SVD Computes Wrong Singular Values for Very Small-Valued Matrices

[Vollkornaffe]

This should be reproducible with nalgebra = { version = "0.32.4", features = ["rand"] }

use nalgebra::Matrix3;

fn main() {
    let id = Matrix3::<f32>::identity();
    loop {
        let r = Matrix3::<f32>::new_random();
        let m = id + r * 0.000000000000000000001;

        if m == id {
            panic!("too small")
        }

        let svd = m.svd_unordered(true, true);
        if svd.singular_values.iter().any(|s| (s - 1.).abs() > 0.1) {
            println!("Input Matrix: {m:#?}");
            println!("Singular Values: {:#?}", svd.singular_values);
            panic!();
        }
    }
}

Example output:

Input Matrix: [
    [
        1.0,
        6.23101e-24,
        2.2194742e-23,
    ],
    [
        7.9115625e-24,
        1.0,
        6.47089e-22,
    ],
    [
        1.922059e-23,
        9.848782e-22,
        1.0,
    ],
]
Singular Values: [
    [
        2.1223645,
        1.0000001,
        1.0,
    ],
]

[Ralith]

There was a similar issue in #1314, but I think the fix for that was included in 0.32.4. Maybe an interesting reference all the same.