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By fixed-Point Iteration, evaluate f(x)=x² –x-e* starting at x1

By fixed-Point Iteration, evaluate f(x)=x² –x-e* starting at x1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Fixed-point iteration. Do not use equations such as x= x^2 - e^-x for it will diverge.

Stopping Criteria: 0.00005.

Kindly show algebra.

Thanks!

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By fixed-Point Iteration, evaluate f(x)=x² - x -x-estarting at x1 = 0.
Transcribed Image Text:By fixed-Point Iteration, evaluate f(x)=x² - x -x-estarting at x1 = 0.
Fixed Point Iteration Method : In this method, we first rewrite the equation f(x) = 0 in the form x = g(x) in such a way that any solution of the equation , which is a fixed point of g, is a solution of equation f (x) = 0. Then consider the following algorithm. Algorithm 1 : Start from any point xo and consider the recursive process xn+I s(xа), п %3D 0, 1, 2, ... If f is continuous and (xn) converges to some lo then it is clear that lo is a fixed point of g and hence it is a solution of the equation f(x) = 0. Here f(x) = x² – x – e* at x¡ = 0 Therfore, x = xr² – e=x p(x) = x² – e* x2 = 4(x1) = 4(0) = –1 X3 = p(x2) = p(-1) = -1.7183 %3D X4 = p(x3) = p(-1.7183) = -2. 6224 x3 = p(x4) = p(-2.6224) = –6. 8922 X6 = 4(xs) = 4(-6. 8922) = -937.0292 x7 = 4(x6) = 4(-937.0292) = Undefined The method lead us away from the solution, so it is divergent.
Transcribed Image Text:Fixed Point Iteration Method : In this method, we first rewrite the equation f(x) = 0 in the form x = g(x) in such a way that any solution of the equation , which is a fixed point of g, is a solution of equation f (x) = 0. Then consider the following algorithm. Algorithm 1 : Start from any point xo and consider the recursive process xn+I s(xа), п %3D 0, 1, 2, ... If f is continuous and (xn) converges to some lo then it is clear that lo is a fixed point of g and hence it is a solution of the equation f(x) = 0. Here f(x) = x² – x – e* at x¡ = 0 Therfore, x = xr² – e=x p(x) = x² – e* x2 = 4(x1) = 4(0) = –1 X3 = p(x2) = p(-1) = -1.7183 %3D X4 = p(x3) = p(-1.7183) = -2. 6224 x3 = p(x4) = p(-2.6224) = –6. 8922 X6 = 4(xs) = 4(-6. 8922) = -937.0292 x7 = 4(x6) = 4(-937.0292) = Undefined The method lead us away from the solution, so it is divergent.
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