## no-std aberth

Aberth's method for finding the zeros of a polynomial

### 4 releases

 0.4.1 May 12, 2024 Sep 20, 2023 Apr 10, 2023

#203 in Math

MIT OR Apache-2.0 OR Zlib

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# aberth

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

``````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.