```
% cholesky.m: Cholesky factorization for solving the linear system
%             of equations  A*x = b.

clear;
format short;

% Step 0: Assign the matrix A and the vector b.

n = 4;
A = [ 0.05, 0.07, 0.06, 0.05;
0.07, 0.10, 0.08, 0.07;
0.06, 0.08, 0.10, 0.09;
0.05, 0.07, 0.09, 0.10 ];
b = [ 0.23, 0.32, 0.33, 0.31 ];

% Step 1: Cholesky factorization: A = L*L' where L is a lower
%         triangular matrix and L' is the tranpose matrix of L.

L = zeros(n,n);       % initialize the n x n matrix L
U = zeros(n,n);       % initialize the n x n matrix U
for k = 1:n
L(k,k) = A(k,k);
for s = 1:(k-1)
L(k,k) = L(k,k)-L(k,s)*L(k,s);
end
L(k,k) = sqrt(L(k,k));
for i = (k+1):n
%     (fill out a few lines here to compute L(i,k))
end
end
L                     % output L

% Step 2: Forward subsitiution to solve  L*y = b.

y = zeros(n,1);       % initialize the vector y
y(1) = b(1)/L(1,1);
for i = 2:n
y(i) = b(i);
for j = 1:(i-1)
y(i) = y(i)-L(i,j)*y(j);
end
y(i) = y(i)/L(i,i);
end
y                     % output y

% Step 3: Back subsitiution to solve  U*x = y.

x = zeros(n,1);       % initialize the vector x
x(n) = y(n)/L(n,n);
for i = (n-1):-1:1
%   (fill out a few lines here to compute x(i))
end
x                     % output x
```