const { sign } = Math;


/** 
 * * `signChanges` is identical to `descartes`
 * 
 * Returns the number of sign changes in the polynomial coefficents when 
 * ordered in descending order; zeros are ignored.
 * 
 * **precondition:** the polynomial leading coefficient must be non-zero.
 *
 * * Descartes' rule of signs states (quoted from Wikipedia):
 * "if the terms of a polynomial are ordered by descending variable 
 * exponent, then the number of positive roots of the polynomial is 
 * either equal to the number of sign differences between consecutive 
 * nonzero coefficients, or is less than it by an even number. Multiple 
 * roots of the same value are counted separately."
 * 
 * * see [Descartes' rule of signs](https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs)
 * 
 * @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`
 * 
 * @example
 * ```typescript
 * signChanges([1,2,-3,0,0,3,-1]); //=> 3
 * ```
 * 
 * @doc
 */
const descartes = signChanges;


/** 
 * Returns the number of sign changes in the polynomial coefficents when 
 * ordered in descending order; zeros are ignored.
 *
 * * Descartes' rule of signs states (quoted from Wikipedia):
 * "if the terms of a polynomial are ordered by descending variable 
 * exponent, then the number of positive roots of the polynomial is 
 * either equal to the number of sign differences between consecutive 
 * nonzero coefficients, or is less than it by an even number. Multiple 
 * roots of the same value are counted separately."
 * 
 * * see [Descartes' rule of signs](https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs)
 * 
 * @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`
 * 
 * @example
 * ```typescript
 * signChanges([1,2,-3,0,0,3,-1]); //=> 3
 * ```
 * 
 * @doc
 */
function signChanges(p: number[]): number {
    const d = p.length - 1;

    if (d < 1) { return 0; }

    let r = 0;
    let j = 0;
    while (p[j] === 0) { j++; }

    let _s = sign(p[j]);
    for (let i=j+1; i<d+1; i++) {
        const s = sign(p[i]);
        
        if (s !== _s && s !== 0) {
            r++;
            _s = s;
        }
    }
    
    return r;
}


export { signChanges, descartes }