Numerical Methods For Engineers, 7 Ed
Numerical Methods For Engineers, 7 Ed
7th Edition
ISBN: 9789352602131
Author: Canale Chapra
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Chapter 6, Problem 1P

Use simple fixed-point iteration to locate the root of

f ( x ) = sin  ( x ) x

Use an initial guess of x 0 = 0.5 and iterate until ε a 0.01 % . Verify that the process is linearly convergent as described in Box 6.1.

Expert Solution & Answer
Check Mark
To determine

To calculate: The root of the function f(x)=sin(x)x by the use of simple fixed-point iteration with x0=0.5 as the initial condition and iterate until εa0.01%. Also, verify that the process is linearly convergent.

Answer to Problem 1P

Solution:

The root of the function f(x)=sin(x)x is 0.7686.

Explanation of Solution

Given:

The function, f(x)=sin(x)x.

The initial condition, x0=0.5 and iterate until εa0.01%.

Formula used:

The simple fixed-point iteration formula for the function x=g(x),

xi+1=g(xi)

And, formula for approximate error is,

εa=|xi+1xixi+1|100%

Calculation:

Consider the function,

f(x)=sin(x)x

The function can be formulated as fixed-point iteration as,

0=sin(x)xx=sin(x)xi+1=sin(xi)

Use initial guess of x0=0.5, the first iteration is,

x0+1=sin(x0)x1=sin(0.5)=sin(0.7071)=0.6496

Therefore, the approximate error is,

εa=|0.64960.50.6496|×100%=|0.14960.6496|×100%=|0.2303|×100%=23.03%

Use x1=0.6496, the second iteration is,

x1+1=sin(x1)x2=sin(0.6496)=sin(0.80598)=0.7215

Therefore, the approximate error is,

εa=|0.72150.64960.7215|×100%=|0.07190.7215|×100%=|0.09965|×100%=9.965%

Use x2=0.7215, the second iteration is,

x2+1=sin(x2)x3=sin(0.7215)=sin(0.8494)=0.7509

Therefore, the approximate error is,

εa=|0.75090.72150.7509|×100%=|0.02940.7509|×100%=|0.03915|×100%=3.915%

Similarly, all the iteration can be summarized as below,

i xi εa=|xi+1xixi+1|100%
0 0.5
1 0.6496 23.03%
2 0.7215 9.965%
3 0.7509 3.915%
4 0.7621 1.47%
5 0.7662 0.535%
6 0.7678 0.208%
7 0.7683 0.0651%
8 0.76852 0.029%
9 0.7686 0.01%

Since, the approximate error in the ninth iteration is 0.01%. So, stop the iteration.

Hence, the root of the function is 0.7686.

Now, to verify that the process is linearly convergent, the condition to be satisfied is |g(x)|<1 for x=0.7686.

The fixed-point iteration is,

xi+1=sin(xi)

Therefore,

g(x)=sin(x)

Differentiate the above function with respect to x,

g(x)=ddx[sin(x)]=cos(x)ddx(x)=cos(x)(12x)=cos(x)2x

Therefore, |g(x)| at x=0.7686 is,

|g(0.7686)|=|cos(0.7686)20.7686|=|0.63972×0.8767|=|0.3648|=0.3648

Since, |g(0.7686)|<1. Hence, it is verified that the process is linearly convergent.

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Numerical Methods For Engineers, 7 Ed

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