Question

Show it in Matlab. thx！

5. Formulate Newton's method for the system x = y v = Note that this system has three real roots (-1,-1), (0,0) and (1,1). By taking various initial points in the square -2 <ry < 2, try to obtain the attraction regions of these three roots. P.S. An attraction region of a root is defined as the set of all initial points which will eventually converge to the root.

Answer 1

Matlab code for Multivariate Newton Raphson method clear all close all syms X Y #steady state equation f(x,y)=x^3-y; g(x,y)=Y3-X; Newton Raphson algorithm f_x(x,y)=diff(f,x); f_y(x,y) diff(fy); g_x(x,y)=diff(g,x); g_y(x, y) =diff(g,y); $initial guess for Newton Raphson x0=-2;y=-2; fprintf('For initial guess x=8f and y=ff.\n',x0, y0) err=1; c=0; #loop for Newton Raphson Algorithm while err>10^-12 c=c+1; Jacobian of equation of 2 variable jac=[f_x (x0, y0) f_y(x0,70);9_x(x0, y0) g_y(x0, 0)); ijac=inv(jac); xx=double([x0;yo ]-ijac*[ f(x0,0); g(x0, y0)]); terror in each loop err(c)=norm( XX-[ x0;y0]); Xiteration count it (C)=c; x0=double(xx(1)); yO=double(xx (2)); fprintf( 'After %d iteration\n',c) fprintf('\t x=$.12f\ty=8.12f\n', 0,70) end fprintf('\nNR solution for solution of equation is x=3.12f y=8.12f. In',xx (1),XX(2)) fprintf('--- \n\n') Xinitial guess for Newton Raphson x0=2;y0=2; fprintf('For initial guess x=$f and y=$f.\n',x0,yo) err=1; c=0; #loop for Newton Raphson Algorithm while err>10°-12 C=C+1; Jacobian of equation of 2 variable jac=[f_x(x0,70) f_y(x0,70);9_x(x0,70) g_y(x0,0));

ijac=inv(jac); xx-double([x0;yo ]-ijac*[ f(x0,70);9(x0, y0)]); $error in each loop err(c)=norm(xx-[x0;y0]); literation count it(c)=c; x0=double(xx(1)); yO=double(xx(2)); fprintf( 'After %d iteration\n',c) fprintf('\t x=$. 12f\ty=%.12f\n', 0, y0) end fprintf("\nNR solution for solution of equation is x=8.12f y=8.12f. \n', xx(1), XX(2)) fprintf('- \n\n') finitial guess for Newton Raphson x0=0.5;y0=-0.5; fprintf('For initial guess x=$f and y=$f.\n',x0, y0) err=1; c=0; 8loop for Newton Raphson Algorithm while err>10^-12 C=C+1; Jacobian of equation of 2 variable jac=[f_x (x0, y0) f_y(x0,0);9_x (x0, y0) g_y(x0, 0)); ijac-inv(jac); xx-double([x0;y0]-ijac*[f(x0, 0);9(x0, 0)); Berror in each loop err(c)=norm(xx-[ x0;y0]); Biteration count it (C)=c; x0=double(xx(1)); yO=double(xx(2)); fprintf('After %d iteration\n',c) fprintf('\t x=$.12f\ty=$.12f\n',x0,yo) end fprintf('\nNR solution for solution of equation is x=$.12f y=$.12f. \n',xx(1), XX(2)) fprintf('----- \n\n') 83333333333 End of Code 33%83%88%% For initial guess x=-2.000000 and y=-2.000000. After 1 iteration x=-1.454545454545 y=-1.454545454545 After 2 iteration x=-1.151046789378 y=-1.151046789378 After 3 iteration x=-1.025325928977 y=-1.025325928977 After 4 iteration

