Answer:
Below is the required code.
Explanation:
%% Newton Raphson Method
clear all;
clc;
x0=input('Initial guess:\n');
x=x0;
f=exp(-x)-sin(x)-0.2;
g=-exp(-x)-cos(x);
ep=10;
i=0;
cc=input('Condition of convergence:\n');
while ep>=cc
i=i+1;
temp=x;
x=x-(f/g);
f=exp(-x)-sin(x)-0.2;
g=-exp(-x)-cos(x);
ep=abs(x-temp);
fprintf('x = %6f and error = %6f at iteration = %2f \n',x,ep,i);
end
fprintf('The solution x = %6f \n',x);
%% End of MATLAB Program
Command Window:
(a) First Root:
Initial guess:
1.5
Condition of convergence:
0.01
x = -1.815662 and error = 3.315662 at iteration = 1.000000
x = -0.644115 and error = 1.171547 at iteration = 2.000000
x = 0.208270 and error = 0.852385 at iteration = 3.000000
x = 0.434602 and error = 0.226332 at iteration = 4.000000
x = 0.451631 and error = 0.017029 at iteration = 5.000000
x = 0.451732 and error = 0.000101 at iteration = 6.000000
The solution x = 0.451732
>>
Second Root:
Initial guess:
3.5
Condition of convergence:
0.01
x = 3.300299 and error = 0.199701 at iteration = 1.000000
x = 3.305650 and error = 0.005351 at iteration = 2.000000
The solution x = 3.305650
>>
(b) Guess x=0.5:
Initial guess:
0.5
Condition of convergence:
0.01
x = 0.450883 and error = 0.049117 at iteration = 1.000000
x = 0.451732 and error = 0.000849 at iteration = 2.000000
The solution x = 0.451732
>>
Guess x=1.75:
Initial guess:
1.75
Condition of convergence:
0.01
x = 227.641471 and error = 225.891471 at iteration = 1.000000
x = 218.000998 and error = 9.640473 at iteration = 2.000000
x = 215.771507 and error = 2.229491 at iteration = 3.000000
x = 217.692636 and error = 1.921130 at iteration = 4.000000
x = 216.703197 and error = 0.989439 at iteration = 5.000000
x = 216.970438 and error = 0.267241 at iteration = 6.000000
x = 216.971251 and error = 0.000813 at iteration = 7.000000
The solution x = 216.971251
>>
Guess x=3.0:
Initial guess:
3
Condition of convergence:
0.01
x = 3.309861 and error = 0.309861 at iteration = 1.000000
x = 3.305651 and error = 0.004210 at iteration = 2.000000
The solution x = 3.305651
>>
Guess x=4.7:
Initial guess:
4.7
Condition of convergence:
0.01
x = -1.916100 and error = 1.051861 at iteration = 240.000000
x = -0.748896 and error = 1.167204 at iteration = 241.000000
x = 0.162730 and error = 0.911626 at iteration = 242.000000
x = 0.428332 and error = 0.265602 at iteration = 243.000000
x = 0.451545 and error = 0.023212 at iteration = 244.000000
x = 0.451732 and error = 0.000187 at iteration = 245.000000
The solution x = 0.451732
>>
Explanation:
The two solutions are x =0.451732 and 3.305651 within the range 0 < x< 5.
The initial guess x = 1.75 fails to determine the solution as it's not in the range. So the solution turns to unstable with initial guess x = 1.75.
" is often considered as the significant fuels for rocket.
