4-th order polynomial equation resolution / julesallegre

julesallegre

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Published

Edited

May 24, 2020

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equationSolver4(1, 0, 0, 0, -1)

function calcmult(a2,b2,c2,d2,e2) {

var real = a2*c2 - b2*d2;

var img = b2*c2 + a2*d2;

if (e2 == 0) {

return real;

} else {

return img;

}

};

function isquareroot(a1,b1,n1) {

var y = Math.sqrt((a1*a1) + (b1*b1));

var y1 = Math.sqrt((y - a1) / 2);

var x1 = b1 / (2*y1);

if (n1 == 0) {

return x1;

} else {

return y1;

}

};

equationSolver4 = (aq, bq, cq, dq, eq) => {

//Permet de résoudre une équation algébrique d'ordre 4.

/* This script and many more are available free online at

The JavaScript Source!! http://www.javascriptsource.com

Created by: Brian Kieffer | http://www.freewebs.com/brianjs/ */

var aq2 = aq; // Keeps Orignial AQ value

var bq2 = bq;// Keeps Orignial BQ Value

var perfect = 0;

var perfectbiquadratic = 0;

// The Bi-Quadratic 2 Perfect Squares that are negative test

if (cq*cq - 4*aq*eq == 0 && cq > 0) {

perfectbiquadratic = 1;

}

// Divide Equation by the X^4 Coefficent to make equation in the form of X^4 + AX^3 + BX^2 + CX + D

bq /= aq;

cq /= aq;

dq /= aq;

eq /= aq;

aq = 1;

var f2 = cq - (3*bq*bq / 8);

var g2 = dq + (bq*bq*bq/8) - (bq*cq/2);

var h2 = eq - (3*bq*bq*bq*bq/256) + (bq*bq*(cq/16)) - (bq*dq/4);

var a = 1;

var b = f2/2;

var c = (f2*f2 - (4*h2)) / 16;

var d = -1*((g2*g2)/64);

if (b == 0 && c == 0 && d == 0) {

perfect = 1;

}

// Cubic routine starts here…..

var f = (((3*c) / a) - ((b*b) / (a*a))) / 3;

var g = (((2*b*b*b) / (a*a*a)) - ((9*b*c) / (a*a)) + ((27*d) / a)) / 27;

var h = eval(((g*g)/4) + ((f*f*f)/27));

var z = 1/3;

var i;

var j;

var k;

var l;

var m;

var n;

var p;

var xoneterm;

var xtwoterm;

var xthreeterm;

var alreadydone;

var alreadydone2 = 0;

var ipart = 0;

var p = 0;

var q = 0;

var r = 0;

var s = 0;

if (h <= 0) {

var exec = 2;

i = Math.sqrt(((g*g) / 4) - h);

j = Math.pow(i,z);

k = Math.acos(-1 * (g / (2*i)));

l = -1*j;

m = Math.cos(k / 3);

n = Math.sqrt(3) * Math.sin(k / 3);

p = (b / (3*a)) * -1;

xoneterm = (2*j) * Math.cos(k/3) - (b / (3*a));

xtwoterm = l * (m + n) + p;

xthreeterm = l * (m - n) + p;

}

if (h > 0) {

var exec = 1;

var R = (-1*(g / 2)) + Math.sqrt(h);

if (R < 0) {

var S = -1*(Math.pow((-1*R),z));

} else {

var S = Math.pow(R,z);

}

var T = (-1*(g / 2)) - Math.sqrt(h);

if (T < 0) {

var U = -1*(Math.pow((-1*T),z));

} else {

var U = Math.pow(T,z);

}

xoneterm = (S + U) - (b / (3*a));

xtwoterm = (-1*(S+U)/2) - (b / (3*a));

var ipart = ((S-U) * Math.sqrt(3)) / 2;

xthreeterm = xtwoterm;

}

if (f == 0 && g == 0 && h == 0) {

if ((d/a) < 0 ) {

xoneterm = (Math.pow((-1*(d/a)),z));

xtwoterm = xoneterm;

xthreeterm = xoneterm;

} else {

xoneterm = -1*(Math.pow((d/a),z));

xtwoterm = xoneterm;

xthreeterm = xoneterm;

}

// ….and ends here.

