{"id":192294,"date":"2025-02-17T12:51:19","date_gmt":"2025-02-17T12:51:19","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=192294"},"modified":"2025-02-17T12:51:21","modified_gmt":"2025-02-17T12:51:21","slug":"write-a-matlab-function-called-fixed_point_iteration-that-inputs-a-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/17\/write-a-matlab-function-called-fixed_point_iteration-that-inputs-a-function\/","title":{"rendered":"Write a MATLAB function, called fixed_point_iteration that inputs a function"},"content":{"rendered":"\n<p>Write a MATLAB function, called fixed_point_iteration that inputs a function<\/p>\n\n\n\n<p>Fixed point iteration function solutions submitted (max: Unlimited) Write a MATLAB function, called fixed_point_iteration that inputs a function, g, an initial guess Xo, an error tolerance, tol, and a maximum number of iterations, N, and outputs the fixed point of g, obtained using the fixed point iteration, starting with Xo. Your function should have an error defined by E = (x2 \u2013 Xn-1), and stop when the error is less than the tolerance, or if the number of iterations exceeds N &#8211; whichever happens first. Your function header should look something like:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-803.png\" alt=\"\" class=\"wp-image-192295\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Here is the MATLAB function for performing fixed point iteration based on the problem description:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>function &#91;c, n, err] = fixed_point_iteration(g, xe, tol, N)\n    % Fixed point iteration method to find the root of g(x) = x\n    % INPUTS:\n    %   g   - the function g(x)\n    %   xe  - initial guess\n    %   tol - error tolerance\n    %   N   - maximum number of iterations\n    % OUTPUTS:\n    %   c   - the computed fixed point\n    %   n   - the number of iterations made\n    %   err - the final error\n\n    % Initialize values\n    x0 = xe; % initial guess\n    n = 0;    % iteration counter\n    err = Inf; % initialize error with a large value\n\n    % Start the iteration loop\n    while n &lt; N\n        % Compute the next approximation using the iteration g(x)\n        x1 = g(x0);\n\n        % Compute the error\n        err = abs(x1 - x0);\n\n        % Check if the error is below the tolerance\n        if err &lt; tol\n            break; % exit the loop if tolerance is satisfied\n        end\n\n        % Update x0 for the next iteration\n        x0 = x1;\n\n        % Increment iteration counter\n        n = n + 1;\n    end\n\n    % Output the result\n    c = x1; % final value is the computed fixed point\nend<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The <strong>fixed_point_iteration<\/strong> function is used to find the fixed point of a given function ( g(x) ). It starts with an initial guess, iteratively updates the guess using the fixed-point formula, and continues until either the error becomes smaller than a specified tolerance or the maximum number of iterations is reached.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Inputs:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>g<\/code>: The function for which we&#8217;re solving ( g(x) = x ).<\/li>\n\n\n\n<li><code>xe<\/code>: The initial guess ( x_0 ).<\/li>\n\n\n\n<li><code>tol<\/code>: The error tolerance. When the error between successive guesses is less than this value, the iteration stops.<\/li>\n\n\n\n<li><code>N<\/code>: The maximum number of iterations.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Outputs:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>c<\/code>: The estimated fixed point (root of ( g(x) = x )).<\/li>\n\n\n\n<li><code>n<\/code>: The number of iterations performed.<\/li>\n\n\n\n<li><code>err<\/code>: The final error after the last iteration.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Process:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The function initializes the first guess and enters a loop to perform the fixed-point iteration.<\/li>\n\n\n\n<li>For each iteration, the new approximation is computed using ( x_1 = g(x_0) ), and the error ( \\text{err} = |x_1 &#8211; x_0| ) is calculated.<\/li>\n\n\n\n<li>The loop continues until the error is smaller than the tolerance, or the maximum number of iterations is reached.<\/li>\n<\/ul>\n\n\n\n<p>The plot above illustrates how the method works: it finds the point where the function ( g(x) ) intersects the line ( y = x ), which represents the fixed point.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Write a MATLAB function, called fixed_point_iteration that inputs a function Fixed point iteration function solutions submitted (max: Unlimited) Write a MATLAB function, called fixed_point_iteration that inputs a function, g, an initial guess Xo, an error tolerance, tol, and a maximum number of iterations, N, and outputs the fixed point of g, obtained using the fixed [&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 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