# I got stuck with a loop after the value gets to NaN

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% Newton-Raphson
clear; clc
% Symbolic math to compute the derivative
syms x y
u = x^2+x*y-10;
v = y+ 3*x*y^2-57;
dudx=diff(u,x);
dudy=diff(u,y);
dvdx=diff(v,x);
dvdy=diff(v,y);
% Covert symbolic functions into regular functions with inputs of (x,y)
u = matlabFunction(u);
v = matlabFunction(v);
dudx = matlabFunction(dudx,'Vars',[x y]);
dudy = matlabFunction(dudy,'Vars',[x y]);
dvdx = matlabFunction(dvdx,'Vars',[x y]);
dvdy = matlabFunction(dvdy,'Vars',[x y]);
% Initial guess for the root
xr=1;
yr=1;
% Maximum number of steps
N=1000;
tol=1e-5;
% Plot function
x=linspace(-5,5,100);
y=linspace(-5,5,100);
U=zeros(100,100);
V=zeros(100,100);
for i=1:100
for j=1:100
U(i,j)=u(x(i),y(j));
V(i,j)=v(x(i),y(j));
end
end
figure(1); clf(1);
% Plot u(x,y)
surf(x,y,U')
hold on
% Plot v(x,y)
surf(x,y,V')
% Iterate
for i=1:1000
% Store the old value
xro=xr;
yro=yr;
% Update xr & yr
xr=xro-(u(xro,yro)*dvdy(xro,yro)-v(xro,yro)*dudy(xro,yro)) ...
/ (dudx(xro,yro)*dvdy(xro,yro)-dudy(xro,yro)*dvdx(xro,yro));
yr=yro-(v(xro,yro)*dudx(xro,yro)-u(xro,yro)*dvdy(xro,yro) ...
/ (dudx(xro,yro)*dvdy(xro,yro)-dudy(xro,yro)*dvdx(xro,yro)));
% Plot current guess for root
hold on
plot3(xr,yr,u(xr,yr),'o','Markersize',10)
plot3(xr,yr,u(xr,yr),'o','Markersize',10)
pause(1);
% Output to command window
fprintf('Iter=%5i, xr=%5.5f, yr=%5.5f, Error=%5.5e \n', ...
i,xr,yr,max(abs(xr-xro),max(abs(yr-yro))))
% Stop loop if converged
if max(abs(xr-xro),abs(yr-yro))<tol
fprintf('The root is (x,y)=(%5.10f,%5.10f) \n',xr,yr)
break %stop the loop
end
end
Paul Hoffrichter on 8 Sep 2021
If xr is NaN, then max(abs(xr-xro),abs(yr-yro)) is also NaN and max(abs(xr-xro),abs(yr-yro))<tol is false. To test for NaN, use isnan. But then you drop into the if-body, and since xr is NaN, you will get NaN for the result.

per isakson on 8 Sep 2021
Edited: per isakson on 8 Sep 2021
The values of xr and yr do not converge to a solution that you expect. xr goes to zero and yr goes to "infinity".
>> Newton_Raphson_multidim
Iter= 100, xr= 1.17, yr=1.57e+02, Error=1.56e+02
Iter= 100, xr= 0.14, yr=-1.37e+07, Error=1.37e+07
Iter= 100, xr=-0.00, yr=1.10e+21, Error=1.10e+21
Iter= 100, xr= 0.00, yr=5.96e+57, Error=5.96e+57
Iter= 100, xr= 0.00, yr=-1.19e+154, Error=1.19e+154
Iter= 100, xr= NaN, yr= NaN, Error= NaN
Iter= 100, xr= NaN, yr= NaN, Error= NaN
Iter= 100, xr= NaN, yr= NaN, Error= NaN
"the loop is suppose to 'break' when the value is NaN" I modified the "ouput" statements
% Output to command window
fprintf('Iter=%5i, xr=%5.2f, yr=%5.2e, Error=%5.2e \n', ...
i,xr,yr,max(abs(xr-xro),max(abs(yr-yro))))
% Stop loop if converged
if max(abs(xr-xro), abs(yr-yro))<tol || isnan(xr) || isnan(yr)
fprintf('The root is (x,y)=(%5.2e,%5.2e) \n',xr,yr)
break %stop the loop
end
Now I get
>> Newton_Raphson_multidim
Iter= 100, xr= 1.17, yr=1.57e+02, Error=1.56e+02
Iter= 100, xr= 0.14, yr=-1.37e+07, Error=1.37e+07
Iter= 100, xr=-0.00, yr=1.10e+21, Error=1.10e+21
Iter= 100, xr= 0.00, yr=5.96e+57, Error=5.96e+57
Iter= 100, xr= 0.00, yr=-1.19e+154, Error=1.19e+154
Iter= 100, xr= NaN, yr= NaN, Error= NaN
The root is (x,y)=( NaN, NaN)
>>
alexander fastiggi on 8 Sep 2021
thank you!

R2019a

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