Question

How do I deduce replacement algorithms forwards and backwards in the solution phase of the Cholesky method?

How do I compare the function choleskiSol?

Here’s my code for choleskisol

function x = choleskiSol(L,b)
% Solves [L][L’]{x} = {b}
% USAGE: x = choleskiSol(L,b)

n = length(b);

if size(b,2) > 1
    b = b’;
end % {b} must be column vector

for k = 1:n % Solution of [L]{y} = {b}
    b(k) = (b(k) - dot(L(k,1:k-1),b(1:k-1)’))/L(k,k);
end

for k = n:-1:1 % Solution of {L}’{x} = {y}
    b(k) = (b(k) - dot(L(k+1:n,k),b(k+1:n)))/L(k,k);
end
x = b;

Answer 1

The standard Cholesky decomposition (chol(A)) in matlab decomposes a symmetric (positive-definite) matrix, A, into upper-triangular form. To solve a linear system of equations, you must simply take the upper-triangular form, and solve it via backward substitution. This will yield the variable values for the system.

To complete the solution in matlab w/ parameter matrix A and output vector B:

L = chol(A); % A must be sym and det(A) > 0
x = (L \ (L' \ b)); % L' is lower-triangular