n = 63;

K = zeros(n+2, n)

for i=1:n+2
    if i<=n
        K(i,i) = 1;
    end
    if i-1 >= 1 && i-1 <= n
        K(i,i-1) = -2;
    end
    if i-2 >= 1 && i-2 <= n
        K(i,i-2) = 1;
    end
        
end

xinit = 2;
xgoal = -2;

e = zeros(n+2,1);
e(1) = -xinit;
e(2) =  xinit;
e(end-1) = xgoal;
e(end) = -xgoal;

A = K'*K

b = K'*e

x = -inv(A)*b
plot(linspace(0, 1, n+2), [xinit; x; xgoal]);


i1 = round(0.25*(n+1))
i2 = round(0.75*(n+1))

C = zeros(2,n);
C(1,i1) = 1;
C(2,i2) = 1;

d = [ +3; 1 ];

M = [ A C' ; C zeros(2,2) ];

AiCt = A\C';
Q = C*AiCt;

m11 = A\(eye(n,n) - C'*(Q\AiCt'));
m12 = (Q\AiCt')';

%Mi = inv(M);
%x = Mi(1:n,:) * [ -b; d ];

x = -m11*b + m12*d;

hold on

plot(linspace(0, 1, n+2), [xinit; x; xgoal], 'r-');
plot([0.25, 0.75], d, 'ro');

hold off