// Return to solving the Quartic.

if (ipart == 0 && xoneterm.toFixed(10) == 0) {

var alreadydone2 = 1;

var p2 = Math.sqrt(xtwoterm);

var q = Math.sqrt(xthreeterm);

var r = -g2 / (8*p2*q);

var s = bq2/(4*aq2);

}

if (ipart == 0 && xtwoterm.toFixed(10) == 0 && alreadydone2 == 0 && alreadydone2 != 1) {

var alreadydone2 = 2;

var p2 = Math.sqrt(xoneterm);

var q = Math.sqrt(xthreeterm);

var r = -g2 / (8*p2*q);

var s = bq2/(4*aq2);

}

if (ipart == 0 && xthreeterm.toFixed(10) == 0 && alreadydone2 == 0 && alreadydone2 != 1 && alreadydone2 != 2) {

var alreadydone2 = 3;

var p2 = Math.sqrt(xoneterm);

var q = Math.sqrt(xtwoterm);

var r = -g2 / (8*p2*q);

var s = bq2/(4*aq2);

}

if (alreadydone2 == 0 && ipart == 0) {

if (xthreeterm.toFixed(10) < 0) {

var alreadydone2 = 4;

var p2 = Math.sqrt(xoneterm);

var q = Math.sqrt(xtwoterm);

var r = -g2 / (8*p2*q);

var s = bq2/(4*aq2);

} else {

var alreadydone2 = 5;

var p2 = Math.sqrt(xoneterm.toFixed(10));

var q = Math.sqrt(xthreeterm.toFixed(10));

var r = -g2 / (8*p2*q);

var s = bq2/(4*aq2);

}

if (ipart != 0) {

var p2 = isquareroot(xtwoterm,ipart,0);

var p2ipart = isquareroot(xtwoterm,ipart,1);

var q = isquareroot(xthreeterm,-ipart,0);

var qipart = isquareroot(xthreeterm,-ipart,1);

var mult = calcmult(p2,p2ipart,q,qipart,0);

var r = -g2/(8*mult);

var s = bq2/(4*aq2);

}

if (ipart == 0 && xtwoterm.toFixed(10) < 0 && xthreeterm.toFixed(10) < 0) {

xtwoterm /= -1

xthreeterm /= -1

var p2 = 0;

var q = 0;

var p2ipart = Math.sqrt(xtwoterm);

var qipart = Math.sqrt(xthreeterm);

var mult = calcmult(p2,p2ipart,q,qipart,0);

var r = -g2/(8*mult);

var s = bq2/(4*aq2);

var ipart = 1;

}

if (xoneterm.toFixed(10) > 0 && xtwoterm.toFixed(10) < 0 && xthreeterm.toFixed(10) == 0 && ipart == 0) {

xtwoterm /= -1;

var p2 = Math.sqrt(xoneterm);

var q = 0;

var p2ipart = 0;

var qipart = Math.sqrt(xtwoterm);

var mult = calcmult(p2,p2ipart,q,qipart,0);

var mult2 = calcmult(p2,p2ipart,q,qipart,1);

var r = -g2/(8*mult);

if (mult2 != 0) {

var ripart = g2/(8*mult2);

var r = 0;

}

var s = bq2/(4*aq2);

var ipart = 1;

}

if (xtwoterm.toFixed(10) == 0 && xthreeterm.toFixed(10) == 0 && ipart == 0) {

var p2 = Math.sqrt(xoneterm);

var q = 0;

var r = 0;

var s = bq2/(4*aq2);

}

if (ipart == 0) {

var x1 = eval((p2 + q + r - s).toFixed(10));

var x2 = eval((p2 - q - r - s).toFixed(10));

var x3 = eval((-p2 + q - r - s).toFixed(10));

var x4 = eval((-p2 - q + r - s).toFixed(10));

var x1i = 0;

var x2i = 0;

var x3i = 0;

var x4i = 0;

