Populating matrix with for loop depending if index is even or odd - matlab

Question

This is a part of a PDE problem that I'm trying to solve numerically. I know that my calculations for the entries are correct, however I'm struggling to assemble the matrix properly in Matlab. The first rows are correct, however the two last rows are not correct since I'm missing the elements a_{11,10}, a_{10,10} and a_{10,11}. I'm missing these because my loop never accesses them. For for example a_{10,10} requires i = 10, which is an even number, however my M = 11 in this case and i only goes from 1 to 9. If I make my for loop go from 1:11, I get out of bounds error since I'm using indecies i+1 and i+2.
The same goes for the vector b. How do I fill in all my b_i's if i only goes to 9 but the vector is 11 elements long?
Any suggestion on how to fix this? I've tried making separate for loops for the odd and even case but I'm still having the same problem of entries missing due to using indecies i+1 and i+2. Or do I manually have to enter (hardcode) these indecies outside of the foor loop?
Here is my loop:
h = 1/10;
x = 0:h:1;
M = length(x);
% Assemble stiffness matrix A and load vector b.
A = zeros(M,M);
b = zeros(M,1);
for i = 1:M-2
if rem(i,2) ~= 0 % if i is odd
A(i,i) = A(i,i) + 1/(3*h);
b(i) = b(i) + f(x(i)) * (-2)*h/3;
elseif rem(i,2) == 0 % if i is even
A(i,i) = A(i,i) + 3/(3*h);
A(i+2,i) = A(i+2,i) - 8/(3*h);
A(i,i+2) = A(i,i+2) - 8/(3*h);
b(i) = b(i) + f(x(i)) * 8*h/3;
end
A(i+1,i) = A(i+1,i) + 2/(3*h);
A(i,i+1) = A(i,i+1) - 3/(3*h);
end
% Enforce the boundary conditions
A(1,1) = 1.e4;
A(end,end) = 1.e4;

Answer 1

I think you can simply pad matrix A with two extra rows and columns, and remove them later:
Look at the following modified version of your code:
h = 1/10;
x = 0:h:1;
M = length(x);
% Assemble stiffness matrix A and load vector b.
%Initialize A to have two more rows and columns.
A = zeros(M+2);
b = zeros(M,1);
for i = 1:M
if rem(i,2) ~= 0 % if i is odd
A(i,i) = A(i,i) + 1/(3*h);
b(i) = b(i) + f(x(i)) * (-2)*h/3;
elseif rem(i,2) == 0 % if i is even
A(i,i) = A(i,i) + 3/(3*h);
A(i+2,i) = A(i+2,i) - 8/(3*h);
A(i,i+2) = A(i,i+2) - 8/(3*h);
b(i) = b(i) + f(x(i)) * 8*h/3;
end
A(i+1,i) = A(i+1,i) + 2/(3*h);
A(i,i+1) = A(i,i+1) - 3/(3*h);
end
%Remove two rows and columns form matrix A.
A = A(1:end-2, 1:end-2);
% Enforce the boundary conditions
A(1,1) = 1.e4;
A(end,end) = 1.e4;
function z = f(t)
% Stab f functionn - f(x) return x
z = t;
end
I hope the code behaves as you expected (it's not so clear what value you are expecting to read outside the bounds of A)...