Question

Using Matlab

1. Solve the following equations set

f1 (x1,x2) = sin (sin (x1)) +x2

f2 (x1,x2) = x1+ e^(x2)

a) Can this equation set be solved by the fixed - point method with the following expressions? And why? Show your analysis with a 2D graph.

g1 (x1,x2) = -e^(x2)

g2 (x1,x2) = -sin⁡(x1)

b) Use Newton Raphson Method with initial values x1 = -2, x2 = 1.5. (8 significant figures. Please submit the code and results.)

Answer 1

%Matlab code for solving nonliear equations using Newton method
clear all
close all
%function for fixed point iterations
g1=@(x1,x2) -exp(x2);
g2=@(x1,x2) -sin(x1);

%plotting both functions
xx1=linspace(-3,3);
yy1=linspace(0,3);
[X,Y]=meshgrid(xx1,yy1);
G1=g1(X,Y);
G2=g2(X,Y);
hold on
surf(X,Y,G1)
surf(X,Y,G2)
shading interp
xlabel('X1')
ylabel('X2')
zlabel('g')
view(3)
box on
fprintf('As the two plots never touch each other, hence fixed point is not possibel.\n')
%Newton method
syms x y
%functions for which intersection have to find
f(x,y)=sin(x)+y;
g(x,y)=x+exp(y);

%Displaying the equation
fprintf('The equations are\n')
disp(f)
disp(vpa(g,2))

%creating Jacobian matrix
f_x(x,y)=diff(f,x);
f_y(x,y)=diff(f,y);
g_x(x,y)=diff(g,x);
g_y(x,y)=diff(g,y);

%Jacobian matrix
jac1=[f_x f_y; g_x g_y];
%displaying the Jacobian Matrix
fprintf('\tThe Jacobian Matrix is \n\n')
disp(vpa(jac1,2))
x1=-2;y1=1.5;
conv=10^-10;

    %loop for Newton iterations
    err=1;k=0;
    fprintf('For initial condition x=%f and y=%f \n',x1,y1)
    cnt=0;

      
    while err>conv
        cnt=cnt+1;
        jac=[f_x(x1,y1) f_y(x1,y1);g_x(x1,y1) g_y(x1,y1)];
        ijac=inv(jac);
        uu=double([x1;y1]-ijac*[f(x1,y1);g(x1,y1)]);
        %uu=double([x1;y1]-0.5*jac*[f(x1,y1);g(x1,y1)]);
        err=norm(uu-[x1;y1]);
        x1=double(uu(1));
        y1=double(uu(2));
        fprintf('After %d iteration \n',cnt)
        fprintf('\tX=%.8f and Y=%.8f \n',x1,y1)
    end
  
    fprintf('\n\n Convergence for order =%.2e occured after %d iteration\n',conv,cnt)
    fprintf('\tThe root occured at x=%f, y=%f\n\n',x1,y1)

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