QuarticRealPolynomial.js

/*global define*/
define([
        './CubicRealPolynomial',
        './DeveloperError',
        './Math',
        './QuadraticRealPolynomial'
    ], function(
        CubicRealPolynomial,
        DeveloperError,
        CesiumMath,
        QuadraticRealPolynomial) {
    'use strict';

    /**
     * Defines functions for 4th order polynomial functions of one variable with only real coefficients.
     *
     * @exports QuarticRealPolynomial
     */
    var QuarticRealPolynomial = {};

    /**
     * Provides the discriminant of the quartic equation from the supplied coefficients.
     *
     * @param {Number} a The coefficient of the 4th order monomial.
     * @param {Number} b The coefficient of the 3rd order monomial.
     * @param {Number} c The coefficient of the 2nd order monomial.
     * @param {Number} d The coefficient of the 1st order monomial.
     * @param {Number} e The coefficient of the 0th order monomial.
     * @returns {Number} The value of the discriminant.
     */
    QuarticRealPolynomial.computeDiscriminant = function(a, b, c, d, e) {
        //>>includeStart('debug', pragmas.debug);
        if (typeof a !== 'number') {
            throw new DeveloperError('a is a required number.');
        }
        if (typeof b !== 'number') {
            throw new DeveloperError('b is a required number.');
        }
        if (typeof c !== 'number') {
            throw new DeveloperError('c is a required number.');
        }
        if (typeof d !== 'number') {
            throw new DeveloperError('d is a required number.');
        }
        if (typeof e !== 'number') {
            throw new DeveloperError('e is a required number.');
        }
        //>>includeEnd('debug');

        var a2 = a * a;
        var a3 = a2 * a;
        var b2 = b * b;
        var b3 = b2 * b;
        var c2 = c * c;
        var c3 = c2 * c;
        var d2 = d * d;
        var d3 = d2 * d;
        var e2 = e * e;
        var e3 = e2 * e;

        var discriminant = (b2 * c2 * d2 - 4.0 * b3 * d3 - 4.0 * a * c3 * d2 + 18 * a * b * c * d3 - 27.0 * a2 * d2 * d2 + 256.0 * a3 * e3) +
            e * (18.0 * b3 * c * d - 4.0 * b2 * c3 + 16.0 * a * c2 * c2 - 80.0 * a * b * c2 * d - 6.0 * a * b2 * d2 + 144.0 * a2 * c * d2) +
            e2 * (144.0 * a * b2 * c - 27.0 * b2 * b2 - 128.0 * a2 * c2 - 192.0 * a2 * b * d);
        return discriminant;
    };

    function original(a3, a2, a1, a0) {
        var a3Squared = a3 * a3;

        var p = a2 - 3.0 * a3Squared / 8.0;
        var q = a1 - a2 * a3 / 2.0 + a3Squared * a3 / 8.0;
        var r = a0 - a1 * a3 / 4.0 + a2 * a3Squared / 16.0 - 3.0 * a3Squared * a3Squared / 256.0;

        // Find the roots of the cubic equations:  h^6 + 2 p h^4 + (p^2 - 4 r) h^2 - q^2 = 0.
        var cubicRoots = CubicRealPolynomial.computeRealRoots(1.0, 2.0 * p, p * p - 4.0 * r, -q * q);

        if (cubicRoots.length > 0) {
            var temp = -a3 / 4.0;

            // Use the largest positive root.
            var hSquared = cubicRoots[cubicRoots.length - 1];

            if (Math.abs(hSquared) < CesiumMath.EPSILON14) {
                // y^4 + p y^2 + r = 0.
                var roots = QuadraticRealPolynomial.computeRealRoots(1.0, p, r);

                if (roots.length === 2) {
                    var root0 = roots[0];
                    var root1 = roots[1];

                    var y;
                    if (root0 >= 0.0 && root1 >= 0.0) {
                        var y0 = Math.sqrt(root0);
                        var y1 = Math.sqrt(root1);

