flo-poly/src/euclidean-division-related/bigint/b-pdiv-trivial.ts

A practical, root-focused JavaScript polynomial utility library.

import { bDegree as bDegree_ } from "../../basic/bigint/b-degree.js";
import { bMultiplyByConst as bMultiplyByConst_ } from "../../basic/bigint/b-multiply-by-const.js";
import { bPdivInternal as bPdivInternal_ } from './b-pdiv-internal.js';

// We *have* to do the below❗ The assignee is a getter❗ The assigned is a pure function❗ Otherwise code is too slow❗
const bDegree = bDegree_;
const bMultiplyByConst = bMultiplyByConst_;
const bPdivInternal = bPdivInternal_;

const abs = (n: bigint) => n >= 0 ? n : -n;


/**
 * Performs a **trivial pseudo-division** and returns the `quotient` and `remainder` 
 * of the pseudo division of `a/b` (a, b both being polynomials) in such a way 
 * that all intermediate calculations and the final result are done in ℤ, i.e. 
 * performs Euclidean (i.e. long) division on the two given polynomials, a/b, 
 * and returns a scaled `r` and `q` in the formula `a = bq + r`, where 
 * `degree(r) < degree(b)`. `q` is called the quotient and `r` the remainder.
 * 
 * * **precondition:** the coefficients must be bigints; if they are not they 
 * can easily be scaled from floating point numbers to bigints by calling 
 * [[scaleFloatsToBigints]] or similar before calling this function (recall that 
 * all floating point numbers are rational).
 * 
 * * **precondition:** b !== [0], i.e. unequal to the zero polynomial.
 * 
 * * see [trivial pseudo-remainder sequence](https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Trivial_pseudo-remainder_sequence) 
 * * see also [Polynomial long division](https://en.wikipedia.org/wiki/Polynomial_long_division)
 * 
 * @param a the polynomial a in the formula a = bq + r; the polynomial is given
 * with coefficients as a dense array of bigints from highest to lowest 
 * power, e.g. `[5n,-3n,0n]` represents the  polynomial `5x^2 - 3x`
 * @param b the polynomial b in the formula a = bq + r
 * @param positiveMultiplier defaults to false - if set to true then the 
 * multiplier (of the coefficients of the dividend) 
 * `leadingCoeff(b)^(deg(a)-deg(b)+1)` will be 
 * modified to `abs(leadingCoeff(b)^(deg(a)-deg(b)+1))`
 * 
 * @doc
 */
function bPdivTrivial(
        a: bigint[], b: bigint[], positiveMultiplier = false): { q: bigint[]; r: bigint[]; } {

    const d = bDegree(a) - bDegree(b) + 1;
    if (d < 1) {
        return { q: [], r: a };
    }
    let m = b[0]**BigInt(d);

    m = positiveMultiplier 
        ? abs(m) 
        : m;

    const a_ = bMultiplyByConst(m, a); 

    return bPdivInternal(a_, b);
}


export { bPdivTrivial }