    % Update x for next iteration\n    x = x_new;\nend\n\nif iter == max_iter\n    fprintf('Maximum iterations reached without convergence\\n');\nend\n<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Function Definition<\/strong>:\n<ul class=\"wp-block-list\">\n<li>The function <code>g(x)<\/code> is specified as <code>cos(x)<\/code> in this example. You can modify this to reflect the specific function you are working with.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Initial Guess and Parameters<\/strong>:\n<ul class=\"wp-block-list\">\n<li>The initial guess (<code>x0<\/code>) is set to 0.5. The tolerance defines how close the iterated value must be to the previous value to stop the iterations. A <code>max_iter<\/code> value ensures the loop terminates after a set number of iterations to avoid infinite loops.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Iteration Loop<\/strong>:\n<ul class=\"wp-block-list\">\n<li>The loop iteratively computes the next value xn+1x_{n+1}xn+1\u200b from g(xn)g(x_n)g(xn\u200b) and checks for convergence by comparing the difference between successive values. If the difference is less than the tolerance, the process stops.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Output<\/strong>:\n<ul class=\"wp-block-list\">\n<li>The script outputs the iteration number and the current value of xxx at each step. It also prints a message if convergence is achieved or if the maximum number of iterations is reached without convergence.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p>For the equation x=cos\u2061(x)x = \\cos(x)x=cos(x), the fixed-point iteration should converge to the value approximately x=0.739085x = 0.739085x=0.739085. Running this MATLAB script will demonstrate how the value approaches the solution with each iteration.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>This script provides a straightforward implementation of the fixed-point iteration method. You can adapt it by modifying the function <code>g(x)<\/code> for different equations, and adjusting the tolerance and maximum iterations as needed for specific problems.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-845.jpeg\" alt=\"\" class=\"wp-image-257929\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>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: The fixed-point iteration method is an iterative numerical technique used to approximate the solution of an equation of the form:x=g(x)x = g(x)x=g(x) Where g(x)g(x)g(x) is a function derived from [&hellip;]<\/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-257928","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257928","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=257928"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257928\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=257928"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=257928"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=257928"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}