sparse-interp

Basic polynomial arithmetic, multi-point evaluation, and sparse interpolation.

This crate is very limited so far in its functionality and under active development. The current functionality isi mostly geared towards sparse interpolation with a known set of possible exponents. Expect frequent breaking changes as things get started.

The [Poly] type is used to represent dense polynomials along with traits for algorithm choices. The [ClassicalPoly] type alias specifies classical arithmetic algorithms via the [ClassicalTraits] trait.

use sparse_interp::ClassicalPoly;

// f represents 4 + 3x^2 - x^3
let f = ClassicalPoly::<f32>::new(vec![4., 0., 3., -1.]);

// g prepresents 2x
let g = ClassicalPoly::<f32>::new(vec![0., 2.]);

// basic arithmetic is supported
let h = f + g;
assert_eq!(h, ClassicalPoly::new(vec![4., 2., 3., -1.]));

Evaluation

Single-point and multi-point evaluation work as follows.

type CP = ClassicalPoly<f32>;
let h = CP::new(vec![4., 2., 3., -1.]);
assert_eq!(h.eval(&0.), Ok(4.));
assert_eq!(h.eval(&1.), Ok(8.));
assert_eq!(h.eval(&-1.), Ok(6.));
let eval_info = CP::mp_eval_prep([0., 1., -1.].iter().copied());
assert_eq!(h.mp_eval(&eval_info).unwrap(), [4.,8.,6.]);

Sparse interpolation

Sparse interpolation should work over any type supporting field operations of addition, subtration, multiplication, and division.

For a polynomial f with at most t terms, sparse interpolation requires eactly 2t evaluations at consecutive powers of some value θ, starting with θ0 = 1.

This value θ must have sufficiently high order in the underlying field; that is, all powers of θ up to the degree of the polynomial must be distinct.

Calling [Poly::sparse_interp()] returns on success a vector of (exponent, coefficient) pairs, sorted by exponent, corresponding to the nonzero terms of the evaluated polynomial.

type CP = ClassicalPoly<f64>;
let f = CP::new(vec![0., -2.5, 0., 0., 0., 7.1]);
let t = 2;
let (eval_info, interp_info) = ClassicalPoly::sparse_interp_prep(
    t,          // upper bound on nonzero terms
    0..8,       // iteration over possible exponents
    &f64::MAX,  // upper bound on coefficient magnitude
);
let evals = f.mp_eval(&eval_info).unwrap();
let mut result = CP::sparse_interp(&evals, &interp_info).unwrap();

// round the coefficients to nearest 0.1
for (_,c) in result.iter_mut() {
    *c = (*c * 10.).round() / 10.;
}

assert_eq!(result, [(1, -2.5), (5, 7.1)]);

Current version: 0.0.3

License

This software was written by Daniel S. Roche in 2021, as part of their job as a U.S. Government employee. The source code therefore belongs in the public domain in the United States and is not copyrightable.

Otherwise, the 0-clause BSD license applies.