const bChangeVariablesTranslateX = bTaylorShift;


/**
 * Returns the Taylor shift `p(x + h)` of the given polynomial computed in
 * `O(n^2)` via repeated synthetic division by `(x - h)` (Horner's scheme).
 *
 * @param p a polynomial with coefficients given densely as an array of bigints
 * from highest to lowest power, e.g. `[5n,-3n,0n]` represents the polynomial `5x^2 - 3x`
 * @param h the shift amount
 * 
 * @doc
 */
function bTaylorShift(
        p: bigint[],
        h: bigint): bigint[] {

    const n = p.length - 1;

    // The successive remainders are the Taylor coefficients of `p` about `h`,
    // i.e. the coefficients of `p(x + h)`.
    const q = p.slice();
    for (let k=0; k<=n; k++) {
        for (let i=1; i<=n - k; i++) {
            q[i] = q[i] + h*q[i - 1];
        }
    }

    return q;
}


export { bTaylorShift, bChangeVariablesTranslateX }