                        return [temp - y1, temp - y0, temp + y0, temp + y1];
                    } else if (root0 >= 0.0 && root1 < 0.0) {
                        y = Math.sqrt(root0);
                        return [temp - y, temp + y];
                    } else if (root0 < 0.0 && root1 >= 0.0) {
                        y = Math.sqrt(root1);
                        return [temp - y, temp + y];
                    }
                }
                return [];
            } else if (hSquared > 0.0) {
                var h = Math.sqrt(hSquared);

                var m = (p + hSquared - q / h) / 2.0;
                var n = (p + hSquared + q / h) / 2.0;

                // Now solve the two quadratic factors:  (y^2 + h y + m)(y^2 - h y + n);
                var roots1 = QuadraticRealPolynomial.computeRealRoots(1.0, h, m);
                var roots2 = QuadraticRealPolynomial.computeRealRoots(1.0, -h, n);

                if (roots1.length !== 0) {
                    roots1[0] += temp;
                    roots1[1] += temp;

                    if (roots2.length !== 0) {
                        roots2[0] += temp;
                        roots2[1] += temp;

                        if (roots1[1] <= roots2[0]) {
                            return [roots1[0], roots1[1], roots2[0], roots2[1]];
                        } else if (roots2[1] <= roots1[0]) {
                            return [roots2[0], roots2[1], roots1[0], roots1[1]];
                        } else if (roots1[0] >= roots2[0] && roots1[1] <= roots2[1]) {
                            return [roots2[0], roots1[0], roots1[1], roots2[1]];
                        } else if (roots2[0] >= roots1[0] && roots2[1] <= roots1[1]) {
                            return [roots1[0], roots2[0], roots2[1], roots1[1]];
                        } else if (roots1[0] > roots2[0] && roots1[0] < roots2[1]) {
                            return [roots2[0], roots1[0], roots2[1], roots1[1]];
                        }
                        return [roots1[0], roots2[0], roots1[1], roots2[1]];
                    }
                    return roots1;
                }

                if (roots2.length !== 0) {
                    roots2[0] += temp;
                    roots2[1] += temp;

                    return roots2;
                }
                return [];
            }
        }
        return [];
    }

    function neumark(a3, a2, a1, a0) {
        var a1Squared = a1 * a1;
        var a2Squared = a2 * a2;
        var a3Squared = a3 * a3;

        var p = -2.0 * a2;
        var q = a1 * a3 + a2Squared - 4.0 * a0;
        var r = a3Squared * a0 - a1 * a2 * a3 + a1Squared;

        var cubicRoots = CubicRealPolynomial.computeRealRoots(1.0, p, q, r);

        if (cubicRoots.length > 0) {
            // Use the most positive root
            var y = cubicRoots[0];

            var temp = (a2 - y);
            var tempSquared = temp * temp;

            var g1 = a3 / 2.0;
            var h1 = temp / 2.0;

            var m = tempSquared - 4.0 * a0;
            var mError = tempSquared + 4.0 * Math.abs(a0);

            var n = a3Squared - 4.0 * y;
            var nError = a3Squared + 4.0 * Math.abs(y);

            var g2;
            var h2;

            if (y < 0.0 || (m * nError < n * mError)) {
                var squareRootOfN = Math.sqrt(n);
                g2 = squareRootOfN / 2.0;
                h2 = squareRootOfN === 0.0 ? 0.0 : (a3 * h1 - a1) / squareRootOfN;
            } else {
                var squareRootOfM = Math.sqrt(m);
                g2 = squareRootOfM === 0.0 ? 0.0 : (a3 * h1 - a1) / squareRootOfM;
                h2 = squareRootOfM / 2.0;
            }

            var G;
            var g;
            if (g1 === 0.0 && g2 === 0.0) {
                G = 0.0;
                g = 0.0;
            } else if (CesiumMath.sign(g1) === CesiumMath.sign(g2)) {
                G = g1 + g2;
                g = y / G;
            } else {
                g = g1 - g2;
                G = y / g;
            }

            var H;
            var h;
            if (h1 === 0.0 && h2 === 0.0) {
                H = 0.0;
                h = 0.0;
            } else if (CesiumMath.sign(h1) === CesiumMath.sign(h2)) {
                H = h1 + h2;
                h = a0 / H;
            } else {
                h = h1 - h2;
                H = a0 / h;
            }

