Question

How to write in matlab program ?

(1) Reduce the following system of equations to one equation in terms of x and solve the resulting equation numerically using Newton-Raphson method.

e^x/10 – y = 0 and 2log_ey – cosx = 2

(2) Solve the above equations numerically using system of equations, first by plotting the graph to obtain an initial approximation of the roots (such as x = 1 with y = 1 or other combinations), then produce the results numerically with tolerance of 1e−4.

Answer 1

Part 1

function file

myFunction.m

function fval= myFunction(x)

fval=10+5*cos(x)-x;

end

using newton raphson method

script file

newtonRaphson.m

clc;clear all;

x0=3.3;

maxIter=500;

tol=1e-16;

x=x0;

xOld=x0;

for i=1:maxIter

df=-5*sin(x)-1;

x=x-(myFunction(x)/df);

err=abs(x-xOld);

xSol=x;

xOld=x;

disp(['Iteration No is ', num2str(i),' Error is =',num2str(err),' ,XSol is =',num2str(x)]);

if err<tol

break;

end

end

Part 2

Plotting is done for initial guess

IntialGuess.m

x=-10:0.1:20;

y1=exp(x/10);

y2=exp(1+0.5*cos(x));

plot(x,y1,'-b'); hold on

plot(x,y2,'--k');

solving of nonlinear equation by fsolve

8 7 4 x 8.2 Y 2.2943 3 2 10 15 20 -10 -5

funNonlinear.m

function fVal= funNonlinear(Y)

x=Y(1);

y=Y(2);

fVal(1,1)=exp(x/10)-y;

fVal(2,1)=2*log(y)-cos(x)-2;

end

Y0=[8.2;2.29];

ySol=fsolve(@(Y) funNonlinear(Y),Y0)

Note: Your given function were not clear either e^x/10 or e^(x/10); so solution may vary