Data Science: Computation of Eigenvectors — Power Method.

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This story is part of my Data Science series.

To exactly compute the eigenvalue/eigenvectors pairs, there exists several methods. One of this is the so called power method at which theory and implementation this article is devoted.

Power Method:

Implementation:

The implementation will proceed what we have started in the previous story (see here).

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The power method is implemented easily like so:

pub fn compute_ev_power_method(a: &Array2<f64>) -> (Array1<f64>, f64) {
    let mut x = Array1::ones(a.nrows());
    let max_iter = 10;
    for _ in 0..max_iter {
        x = a.dot(&x);
        x /= *x.mapv(f64::abs).max().unwrap();
    }
    let lambda = &x.t().dot(&a.dot(&x)) / &x.t().dot(&x); 
    (x, lambda)
}

Of course, in practice one would apply in addition some convergence criterion to break out early from the loop. For instance |Ax — lambda*x| < epsilon.

Our complete program now looks like this:

let data = load_data();
let cov = compute_covariance(&data);
let ev_circles = compute_ev_circles(&cov);
let (e_vec, e_val) = compute_ev_power_method(&cov);

println!("{}", e_val);

This finds the eigenvalue 13.242.

Having this, the question arises how to obtain the remaining eigenvalue/eigenvector pairs. More on this, in our next story.

Thank you for reading!