Condition number and 1-norm estimation in Rust

This crate implements the matrix 1-norm estimator by Higham and Tisseur, Algorithm 2.4 on page 7 (1190) in the linked document. It allows for 1-norm estimation of a single matrices, A, matrix powers, A^m, and matrix products, A1 * A2 * ... * An, which can be cheaper than explicitly calculating them.

It uses the excellent rust-ndarray crate for matrix storage.

Example usage

The example below generates a random matrix a and estimates its 1-norm. On average, this gives pretty good results. Of course, there are some matrices where this algorithm severely underestimates the actual 1-norm. See Higham and Tisseur for more.

condest::normest1 creates a Normest1 struct, uses it to estimate the 1-norm, and throws it away. If you want to repeatedly estimate 1-norms of matrices of the same dimensions, initialize Normest1 and call normest1, normest1_pow or normest1_prod on it.

Important: You need to explicitly link to a BLAS + LAPACK provider such as openblas_src. See the explanations given at the blas-lapack-rs organization.

extern crate openblas_src; // Need to declare `openblas_src` (or some other BLAS provider) explicitly to link to a BLAS library.

use ndarray::{
    prelude::*,
    Zip,
};
use ndarray_rand::RandomExt;
use rand::{
    SeedableRng,
};
use rand::distributions::StandardNormal;
use rand_xoshiro::Xoshiro256Plus;

fn main() {
    let t = 2;
    let n = 100;
    let itmax = 5;

    let mut rng = Xoshiro256Plus::seed_from_u64(1234);
    let distribution = StandardNormal;

    let mut a = Array::random_using((n, n), distribution, &mut rng);
    a.mapv_inplace(|x| 1.0/x);

    let estimated = condest::normest1(&a, t, itmax);
    let expected = {
        let (layout, a_slice) = if let Some(a_slice) = a.as_slice() {
            (cblas::Layout::RowMajor, a_slice)
        } else if let Some(a_slice) = a.as_slice_memory_order() {
            (cblas::Layout::ColumnMajor, a_slice)
        } else {
            panic!("Matrix not contiguous!")
        };

        unsafe {
            lapacke::dlange(
            layout,
            b'1',
            n as i32,
            n as i32,
            a_slice,
            n as i32,
        )}
    };

    assert_eq!(estimated, expected);
}

Todo

Right now, only 1-norm estimates are exposed. The vectors needed to estimate the condition number are implemented, but are not yet accessible through an API. Outstanding points are:

• Return vectors required for calculating the 1-norm;
• Create a struct holding the necessary temporaries to repeatedly call normest1 without extra allocation.
• Implement extra tests to mimic the numerical experiments in Higham and Tisseur.
• Make some nice docs.