# Crate aberth

Expand description

## §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>`:

``````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.

## Modules§

• The `internal` module may experience breaking changes even in minor and patch releases.

## Structs§

• A solver for polynomials with Float or ComplexFloat coefficients. Will find all complex-roots, using the Aberth-Ehrlich method.
• A complex number in Cartesian form.
• The roots of a polynomial

## Enums§

• The reason the solver terminated and the number of iterations it took.

## Functions§

• Find all of the roots of a polynomial using Aberth’s method