{"id":2774,"date":"2025-06-02T05:41:03","date_gmt":"2025-06-02T05:41:03","guid":{"rendered":"https:\/\/diznr.com\/?p=2774"},"modified":"2025-06-02T05:41:03","modified_gmt":"2025-06-02T05:41:03","slug":"computer-science-numerical-methods-iteration-method-with-its-convergence-condition","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/computer-science-numerical-methods-iteration-method-with-its-convergence-condition\/","title":{"rendered":"Computer Science\/Numerical Methods\/ Iteration method ( with it’s convergence condition)."},"content":{"rendered":"

Computer Science\/Numerical Methods\/ Iteration method ( with it’s convergence condition).<\/p>\n

[fvplayer id=”106″]<\/p>\n

Iteration Method in Numerical Methods<\/strong><\/h3>\n

The Iteration Method<\/strong> is a technique used to find the root of an equation f(x)=0f(x) = 0<\/span>f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>. It is based on approximating the root through successive iterations, refining the value until it reaches the desired accuracy.<\/p>\n

1. General Form of Iteration Method<\/strong><\/h3>\n

The given equation f(x)=0f(x) = 0<\/span>f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span> is rewritten in the form:<\/p>\n

x=g(x)x = g(x)<\/span>x<\/span>=<\/span><\/span>g<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

where g(x)g(x)<\/span>g<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span> is a function derived from f(x)f(x)<\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span>.<\/p>\n

An initial guess x0x_0<\/span>x<\/span>0<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is selected, and the next approximation is obtained using the formula:<\/p>\n

xn+1=g(xn)x_{n+1} = g(x_n)<\/span>x<\/span>n<\/span>+<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>g<\/span>(<\/span>x<\/span>n<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

This process continues until the absolute difference between successive approximations is less than a predefined tolerance \u03f5\\epsilon<\/span>\u03f5<\/span><\/span><\/span><\/span>, i.e.,<\/p>\n

\u2223xn+1\u2212xn\u2223<\u03f5|x_{n+1} – x_n| < \\epsilon<\/span>\u2223<\/span>x<\/span>n<\/span>+<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span>n<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2223<\/span><<\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span><\/p>\n

2. Convergence Condition<\/strong><\/h2>\n

For the iteration method to converge to a root \u03b1\\alpha<\/span>\u03b1<\/span><\/span><\/span><\/span>, the function g(x)g(x)<\/span>g<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span> must satisfy:<\/p>\n

\u2223g\u2032(\u03b1)\u2223<1| g'(\\alpha) | < 1<\/span>\u2223<\/span>g<\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b1<\/span>)<\/span>\u2223<\/span><<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/p>\n

where g\u2032(\u03b1)g'(\\alpha)<\/span>g<\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span> is the derivative of g(x)g(x)<\/span>g<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span> evaluated at the root \u03b1\\alpha<\/span>\u03b1<\/span><\/span><\/span><\/span>.<\/p>\n

Explanation of Convergence Condition<\/strong><\/h3>\n