trying to stop my loop with below an error

trying to stop my loop when you substract 2 matrix and the error between both is below 0.001

clear
close all
clc



%initialisation
a=0;
b=3;
c=0;
d=3;
h=1;
k=1;
n=(d-c)/k+1;
m=(b-a)/h+1;
u=zeros(n,m);
errmax=0.001;

x=a:h:b;
y= fliplr(c:k:d);



%CL
u(:,1)=0;
u(4,:)=x.^3;
u(:,4)= 27-9*y.^2;
u(1,:)= x.^3 -27*x;
lmax=100;
usup = i-1;

for l=1:lmax
    
    for i=2:m-1
        for j=2:n-1
            u(i,j)=(1/4)*(u(i-1,j)+u(i,j-1)+u(i+1,j)+u(i,j+1))
        end
        if  max(max(abs(u-usup)))< errmax
            break
        end
    end
end

%end
%plot(u)
%contourf(u)

thanks

Answer 1

[Walter Roberson]

u(i,j)=(1/4)*(u(i-1,j)+u(i,j-1)+u(i+1,j)+u(i,j+1))

You always write additional entries into u, never overwriting existing entries.

        if  max(max(abs(u-usup)))< errmax

You test all of u. But max() over all of u can only ever stay the same or increase because all of the old entries are still there and you are testing all of them.

[felix belhumeur]

thank you for the answer but I still dont understand .

for the first line: this line of code seams to work on my end. I dont understand the mistake

for the second line: how can i test just 2 matrix between them and make sure the error is below the limit without keeping all the old one in memory

[Walter Roberson]

        for j=2:n-1
            u(i,j)=(1/4)*(u(i-1,j)+u(i,j-1)+u(i+1,j)+u(i,j+1))
        end

That loop writes in into column 2 through to the second last column of u, for the current row.

        if  max(max(abs(u-usup)))< errmax

If you want to test only the entries you just wrote, then test

       if  max(max(abs(u(i,2:n-1)-usup)))< errmax

However, I realized that you are doing a Finite Element iteration, and that really you do want to test the entire matrix -- but you want to test it after you have completed the for i as well. Like

for l=1:lmax
    for i=2:m-1
        for j=2:n-1
            u(i,j)=(1/4)*(u(i-1,j)+u(i,j-1)+u(i+1,j)+u(i,j+1))
        end
    end
    if  max(max(abs(u-usup)))< errmax
        break
    end
end

I have to ask about your stopping condition, u - usup . You define

usup = i-1;

and you do that before you have for i, so you are defining usup as a complex constant -1+1i . Then you are comparing abs() of the value minus the complex constant to all of the entries. Does that make sense to do?

u(:,1)=0;
u(4,:)=x.^3;
u(:,4)= 27-9*y.^2;
u(1,:)= x.^3 -27*x;

You initialize column 1 and column 4 of u, and row 1 and row 4 of u. But u is m x n, and you never write into the last row or last column of u unless m and n both happen to be 4... which is true with those particular constants but is not going to be true in general. You should be initializing u(end,:) not u(4,:) and u(:,end) not u(:,4) .

You never write to the outside row or outside column of u, but you are testing all of u when you test abs(u-usup) . But suppose those initial values like 27-9*y.^2 are outside of the range of usup, then the max() comparison over the whole u matrix can never hold.

I do not think you should be comparing all of u to a particular constant -- not unless your situation is something like you having an initial configuration that you are allowing to dampen out.

What might make sense instead is to keep an "old u" and an "updated u" and to compare the two of them, and keep iterating if any location has too large of a difference.

[felix belhumeur]

that makes more sense! thank you so much!