}

if (perfect == 1) {

var x1 = -bq/4;

var x2 = -bq/4;

var x3 = -bq/4;

var x4 = -bq/4;

var x1i = 0;

var x2i = 0;

var x3i = 0;

var x4i = 0;

}

if (ipart == 0 && xtwoterm.toFixed(10) < 0 && xthreeterm.toFixed(10) < 0) {

xtwoterm /= -1;

xthreeterm /= -1;

var p2 = 0;

var q = 0;

var p2ipart = Math.sqrt(xtwoterm);

var qipart = Math.sqrt(xthreeterm);

var mult = calcmult(p2,p2ipart,q,qipart,0);

var r = -g2/(8*mult);

var s = bq2/(4*aq2);

var ipart = 1;

}

if (xoneterm.toFixed(10) > 0 && xtwoterm.toFixed(10) < 0 && xthreeterm.toFixed(10) == 0 && ipart == 0) {

xtwoterm /= -1

var p2 = Math.sqrt(xoneterm);

var q = 0;

var p2ipart = 0;

var qipart = Math.sqrt(xtwoterm);

var mult = calcmult(p2,p2ipart,q,qipart,0);

var mult2 = calcmult(p2,p2ipart,q,qipart,1);

var r = -g2/(8*mult);

if (mult2 != 0) {

var ripart = g2/(8*mult2);

var r = 0;

}

var s = bq2/(4*aq2);

var ipart = 1;

}

if (xtwoterm.toFixed(10) == 0 && xthreeterm.toFixed(10) == 0 && ipart == 0) {

var p2 = Math.sqrt(xoneterm);

var q = 0;

var r = 0;

var s = bq2/(4*aq2);

}

if (ipart != 0) {

var x1 = eval((p2 + q + r - s).toFixed(10));

var x2 = eval((p2 - q - r - s).toFixed(10));

var x3 = eval((-p2 + q - r - s).toFixed(10));

var x4 = eval((-p2 - q + r - s).toFixed(10));

var x1i = eval((p2ipart + qipart).toFixed(10));

var x2i = eval((p2ipart - qipart).toFixed(10));

var x3i = eval((-p2ipart + qipart).toFixed(10));

var x4i = eval((-p2ipart - qipart).toFixed(10));

}

if (perfectbiquadratic == 1) {

var x1 = 0;

var x2 = 0;

var x3 = 0;

var x4 = 0;

var x1i = eval(Math.sqrt(cq/2).toFixed(10));

var x2i = eval(Math.sqrt(cq/2).toFixed(10));

var x3i = -eval(Math.sqrt(cq/2).toFixed(10));

var x4i = -eval(Math.sqrt(cq/2).toFixed(10));

}

return ({'x1' : x1 + ' ' + x1i,

'x2' : x2 + ' ' + x2i,

'x3' : x3 + ' ' + x3i,

'x4' : x4 + ' ' + x4i})

};

solutionFilter = (solution) => {

var newSolution;

let x1 = solution.x1.split(" ");

let x2 = solution.x2.split(" ");

let x3 = solution.x3.split(" ");

let x4 = solution.x4.split(" ");

let n = 0; //to get one solution

let x = [x1, x2, x3, x4];

x.forEach((d, j) => {

if (+d[1] == 0){

if (+d[0] >= 0 && n == 0){

n++;

newSolution = +d[0];

}

});

return newSolution;

};

distanceTime = (x0, y0, xf, yf, vx, vy, F) => {

let X = x0 - xf;

let Y = y0 - yf;

let a = 1;

let b = 0;

let c = -4 * (vx**2 + vy**2) / (F**2);

let d = -8 * (vx * X + vy * Y) / (F**2);

let e = -4 * (X**2 + Y**2) / (F**2);

var temp = equationSolver4(a, b, c, d, e);

temp = solutionFilter(temp); // on ne garde que les valeurs réels positives.

return temp;

};

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