Instability in the normal equationsΒΆ

Because the functions \(\sin^2(t)\), \(\cos^2(t)\), and \(1\) are linearly dependent, we should find that the following matrix is somewhat ill-conditioned.

using LinearAlgebra
t = LinRange(0,3,400)
A = [ sin.(t).^2 cos.((1+1e-7)*t).^2 t.^0 ]
kappa = cond(A)

Now we set up an artificial linear least squares problem with a known exact solution that actually makes the residual zero.

x = [1.,2,1]
b = A*x;

Using backslash to find the solution, we get a relative error that is about \(\kappa\) times machine epsilon.

x_BS = A\b
@show observed_err = norm(x_BS-x)/norm(x);
@show max_err = kappa*eps();
observed_err = norm(x_BS - x) / norm(x) = 1.144699458444126e-11
max_err = kappa * eps() = 4.053030231047602e-9

If we formulate and solve via the normal equations, we get a much larger relative error. With \(\kappa^2\approx 10^{14}\), we may not be left with more than about 2 accurate digits.

N = A'*A
x_NE = N\(A'*b)
@show observed_err = norm(x_NE-x)/norm(x);
@show digits = -log(10,observed_err);
observed_err = norm(x_NE - x) / norm(x) = 0.007289760380260753
digits = -(log(10, observed_err)) = 2.137286747023705