Use simplified fixedpoint iteration to locate the root of Fx

Use simplified fixed-point iteration to locate the root of F(x) = Sin(Sqrt(x))-x

Use an initial guess of X_0 = 0.5 and itterate until Ea less or equal to 0.01%. Verify that the process is linearly convergent as described in Box 6.1.

Book is Numerical Methods for Engineers 7th Edition ISBN 978-0-07-339792-4

Solution

x sin(sqrtx) f(x)= sin(sqrtx)-x f\'(x)={cos(sqrt(x))/2x^1/2} - 1 0.5 0.707107 0.149636939 -0.731212945 0.6 0.774597 0.09942787 -0.723196644 0.7 0.83666 0.042409663 -0.719741403 0.8 0.894427 -0.02014935 -0.720059662 0.9 0.948683 -0.087351103 -0.723575691 1 1 -0.158529015 -0.729848847 x sin(sqrtx) f(x)= sin(sqrtx)-x f\'(x)={cos(sqrt(x))/2x^1/2} - 1 0.5 0.707107 0.149636939 -0.731212945 0.6 0.774597 0.09942787 -0.723196644 0.7 0.83666 0.042409663 -0.719741403 0.8 0.894427 -0.02014935 -0.720059662 0.9 0.948683 -0.087351103 -0.723575691 1 1 -0.158529015 -0.729848847