flo-poly/src/evaluate/double/eft-horner.ts

A practical, root-focused JavaScript polynomial utility library.

import { twoProduct as twoProduct_ } from "big-float-ts";
import { twoSum as twoSum_ } from "big-float-ts";

// We *have* to do the below❗ The assignee is a getter❗ The assigned is a pure function❗ Otherwise code is too slow❗
const twoSum = twoSum_;
const twoProduct = twoProduct_;


/**
 * Returns an EFT (error free transformation) for the Horner evaluation of a
 * polymial at a specified x. The result is returned as an object with
 * properties: r̂ -> the calculated evaluation, pπ and pσ -> two polynomials
 * with coefficients around 2^53 times smaller than the input polynomial.
 * 
 * * r̂ + pπ(x) + pσ(x) = the *exact* evaluation (no error)
 * 
 * * see [Algorithms for Accurate, Validated and Fast Polynomial Evaluation, *Stef Graillat, Philippe Langlois and Nicolas Louvet*](https://projecteuclid.org/download/pdf_1/euclid.jjiam/1265033778)
 * * see also [*Philippe Langlois, Nicolas Louvet.* Faithful Polynomial Evaluation with Compensated Horner Algorithm. ARITH18: 18th IEEE International Symposium on Computer Arithmetic, Jun 2007, Montpellier, France. pp.141–149. ffhal-00107222f](https://hal.archives-ouvertes.fr/hal-00107222/document)
 * * see also [Horner's Method](https://en.wikipedia.org/wiki/Horner%27s_method)
 * 
 * @param p a polynomial with coefficients given densely as an array of double
 * floating point numbers from highest to lowest power, e.g. `[5,-3,0]` 
 * represents the polynomial `5x^2 - 3x`
 * @param x the value at which to evaluate the polynomial
 * 
 * @doc
 */
function EFTHorner(
		p: number[], x: number): { r̂: number, pπ: number[], pσ: number[] } {

	const pπ: number[] = []; // A polynomial containing part of the error
	const pσ: number[] = []; // Another polynomial containing part of the error
	let σ: number;
	
	let r̂ = p[0];
	for (let i=1; i<p.length; i++) {
		const [π,pi] = twoProduct(r̂,x);
		[σ,r̂] = twoSum(pi, p[i]);
		// inlined
		//r̂ = pi + p[i]; const bv = r̂ - pi; σ = (pi - (x-bv)) + (p[i]-bv);
		pπ.push(π);
		pσ.push(σ);
	}

	return { r̂, pπ, pσ }
}
// inlined
//const pπ: number[] = []; const pσ: number[] = []; const σ: number; const r̂ = p[0];	for (const i=1; i<p.length; i++) { const [π,pi] = twoProduct(r̂,x); [σ,r̂] = twoSum(pi, p[i]); pπ.push(π); pσ.push(σ); } return { r̂, pπ, pσ }


export { EFTHorner }