31:T1d19,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}) };32:[] 2f:{"title":"4-th order polynomial equation resolution","cells":[{"id":0,"value":"md`# 4-th order polynomial equation resolution\n\nThe code is copy and paste (and little adapt) from the code of [Brian Kieffer](https://javascriptsource.com/quartic-equation-solver/).\n\nThose functions allow us to compute the 4 solutions of an equation like : \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"tex.block\n `ax^4 + bx^3 + cx^2 + dx + e = 0`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":34,"value":"md`For instance, the solutions of ${tex `x^4 - 1 = 0`} can be obtain as follow : `","pinned":false,"hidden":false,"mode":"ojs"},{"id":37,"value":"equationSolver4(1, 0, 0, 0, -1)","pinned":true,"hidden":false,"mode":"ojs"},{"id":39,"value":"md`Where each solution is a composition of the real part and the imaginary part separate by a space. Therefore the translation of the solutions are : `","pinned":false,"hidden":false,"mode":"ojs"},{"id":42,"value":"tex.block` x_1 = i, \\ x_2 = 1, \\ x_3 = -1, \\ \\text{and} \\ x_4 = -i`","pinned":false,"hidden":false,"mode":"ojs"},{"id":4,"value":"function calcmult(a2,b2,c2,d2,e2) {\n var real = a2*c2 - b2*d2;\n var img = b2*c2 + a2*d2;\n\n if (e2 == 0) {\n return real;\n } else {\n return img;\n }\n};","pinned":true,"hidden":false,"mode":"ojs"},{"id":5,"value":"function isquareroot(a1,b1,n1) {\n var y = Math.sqrt((a1*a1) + (b1*b1));\n var y1 = Math.sqrt((y - a1) / 2);\n var x1 = b1 / (2*y1);\n\n if (n1 == 0) {\n return x1;\n } else {\n return y1;\n }\n};","pinned":true,"hidden":false,"mode":"ojs"},{"id":7,"value":"$31","pinned":true,"hidden":false,"mode":"ojs"},{"id":23,"value":"md`This function filters the real and positive solutions.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":9,"value":"solutionFilter = (solution) => {\n \n var newSolution;\n \n let x1 = solution.x1.split(\" \");\n let x2 = solution.x2.split(\" \");\n let x3 = solution.x3.split(\" \");\n let x4 = solution.x4.split(\" \");\n \n let n = 0; //to get one solution\n let x = [x1, x2, x3, x4];\n x.forEach((d, j) => {\n if (+d[1] == 0){\n if (+d[0] >= 0 && n == 0){\n n++;\n newSolution = +d[0];\n }\n }\n \n });\n \n return newSolution;\n};","pinned":true,"hidden":false,"mode":"ojs"},{"id":13,"value":"md`## Application\n\nWe use the resolution of a 4-th order polynomial equation to compute the time needed to reach (xf, yf) from (x0, y0) with a initial speed (vx, vy) and with a constant force F. \nTo know more about this model see : [Gabin Rolland et al.](https://hal.archives-ouvertes.fr/hal-02482706/file/characterization_of_Space_and_Time-Dependece_of_3-Point_Shots_in_Basketball.pdf)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"distanceTime = (x0, y0, xf, yf, vx, vy, F) => {\n\n let X = x0 - xf;\n let Y = y0 - yf;\n let a = 1;\n let b = 0;\n let c = -4 * (vx**2 + vy**2) / (F**2);\n let d = -8 * (vx * X + vy * Y) / (F**2);\n let e = -4 * (X**2 + Y**2) / (F**2);\n var temp = equationSolver4(a, b, c, d, e);\n temp = solutionFilter(temp); // on ne garde que les valeurs réels positives.\n \n return temp;\n};","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}