NEWTON'S 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 exp1.jpg for (cos[x])-(x * exp[x])

The graph of this equation is given in the figure.

Let the initial guess x₀ be 2.0

i	0	1	2	3	4	5	6	7
x_i	2	1.34157	0.8477	0.58756	0.52158	0.51777	0.51776	0.51776

So the iterative process converges at 0.51776

2. Find the root of x⁴-x-10 = 0 exp2.jpg for x^4-x-10

The graph of this equation is given in the figure.

Let the initial guess x₀ be 2.0

i	0	1	2	3	4
x_i	2	1.87097	1.85578	1.85558	1.85558

So the iterative process converges at 1.85558.

3. Find the root of x-exp[-x] = 0 exp3.jpg for x-exp[-x]

The graph of this equation is given in the figure.l

Let the initial guess x₀ be 2.0

i	0	1	2	3	4	5
x_i	2	0.35761	0.55871	0.56713	0.56714	0.56714

So one of the roots of x-exp[-x] = 0 is approximately 0.56714.

4. Find the root of (exp[-x] * (x²+5x+2)) + 1= 0 exp4.jpg for exp[-x]*(x^2-5x+2)+1

The graph of this equation is given in the figure.

let the initial guess x₀ be -2.0

i	0	1	2	3	4	5	6
x_i	-2	-1.22707	-0.77562	-0.60291	-0.57955	-0.57916	-0.57916

So one of the roots of (exp[-x] * (x²+5x+2)) + 1= 0 is approximately -0.57916.

5. Find the root of x-sin[x]-(1/2)= 0 exp5.jpg for x-sin[x]-1/2

The graph of this equation is given in the figure.

let the initial guess x₀ be 2.0

i	0	1	2	3	4
x_i	2	1.58228	1.50087	1.49731	1.4973

So one of the roots of x-sin[x]-(1/2) = 0 is approximately 1.4973.

6. Find the root of exp[-x]=3log[x] exp6.jpg for exp[-x]=3log[x]

The graph of this equation is given in the figure.

let the initial guess x₀ be 2.0

i	0	1	2	3	4	5
x_i	2	1.02418	1.22523	1.24663	1.24682	1.24682

So one of the roots of exp[-x]=3log[x] is approximately 1.24682.

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]