Question

Please use comments to show steps clearly

Consider the following system: f(x) = 5x2 + 2y2-10 g(x) = x2 + y -2 Use Newton's Method to find the roots of the system to a tolerance of 10-3 (with MATLAB).

Answer 1

warning('off', 'all'); %to suppress any worning

f = @(x,y) 5*x*x+2*y*y-10; % function

fy = @(x,y) 4*y; % y derivative

fx = @(x,y) 10*x; % x derivative

g = @(x,y) x*x +y -2; % function

gy = @(x,y) 1 ; %y derivative

gx = @(x,y) 2*x ; % x derivative

x0=1; % initial values assumed any value

y0=1;

count=0; % to count number of iterations

while true

x1 = x0 + (fy(x0,y0)*g(x0,y0)-f(x0,y0)*gy(x0,y0))/(fx(x0,y0)*gy(x0,y0)-fy(x0,y0)*gx(x0,y0));

y1= y0+ (f(x0,y0)*gx(x0,y0)-fx(x0,y0)*g(x0,y0))/(fx(x0,y0)*gy(x0,y0)-fy(x0,y0)*gx(x0,y0));

  

%condition to break the loop

if (abs(x1-x0) < 10^(-3)) & ( abs(y1-y0) < 10^(-3))

break;

end

if count>10000 % to prevent divergence, so loop doesnot go infinitely

break;

end

   count = count+1;

x0 = x1;

y0=y1;

end

%show values obtained

disp(x0)

disp(y0)

disp(f(x0,y0))

disp(g(x0,y0))

---------------------------------------------------------------------------------------------------------------------------------------------------------

1.4142 -0.0000057946 0.000065679 0.0000073412

1 2 3 4 5 6 7 warningcoff', 'all'); %to suppress any worning f = @(x,y) 5*x*x+2*y*y-10; % function fy = @(x,y) 4*y; % y derivative fx = @(x,y) 10*x; % x derivative g = @(x,y) x x +y -2; % function gy = @(x,y) 1; %y derivative gx = @(x,y) 28X; % x derivative % initial values assumed any value x0=1; y0=1; count-; % to count number of iterations while true x1 = x + (fy(x0,70) *g(x,y)-f(x0,70) *gy(x0,70))/(fx(x0, 0)* yl= you (f(x0,70)*gx(x0,70)-fx(x0,70)*(0,0))/(fx(x0,70)*gy %condition to break the Loop if (abs(x1-x0) < 10^(-3)) & (abs(y1-yo) < 10^(-3)) break; end if count>10000 % to prevent divergence, so loop doesnot go in break; end count = count+1; x0 = x1; yo=y1; end 31 %show values obtained

line 11,23-26 can be removed. It is just added to ensure no divergence occur and loop doesnot go into infinite loop.

thanks