MatLab - algorithm for finding inverse of matrix

Question

I am trying to write an algorithm in MatLab which takes as its input a lower triangular matrix. The output should be the inverse of this matrix (which also should be in lower triangular form). I have almost managed to solve this, but one part of my algorithm still leaves me scratching my head. So far I have:

function AI = inverse(A)
n = length(A);
I = eye(n);
AI = zeros(n);
for k = 1:n
    AI(k,k) = (I(k,k) - A(k,1:(k-1))*AI(1:(k-1),k))/A(k,k);
    for i = k+1:n
        AI(i,k) = (I(i,k) - (??????????????))/A(i,i);
    end
end

I have marked with question marks the part I am unsure of. I have tried to find a pattern for this part of the code by writing out the procedure on paper, but I just can't seem to find a proper way to solve this part.

If anyone can help me out, I would be very grateful!

Answer 1

Here is my code to get the inverse of a lower triangular matrix by using row transformation:

function AI = inverse(A)
    len = length(A);
    I  = eye(len);
    M  = [A I];
    for row = 1:len
        M(row,:) = M(row,:)/M(row,row);
        for idx = 1:row-1
            M(row,:) = M(row,:) - M(idx,:)*M(row,idx);
        end
    end
    AI = M(:,len+1:end);
end

Answer 2

You can see how it's done on Octave's source . This seems to be implemented in different places depending on the class of the matrix. For Float type Diagonal Matrix it's on liboctave/array/fDiagMatrix.cc , for Complex Diagonal matrix it's on liboctave/array/CDiagMatrix.cc , etc...

One of the advantages of free (as in freedom) software is that you are free to study how things are implemented ;)

Answer 3

Thanks for all the input! I was actually able to find a very nice and easy way to solve this problem today, given that the input is a lower triangular matrix:

function AI = inverse(A)
n = length(A);
I = eye(n);
AI = zeros(n);
for k = 1:n
    for i = 1:n
        AI(k,i) = (I(k,i) - A(k,1:(k-1))*AI(1:(k-1),i))/A(k,k);
    end
end