{"id":257933,"date":"2025-07-17T18:35:22","date_gmt":"2025-07-17T18:35:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=257933"},"modified":"2025-07-17T18:35:24","modified_gmt":"2025-07-17T18:35:24","slug":"use-matlab-to-code-a-script-for-the-fixed-point-iteration-method-2","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-2\/","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 how you can implement the Fixed-Point Iteration method in MATLAB:<\/p>\n\n\n\n

MATLAB Code<\/h3>\n\n\n\n
matlabCopyEdit% Define the fixed-point function\ng = @(x) cos(x);  % Example: g(x) = cos(x)\n\n% Fixed-point iteration method\nmaxIter = 100;  % Maximum number of iterations\ntolerance = 1e-6;  % Convergence tolerance\nx0 = 0.5;  % Initial guess\n\n% Initialize variables\nx = x0;\n\n% Fixed-point iteration loop\nfor k = 1:maxIter\n    x_new = g(x);  % Compute the next approximation\n    if abs(x_new - x) < tolerance  % Check for convergence\n        fprintf('Converged to %.6f after %d iterations.\\n', x_new, k);\n        break;\n    end\n    x = x_new;  % Update the current approximation\nend\n\n% If the loop completes without convergence\nif k == maxIter\n    fprintf('Did not converge after %d iterations.\\n', maxIter);\nend\n<\/code><\/pre>\n\n\n\n
Explanation<\/h3>\n\n\n\n
    \n
  1. Define the Fixed-Point Function:<\/strong>
    The function g(x)<\/code> represents the function that we use for fixed-point iteration. In this case, we choose the example g(x) = cos(x)<\/code>.<\/li>\n\n\n\n
  2. Set Parameters:<\/strong>
    The initial guess x0<\/code> is set to 0.5. The iteration will continue until either the maximum number of iterations (maxIter<\/code>) is reached or the absolute difference between consecutive approximations is smaller than the tolerance (tolerance<\/code>).<\/li>\n\n\n\n
  3. Iterative Process:<\/strong>
    We compute the next approximation x_new<\/code> using the function g(x)<\/code>. If the absolute difference between x_new<\/code> and x<\/code> (the current approximation) is smaller than the tolerance, the process is considered to have converged, and the loop terminates.<\/li>\n\n\n\n
  4. Convergence:<\/strong>
    If the algorithm converges, a message with the final value and the number of iterations is displayed. If it doesn’t converge within the maximum iterations, an error message is printed.<\/li>\n<\/ol>\n\n\n\n
    Example<\/h3>\n\n\n\n
    Let\u2019s use the function g(x) = cos(x)<\/code> and an initial guess of x0 = 0.5<\/code>. The expected outcome is that the function will converge to the solution x \u2248 0.739085<\/code>, which is the fixed point of the cosine function.<\/p>\n\n\n\n
    Analysis<\/h3>\n\n\n\n
    The fixed-point iteration method is useful for solving equations of the form x = g(x)<\/code>. For it to converge, the function g(x)<\/code> must satisfy certain conditions, such as the derivative of g(x)<\/code> being less than 1 in magnitude at the fixed point.<\/p>\n\n\n\n
    In this case, since |g'(x)| < 1<\/code> near the fixed point, the method will converge. The convergence is guaranteed if these conditions hold, and the accuracy can be adjusted using the tolerance value.<\/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 how you can implement the Fixed-Point Iteration method in MATLAB: MATLAB Code matlabCopyEdit% Define the fixed-point function g = @(x) cos(x); % Example: g(x) = cos(x) % 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-257933","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257933","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=257933"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257933\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=257933"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=257933"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=257933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}