x=-1.000908451943 y=-1.000908451943 After 5 iteration x=-1.000001235309 y=-1.000001235309 After 6 iteration x=-1.000000000002 y=-1.000000000002 After 7 iteration x=-1.000000000000 y=-1.000000000000 After 8 iteration x=-1.000000000000 y=-1.000000000000 NR Solution for solution of equation is x=-1.000000000000 y=-1.000000000000. For initial guess x=2.000000 and y=2.000000. After 1 iteration x=1.454545454545 y=1.454545454545 After 2 iteration x=1.151046789378 y=1.151046789378 After 3 iteration x=1.025325928977 y=1.025325928977 After 4 iteration x=1.000908451943 y=1.000908451943 After 5 iteration x=1.000001235309 y=1.000001235309 After 6 iteration x=1.000000000002 y=1.000000000002 After 7 iteration x=1.000000000000 y=1.000000000000 After 8 iteration x=1.000000000000 y=1.000000000000 NR solution for solution of equation is x=1.000000000000 y=1.000000000000. For initial guess x=0.500000 and y=-0.500000. After 1 iteration x=0.142857142857 y=-0.142857142857 After 2 iteration x=0.005494505495 y=-0.005494505495 After 3 iteration x=0.000000331724 y=-0.000000331724 After 4 iteration x=0.000000000000 y=-0.000000000000 After 5 iteration x=0.000000000000 y=-0.000000000000 NR solution for solution of equation is x=0.000000000000 y=-0.000000000000.

%Matlab code for Multivariate Newton Raphson method
clear all
close all

syms x y
%steady state equation
f(x,y)=x^3-y;
g(x,y)=y^3-x;

%Newton Raphson algorithm
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);
%initial guess for Newton Raphson
x0=-2;y0=-2;
fprintf('For initial guess x=%f and y=%f.\n',x0,y0)
err=1; c=0;
  
    %loop for Newton Raphson Algorithm
    while err>10^-12
        c=c+1;
        %Jacobian of equation of 2 variable
        jac=[f_x(x0,y0) f_y(x0,y0);g_x(x0,y0) g_y(x0,y0)];
        ijac=inv(jac);
        xx=double([x0;y0]-ijac*[f(x0,y0);g(x0,y0)]);
        %error in each loop
        err(c)=norm(xx-[x0;y0]);
        %iteration count
        it(c)=c;
        x0=double(xx(1));
        y0=double(xx(2));
        fprintf('After %d iteration\n',c)
        fprintf('\t x=%.12f\ty=%.12f\n',x0,y0)
    end

fprintf('\nNR solution for solution of equation is x=%.12f y=%.12f.\n',xx(1),xx(2))
fprintf('------------------------------------------------------------\n\n')

%initial guess for Newton Raphson
x0=2;y0=2;
fprintf('For initial guess x=%f and y=%f.\n',x0,y0)
err=1; c=0;
  
    %loop for Newton Raphson Algorithm
    while err>10^-12
        c=c+1;
        %Jacobian of equation of 2 variable
        jac=[f_x(x0,y0) f_y(x0,y0);g_x(x0,y0) g_y(x0,y0)];
        ijac=inv(jac);
        xx=double([x0;y0]-ijac*[f(x0,y0);g(x0,y0)]);
        %error in each loop
        err(c)=norm(xx-[x0;y0]);
        %iteration count
        it(c)=c;
        x0=double(xx(1));
        y0=double(xx(2));
        fprintf('After %d iteration\n',c)
        fprintf('\t x=%.12f\ty=%.12f\n',x0,y0)
    end

fprintf('\nNR solution for solution of equation is x=%.12f y=%.12f.\n',xx(1),xx(2))
fprintf('------------------------------------------------------------\n\n')

%initial guess for Newton Raphson
x0=0.5;y0=-0.5;
fprintf('For initial guess x=%f and y=%f.\n',x0,y0)
err=1; c=0;
  
    %loop for Newton Raphson Algorithm
    while err>10^-12
        c=c+1;
        %Jacobian of equation of 2 variable
        jac=[f_x(x0,y0) f_y(x0,y0);g_x(x0,y0) g_y(x0,y0)];
        ijac=inv(jac);
        xx=double([x0;y0]-ijac*[f(x0,y0);g(x0,y0)]);
        %error in each loop
        err(c)=norm(xx-[x0;y0]);
        %iteration count
        it(c)=c;
        x0=double(xx(1));
        y0=double(xx(2));
        fprintf('After %d iteration\n',c)
        fprintf('\t x=%.12f\ty=%.12f\n',x0,y0)
    end

fprintf('\nNR solution for solution of equation is x=%.12f y=%.12f.\n',xx(1),xx(2))
fprintf('------------------------------------------------------------\n\n')

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