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.

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

Answer 1

[Alan Stevens]

Try taking

Y = (-eta.^2+1); 

outside the while command.. i.e. have

Y = (-eta.^2+1);  
while change >1e-15
...etc 

[Tanya Sharma]

It works now Alan, thanks.

I updated the code with few more things.

But to cross check my results I solved my problem using bvp4c too.

Now the thing is the iterative result is not matching with my bvp4c result.

Is there something wrong with my iterations?

[Alan Stevens]

I can't tell. I don't have your cheb function, so I can't run your code. Also, I don't know the context (the maths/science).

[Tanya Sharma]

I have attached the cheb.m file in my question. Attaching again.

Give it a try if you can just to see the iteration part. Thanks anyways :)

[Alan Stevens]

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.