Homework SourseConsider the problem sinxx 1 Set up newtons method to obtain
Consider the problem sin(x)=x
1. Set up newtons method to obtain a numerical approximation using X0= 1 obtain at least x > 0
2. In newtons method, you have the term f(x)/f\'(x) and simplify to get N(x)/M(x) =f(x)/f\'(x) after simplification
3. Consider Xn+1 = Xn - N(Xn)/M(Xn)
Implement the method in using matlab and obtain several iterations using X0=0.05 on a table list: n, Xn, f(Xn), f\'(Xn), N(Xn) and M(Xn)
What can you say about this approach?
Solution
% Function Given As Sin(x)=x that can be write as f(x) = sin(x)-x
% Now Solve f(x) using newton Raphson Method
clc;
clear all;
x0 = 1;% initial guess
syms x
func=sin(x)-x; % function definition
df_func=diff(func,x); % function differentiation
iter = 1; % initialize iteration as 1
maxIters=30; % initialize maxIters = 30
fprintf(\'n\\t\\t x0\\t\\t\\t\\t\\t Xnew\\t\\t Func(Xn)\\t\\t Diff(Func(Xn)) \\t N(x_new) \\t\\t\\t M(x_new) \ \ \');
while(iter<=maxIters) % loop till iterations exceeds
func_val=double(subs(func,x0)); % function value calculations
func_dff_val=double(subs(df_func,x0)); % function differentiation calculations
x_new=x0-(func_val/func_dff_val); % new value of x updated
fprintf(\'%d\\t\\t %.6f\\t\\t %.6f\\t\\t %.6f\\t\\t %.6f \\t\\t\\t %.6f \\t\\t\\t%.6f \ \',iter,x0,x_new,double(subs(func,x0)),double(subs(df_func,x0)),double(subs(func,x_new)),double(subs(df_func,x_new)));
x0=x_new;
iter=iter+1; %increment in iteration
end
% this approach minimize the value of given function f=sinx -x, this value tends to zero and we have to calculate the value of x.
%results
n x0 Xnew Func(Xn) Diff(Func(Xn)) N(x_new) M(x_new)
1 1.000000 0.655145 -0.158529 -0.459698 -0.045871 -0.207040
2 0.655145 0.433590 -0.045871 -0.207040 -0.013459 -0.092537
3 0.433590 0.288148 -0.013459 -0.092537 -0.003971 -0.041228
4 0.288148 0.191832 -0.003971 -0.041228 -0.001174 -0.018343
5 0.191832 0.127810 -0.001174 -0.018343 -0.000348 -0.008157
6 0.127810 0.085183 -0.000348 -0.008157 -0.000103 -0.003626
7 0.085183 0.056782 -0.000103 -0.003626 -0.000031 -0.001612
8 0.056782 0.037853 -0.000031 -0.001612 -0.000009 -0.000716
9 0.037853 0.025234 -0.000009 -0.000716 -0.000003 -0.000318
10 0.025234 0.016823 -0.000003 -0.000318 -0.000001 -0.000141
11 0.016823 0.011215 -0.000001 -0.000141 -0.000000 -0.000063
12 0.011215 0.007477 -0.000000 -0.000063 -0.000000 -0.000028
13 0.007477 0.004984 -0.000000 -0.000028 -0.000000 -0.000012
14 0.004984 0.003323 -0.000000 -0.000012 -0.000000 -0.000006
15 0.003323 0.002215 -0.000000 -0.000006 -0.000000 -0.000002
16 0.002215 0.001477 -0.000000 -0.000002 -0.000000 -0.000001
17 0.001477 0.000985 -0.000000 -0.000001 -0.000000 -0.000000
18 0.000985 0.000656 -0.000000 -0.000000 -0.000000 -0.000000
19 0.000656 0.000438 -0.000000 -0.000000 -0.000000 -0.000000
20 0.000438 0.000292 -0.000000 -0.000000 -0.000000 -0.000000
21 0.000292 0.000194 -0.000000 -0.000000 -0.000000 -0.000000
22 0.000194 0.000130 -0.000000 -0.000000 -0.000000 -0.000000
23 0.000130 0.000086 -0.000000 -0.000000 -0.000000 -0.000000
24 0.000086 0.000058 -0.000000 -0.000000 -0.000000 -0.000000
25 0.000058 0.000038 -0.000000 -0.000000 -0.000000 -0.000000
26 0.000038 0.000026 -0.000000 -0.000000 -0.000000 -0.000000
27 0.000026 0.000017 -0.000000 -0.000000 -0.000000 -0.000000
28 0.000017 0.000011 -0.000000 -0.000000 -0.000000 -0.000000
29 0.000011 0.000008 -0.000000 -0.000000 -0.000000 -0.000000
30 0.000008 0.000005 -0.000000 -0.000000 -0.000000 -0.000000
>>
For x0=0.05
n x0 Xnew Func(Xn) Diff(Func(Xn)) N(x_new) M(x_new)
1 0.050000 0.033332 -0.000021 -0.001250 -0.000006 -0.000555
2 0.033332 0.022221 -0.000006 -0.000555 -0.000002 -0.000247
3 0.022221 0.014814 -0.000002 -0.000247 -0.000001 -0.000110
4 0.014814 0.009876 -0.000001 -0.000110 -0.000000 -0.000049
5 0.009876 0.006584 -0.000000 -0.000049 -0.000000 -0.000022
6 0.006584 0.004389 -0.000000 -0.000022 -0.000000 -0.000010
7 0.004389 0.002926 -0.000000 -0.000010 -0.000000 -0.000004
8 0.002926 0.001951 -0.000000 -0.000004 -0.000000 -0.000002
9 0.001951 0.001301 -0.000000 -0.000002 -0.000000 -0.000001
10 0.001301 0.000867 -0.000000 -0.000001 -0.000000 -0.000000
11 0.000867 0.000578 -0.000000 -0.000000 -0.000000 -0.000000
12 0.000578 0.000385 -0.000000 -0.000000 -0.000000 -0.000000
13 0.000385 0.000257 -0.000000 -0.000000 -0.000000 -0.000000
14 0.000257 0.000171 -0.000000 -0.000000 -0.000000 -0.000000
15 0.000171 0.000114 -0.000000 -0.000000 -0.000000 -0.000000
16 0.000114 0.000076 -0.000000 -0.000000 -0.000000 -0.000000
17 0.000076 0.000051 -0.000000 -0.000000 -0.000000 -0.000000
18 0.000051 0.000034 -0.000000 -0.000000 -0.000000 -0.000000
19 0.000034 0.000023 -0.000000 -0.000000 -0.000000 -0.000000
20 0.000023 0.000015 -0.000000 -0.000000 -0.000000 -0.000000
21 0.000015 0.000010 -0.000000 -0.000000 -0.000000 -0.000000
22 0.000010 0.000007 -0.000000 -0.000000 -0.000000 -0.000000
23 0.000007 0.000004 -0.000000 -0.000000 -0.000000 -0.000000
24 0.000004 0.000003 -0.000000 -0.000000 -0.000000 -0.000000
25 0.000003 0.000002 -0.000000 -0.000000 -0.000000 -0.000000
26 0.000002 0.000001 -0.000000 -0.000000 -0.000000 -0.000000
27 0.000001 0.000001 -0.000000 -0.000000 -0.000000 -0.000000
28 0.000001 0.000001 -0.000000 -0.000000 -0.000000 -0.000000
29 0.000001 0.000000 -0.000000 -0.000000 -0.000000 -0.000000
30 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000
>>
