FIXED POINT ITERATION METHOD

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Iteration scheme is simple to implement.
Iterations always converge for any continuous function f(x).

Disadvantages

Convergence process is slow.
Always one has to find the interval [a,b] which brackets the zero of f(x)= 0 more importantly f(a) * f(b) must be less than zero.

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1. Find the root of (cos[x])-(x * exp[x]) = 0 fpi1.jpg for (cos[x])-(x * exp[x])

Consider g(x) = cos[x]/exp[x]

The graph of g(x) and x are given in the figure.

let the initial guess x₀ be 2.0

i	0	1	2	3	4	5	6	7	8	9	10	11	12	13	...	31	32
x_i	1	0.199	0.803	0.311	0.698	0.381	0.634	0.427	0.594	0.458	0.567	0.478	0.551	0.491	...	0.518	0.518

That is for g(x) = cos[x]/exp[x] the itirative process is converged to 0.518.

2. Find the root of x⁴-x-10 = 0 fpi2.jpg for x^4-x-10

Consider g(x) = (x + 10)^1/4

The graph of g(x) and x are given in the figure.

let the initial guess x₀ be 4.0

i	0	1	2	3	4	5	6
x_i	4.0	1.93434	1.85866	1.8557	1.85559	1.85558	1.85558

That is for g(x) = (x + 10)^1/4 the itirative process is converged to 1.85558.

3. Find the root of x-exp[-x] = 0 fpi3.jpg for x-exp[-x]

Consider g(x) = exp[-x]

The graph of g(x) and x are given in the figure..

let the initial guess x₀ be 3.0

i	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
x_i	3.0	0.05	0.951	0.386	0.68	0.507	0.602	0.548	0.578	0.561	0.571	0.565	0.568	0.567	0.567

That is for g(x) = exp[-x] the itirative process is converged to 0.567.

4. Find the root of (exp[-x] * (x²+5x+2)) + 1= 0 fpi4.jpg for exp[-x]*(x^2-5x+2)+1

Consider g(x) = (exp[x]+x^3+4x^2+2x+2)/(x^2+3x-3)

The graph of g(x) and x are given in the figure.

let the initial guess x₀ be -2

i	0	1	2	3	4	5	6
x_i	-2	-1.227	-0.776	-0.603	-0.58	-0.579	-0.579

That is for g(x) = (exp[x]+x^3+4x^2+2x+2)/(x^2+3x-3) the itirative process is converged to -0.579.

5. Find the root of x-sin[x]-(1/2)= 0 fpi5.jpg for x-sin[x]-1/2

Consider g(x) = sin[x]+(1/2)

The graph of g(x) and x are given in the figure.

let the initial guess x₀ be 2

i	0	1	2	3	4	5
x_i	2	1.409	1.487	1.496	1.497	1.497

That is for g(x) = sin[x]+(1/2) the itirative process is converged to 1.497.

6. Find the root of exp[-x]=3log[x] fpi6.jpg for exp[-x]=3log[x]

Consider g(x) = exp[(exp[-x]/3)]

The graph of g(x) and x are given in the figure.

let the initial guess x₀ be 2

i	0	1	2	3	4	5
x_i	2	1.046	1.124	1.114	1.116	1.115

That is for g(x) = exp[(exp[-x]/3)] the itirative process is converged to 1.115.

Problems to Work-Out:

7. Find the root of x * cos[(x)/ (x-2)]=0 [Graph]

8. Find the root of x² = (exp[-2x] - 1) / x [Graph]

9. Find the root of exp[x²-1]+10sin2x-5 = 0 [Graph]

10. Find the root of exp[x]-3x²=0 [Graph]

11. Find the root of tan[x]-x-1 = 0 [Graph]

12. Find the root of sin[2x]-exp[x-1] = 0 [Graph]