            // Now solve the two quadratic factors:  (y^2 + G y + H)(y^2 + g y + h);
            var roots1 = QuadraticRealPolynomial.computeRealRoots(1.0, G, H);
            var roots2 = QuadraticRealPolynomial.computeRealRoots(1.0, g, h);

            if (roots1.length !== 0) {
                if (roots2.length !== 0) {
                    if (roots1[1] <= roots2[0]) {
                        return [roots1[0], roots1[1], roots2[0], roots2[1]];
                    } else if (roots2[1] <= roots1[0]) {
                        return [roots2[0], roots2[1], roots1[0], roots1[1]];
                    } else if (roots1[0] >= roots2[0] && roots1[1] <= roots2[1]) {
                        return [roots2[0], roots1[0], roots1[1], roots2[1]];
                    } else if (roots2[0] >= roots1[0] && roots2[1] <= roots1[1]) {
                        return [roots1[0], roots2[0], roots2[1], roots1[1]];
                    } else if (roots1[0] > roots2[0] && roots1[0] < roots2[1]) {
                        return [roots2[0], roots1[0], roots2[1], roots1[1]];
                    } else {
                        return [roots1[0], roots2[0], roots1[1], roots2[1]];
                    }
                }
                return roots1;
            }
            if (roots2.length !== 0) {
                return roots2;
            }
        }
        return [];
    }

    /**
     * Provides the real valued roots of the quartic polynomial with the provided coefficients.
     *
     * @param {Number} a The coefficient of the 4th order monomial.
     * @param {Number} b The coefficient of the 3rd order monomial.
     * @param {Number} c The coefficient of the 2nd order monomial.
     * @param {Number} d The coefficient of the 1st order monomial.
     * @param {Number} e The coefficient of the 0th order monomial.
     * @returns {Number[]} The real valued roots.
     */
    QuarticRealPolynomial.computeRealRoots = function(a, b, c, d, e) {
        //>>includeStart('debug', pragmas.debug);
        if (typeof a !== 'number') {
            throw new DeveloperError('a is a required number.');
        }
        if (typeof b !== 'number') {
            throw new DeveloperError('b is a required number.');
        }
        if (typeof c !== 'number') {
            throw new DeveloperError('c is a required number.');
        }
        if (typeof d !== 'number') {
            throw new DeveloperError('d is a required number.');
        }
        if (typeof e !== 'number') {
            throw new DeveloperError('e is a required number.');
        }
        //>>includeEnd('debug');

        if (Math.abs(a) < CesiumMath.EPSILON15) {
            return CubicRealPolynomial.computeRealRoots(b, c, d, e);
        }
        var a3 = b / a;
        var a2 = c / a;
        var a1 = d / a;
        var a0 = e / a;

        var k = (a3 < 0.0) ? 1 : 0;
        k += (a2 < 0.0) ? k + 1 : k;
        k += (a1 < 0.0) ? k + 1 : k;
        k += (a0 < 0.0) ? k + 1 : k;

        switch (k) {
        case 0:
            return original(a3, a2, a1, a0);
        case 1:
            return neumark(a3, a2, a1, a0);
        case 2:
            return neumark(a3, a2, a1, a0);
        case 3:
            return original(a3, a2, a1, a0);
        case 4:
            return original(a3, a2, a1, a0);
        case 5:
            return neumark(a3, a2, a1, a0);
        case 6:
            return original(a3, a2, a1, a0);
        case 7:
            return original(a3, a2, a1, a0);
        case 8:
            return neumark(a3, a2, a1, a0);
        case 9:
            return original(a3, a2, a1, a0);
        case 10:
            return original(a3, a2, a1, a0);
        case 11:
            return neumark(a3, a2, a1, a0);
        case 12:
            return original(a3, a2, a1, a0);
        case 13:
            return original(a3, a2, a1, a0);
        case 14:
            return original(a3, a2, a1, a0);
        case 15:
            return original(a3, a2, a1, a0);
        default:
            return undefined;
        }
    };

    return QuarticRealPolynomial;
});