aberth

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An implementation of the Aberth-Ehrlich method for finding the zeros of a polynomial.

Aberth's method uses an electrostatics analogy to model the approximations as negative charges and the true zeros as positive charges. This enables finding all complex roots simultaneously, converging cubically (worst-case it converges linearly for zeros of multiplicity).

This crate is #![no_std] and tries to have minimal dependencies. It uses arrayvec to avoid allocations, which will be removed when rust stabilises support for const-generics.

Usage

Add it to your project:

cargo add aberth

Specify the coefficients of your polynomial in an array in ascending order and then call the aberth method on your polynomial.

use aberth::aberth;
const EPSILON: f32 = 0.001;
const MAX_ITERATIONS: u32 = 10;

// 0 = -1 + 2x + 4x^4 + 11x^9
let polynomial = [-1., 2., 0., 0., 4., 0., 0., 0., 0., 11.];

let roots = aberth(&polynomial, MAX_ITERATIONS, EPSILON);
// [
//   Complex { re:  0.4293261, im:  1.084202e-19 },
//   Complex { re:  0.7263235, im:  0.4555030 },
//   Complex { re:  0.2067199, im:  0.6750696 },
//   Complex { re: -0.3448952, im:  0.8425941 },
//   Complex { re: -0.8028113, im:  0.2296336 },
//   Complex { re: -0.8028113, im: -0.2296334 },
//   Complex { re: -0.3448952, im: -0.8425941 },
//   Complex { re:  0.2067200, im: -0.6750695 },
//   Complex { re:  0.7263235, im: -0.4555030 },
// ]

The above method does not require any allocation, instead doing all the computation on the stack. It is generic over any size of polynomial, but the size of the polynomial must be known at compile time.

The coefficients of the polynomial may be f32, f64, or even complex numbers Complex<f32>, Complex<f64>:

# use aberth::{aberth, Complex};
#
# let p1 = [1_f32, 2_f32];
# let r1 = aberth(&p1, 10, 0.001);
#
# let p2 = [1_f64, 2_f64];
# let r2 = aberth(&p2, 10, 0.001);
#
# let p3 = [Complex::new(1_f32, 2_f32), Complex::new(3_f32, 4_f32)];
# let r3 = aberth(&p3, 10, 0.001);
#
# const MAX_ITERATIONS: u32 = 10;
# const EPSILON: f64 = 0.001;
let polynomial = [Complex::new(1_f64, 2_f64), Complex::new(3_f64, 4_f64)];
let roots = aberth(&polynomial, MAX_ITERATIONS, EPSILON);

If std is available then there is also an AberthSolver struct which allocates some memory to support dynamically sized polynomials at run time. This may also be good to use when you are dealing with polynomials with many terms, as it uses the heap instead of blowing up the stack.

use aberth::AberthSolver;

let mut solver = AberthSolver::new();
solver.epsilon = 0.001;
solver.max_iterations = 10;

// 0 = -1 + 2x + 4x^3 + 11x^4
let a = [-1., 2., 0., 4., 11.];
// 0 = -28 + 39x^2 - 12x^3 + x^4
let b = [-28., 0., 39., -12., 1.];

for polynomial in [a, b] {
  let roots = solver.find_roots(&polynomial);
  // ...
}

Note that the returned values are not sorted in any particular order.

The coefficient of the highest degree term should not be zero.

#![no_std]

To use in a no_std environment you must disable default-features and enable the libm feature:

[dependencies]
aberth = { version = "0.4.1", default-features = false, features = ["libm"] }

Stability Guarantees

mod internal may experience breaking changes even in minor and patch releases:

We expose the internal module, for those interested in supplying their own initial guesses to internal::aberth_raw. However this internal module is considered an implementation detail and does not follow the semantic versioning scheme of the rest of the project.

License

This crate is licensed under any of the Apache license, Version 2.0, or the MIT license, or the Zlib license at your option.