nalgebra/src/linalg/balancing.rs

//! Functions for balancing a matrix.

use simba::scalar::RealField;
use std::ops::{DivAssign, MulAssign};

use crate::allocator::Allocator;
use crate::base::dimension::{Dim, U1};
use crate::base::storage::Storage;
use crate::base::{DefaultAllocator, MatrixN, VectorN};

/// Applies in-place a modified Parlett and Reinsch matrix balancing with 2-norm to the matrix `m` and returns
/// the corresponding diagonal transformation.
///
/// See https://arxiv.org/pdf/1401.5766.pdf
pub fn balance_parlett_reinsch<N: RealField, D: Dim>(m: &mut MatrixN<N, D>) -> VectorN<N, D>
where DefaultAllocator: Allocator<N, D, D> + Allocator<N, D> {
    assert!(m.is_square(), "Unable to balance a non-square matrix.");

    let dim = m.data.shape().0;
    let radix: N = crate::convert(2.0f64);
    let mut d = VectorN::from_element_generic(dim, U1, N::one());

    let mut converged = false;

    while !converged {
        converged = true;

        for i in 0..dim.value() {
            let mut c = m.column(i).norm_squared();
            let mut r = m.row(i).norm_squared();
            let mut f = N::one();

            let s = c + r;
            c = c.sqrt();
            r = r.sqrt();

            if c.is_zero() || r.is_zero() {
                continue;
            }

            while c < r / radix {
                c *= radix;
                r /= radix;
                f *= radix;
            }

            while c >= r * radix {
                c /= radix;
                r *= radix;
                f /= radix;
            }

            let eps: N = crate::convert(0.95);
            if c * c + r * r < eps * s {
                converged = false;
                d[i] *= f;
                m.column_mut(i).mul_assign(f);
                m.row_mut(i).div_assign(f);
            }
        }
    }

    d
}

/// Computes in-place `D * m * D.inverse()`, where `D` is the matrix with diagonal `d`.
pub fn unbalance<N: RealField, D: Dim>(m: &mut MatrixN<N, D>, d: &VectorN<N, D>)
where DefaultAllocator: Allocator<N, D, D> + Allocator<N, D> {
    assert!(m.is_square(), "Unable to unbalance a non-square matrix.");
    assert_eq!(m.nrows(), d.len(), "Unbalancing: mismatched dimensions.");

    for j in 0..d.len() {
        let mut col = m.column_mut(j);
        let denom = N::one() / d[j];

        for i in 0..d.len() {
            col[i] *= d[i] * denom;
        }
    }
}