# How to use iterations in solving system of equations? want to update the vector Y at each iteration but ending up with same Y defined in the beginning.

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Tanya Sharma on 2 Feb 2021
Commented: Alan Stevens on 3 Feb 2021
Hi,
I am runnig the code for an iterative solution.
The problem is that I want to update the vector Y at each iteration , and solve the system ynew=A\R with new values of Y .
But in my case I am unable to update the Y after the iteration and ending up using the same Y as defined in the beginning of the code.
I have attached the cheb function too.
clear;
clc;
%%%%----------diff matrix---------------%%
N=5;
[D,x]=cheb(N);
D1=2*D; %%derivative matrix
D2=D1*D1;
D3=D1*D1*D1;
I=eye(N+1);
%%%%---parameters-----------------%%%
Re=5;
alpha=-1;
eta= (1+x)/2;
%-------------initial guess--------%%%
change =1;
it=0;
while change >1e-15
Y = (-eta.^2+1); %%this is the initial approximation for Y %%----NEED to UPDATE AT EACH ITERATION LEVEL
dY = D1*Y; %%derivative for Y
a0 = diag(2*alpha*Re*Y+4*alpha*alpha); %%%----a0 and a1 contain Y and dY
a1 = diag(2*alpha*Re*dY);
A = D3 + a0.*D1 + a1.*I;
A(1,:)=0;
A(1,1)=1;
A(N+1,:)=0;
A(N+1,N+1)=1;
A(N,:)=D(N+1,:);
R = zeros(N+1,1)+2*alpha*Re*Y.*dY; %%----R contains Y and dY
R(1)=0;
R(N+1)=1;
R(N)=0;
ynew = A\R;
change = norm(ynew-Y,inf);
Y = ynew; %%updated Y, it must update the Y so that all the calculation can run with new Y at each iteration
it = it+1;
end
figure(1)
plot(eta,ynew,'r-');
hold on

Alan Stevens on 2 Feb 2021
Try taking
Y = (-eta.^2+1);
outside the while command.. i.e. have
Y = (-eta.^2+1);
while change >1e-15
...etc
Alan Stevens on 3 Feb 2021
Ok. It works just fine (though I suggest you reduce your change test to, say, 10^-8, as this makes little difference to the accuracy, but speeds things up enormously as you increase the size of N). However, I can't comment on the validity of the results as I know nothing of the background to the problem you are trying to solve.