function S = symplectic_red_naive(w,k)
% S=SYMPLECTIC_RED_NAIVE(w,k) constructs an elementary symplectic matrix 
% which reduces the vector w
% w: a vector of length 2n
% k: an integer less than or equal to n
% S: elementary symplectic matrix
n = length(w)/2;
% Constructing the first Householder matrix
[u1,b1] = householder(w(n+k:2*n));
Q1 = eye(2*n);
Q1(k:n,k:n) = eye(n-k+1) - b1*u1*u1';
Q1(n+k:2*n,n+k:2*n) = Q1(k:n,k:n);
w = Q1*w;
% Constructing the Givens matrix
G = givens([w(k) w(n+k)]);
Q2 = eye(2*n);
Q2([k n+k],[k n+k]) = G;
w = Q2*w;
% Constructing the second Householder matrix
[u3,b3] = householder(w(k:n));
Q3 = eye(2*n);
Q3(k:n,k:n) = eye(n-k+1) - b3*u3*u3';
Q3(n+k:2*n,n+k:2*n) = Q3(k:n,k:n);
% Generating the output
S = Q3*Q2*Q1;