Crate nutils_poly

Low-level functions for evaluating and manipulating polynomials.

The polynomials considered in this crate are power series in zero or more variables centered at zero and truncated to order $p$,

$$ \sum_{\substack{k ∈ ℤ^n \\ \sum_i k_i ≤ p}} c_k \prod_i x_i^{k_i} $$

where $c$ is a vector of coefficients, $x$ a vector of $n$ variables and $p$ a nonnegative integer degree.

This crate requires the coefficients to be stored in a linear array in reverse lexicographic order: the coefficient for powers $j ∈ ℤ^n$ comes before the coefficient for powers $k ∈ ℤ^n \setminus \{j\}$ iff $j_i > k_i$, where $i = \max_l(j_l ≠ k_l)$, the index of the last non-matching power.

This crate provides functions for evaluating polynomials, computing coefficients for the partial derivativative and products of polynomials.

Examples

The vector of coefficients for the polynomial $f(x, y) = 3 x y + x^2$ is [0, 3, 0, 1, 0, 0].

With eval() we can evaluate this polynomial:

use nutils_poly;

let coeffs = [0, 3, 0, 1, 0, 0];
assert_eq!(nutils_poly::eval(&coeffs, &[1, 0], 2), Ok( 1)); // f(1, 0) =  1
assert_eq!(nutils_poly::eval(&coeffs, &[1, 1], 2), Ok( 4)); // f(1, 1) =  4
assert_eq!(nutils_poly::eval(&coeffs, &[2, 3], 2), Ok(22)); // f(2, 3) = 22

PartialDerivPlan::apply() computes the coefficients for the partial derivative of a polynomial to one of the variables. The partial derivative of $f$ to $x$, the first variable, is $∂_x f(x, y) = 3 y + 2 x$ (coefficients: [3, 2, 0]):

use nutils_poly::PartialDerivPlan;

let coeffs = [0, 3, 0, 1, 0, 0];
let pd = PartialDerivPlan::new(
    2, // number of variables
    2, // degree
    0, // variable to compute the partial derivative to
).unwrap();
assert_eq!(Vec::from_iter(pd.apply(coeffs)?), vec![3, 2, 0]);

Nutils project

This crate is part of the Nutils project.

Structs

MapDegree

Index map relating coefficients from one degree to another.

Mul

The product of two polynomials.

MulPlan

Plan for computing products of polynomials.

PartialDeriv

The partial derivative of a polynomial.

PartialDerivIter

An iterator of the coefficients of a partial derivative.

PartialDerivPlan

Plan for computing coefficients for partial derivatives.

Enums

Error

The error type for fallible operations in this crate.

MulVar

Existence of a variable in the operands of a product polynomial.

ValueMoreOrLess

Enum used by Error::IncorrectNumberOfCoefficients.

Traits

EvalOps

Operations for evaluating polynomials.

Multiple

Interface for computing a multiple.

Functions

composition_with_inner_matrix

Returns a matrix that flattens coefficients for a composition of polynomials.

degree

Returns the degree of a polynomial given the number of variables and coefficients.

degree_ncoeffs_iter

Returns an iterator of degrees and number of coefficients.

eval

Evaluates a polynomial for the given values.

ncoeffs

Returns the number of coefficients for a polynomial of given degree and number of variables.

ncoeffs_iter

Returns an iterator of the number of coefficients for a polynomial of degree zero and up.

Type Aliases

Power