/// <summary>
/// A method for computing a solution to a third order IVP ODE using RK4
/// </summary>
/// <param name="f1">The first function to be solved</param>
/// <param name="f2">The second function to be solved</param>
/// <param name="f3">The third function to be solved</param>
/// <param name="y0">The initial values for the system</param>
/// <param name="a">The lower bound to the range</param>
/// <param name="b">The upper bound to the range</param>
/// <param name="NumSteps">The number of steps to take in the range</param>
/// <returns>A list of points calculated by the function</returns>
public List <Vector>[] RK4(double[] y0, double a, double b, int NumSteps, int choice)
{
List <Vector> odesol1 = new List <Vector>();
double h = (Math.Max(a, b) - Math.Min(a, b)) / (NumSteps - 1.0);
double xt = a;
double[] yNew = y0, yOld = new double[yNew.Length];
int i = 0;
double deriv = 0;
Function udash = x => x[2];
Function u2dash = x => x[3];
Function u3dash = x => x[0];
Function kappa1 = x => (1.0 / 3.0) + Math.Sin(x[0]);
Function kappa2 = x => (1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0);
Function kappa3 = x => (1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0);
Function kappa4 = x => (1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0);
Function kappa5 = x => (1.0 / 10.0) + Math.Sin(x[0]);
Function kappa6 = x => 1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3);
Function kappa7 = x => 1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4);
while (i < NumSteps)
{
switch (choice)
{
case (1):
deriv = DrawCurve.RichardsonWithError(kappa1, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 3.0) + Math.Sin(x[0]))) * x[3] - (((1.0 / 3.0) + Math.Sin(x[0])) * ((1.0 / 3.0) + Math.Sin(x[0])) * x[2]);
break;
case (2):
deriv = DrawCurve.RichardsonWithError(kappa2, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0))) * x[3] - (((1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0)) * ((1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0)) * x[2]);;
break;
case (3):
deriv = DrawCurve.RichardsonWithError(kappa3, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0))) * x[3] - (((1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0)) * ((1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0)) * x[2]);
break;
case (4):
deriv = DrawCurve.RichardsonWithError(kappa4, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0))) * x[3] - (((1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0)) * ((1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0)) * x[2]);
break;
case (5):
deriv = DrawCurve.RichardsonWithError(kappa5, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 10.0) + Math.Sin(x[0]))) * x[3] - (((1.0 / 10.0) + Math.Sin(x[0])) * ((1.0 / 10.0) + Math.Sin(x[0])) * x[2]);
break;
case (6):
deriv = DrawCurve.RichardsonWithError(kappa6, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3))) * x[3] - ((1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3)) * (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3)) * x[2]);
break;
case (7):
deriv = DrawCurve.RichardsonWithError(kappa7, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4))) * x[3] - ((1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4)) * (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4)) * x[2]);
break;
default:
deriv = DrawCurve.RichardsonWithError(kappa1, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 3.0) + Math.Sin(x[0]))) * x[3] - (((1.0 / 3.0) + Math.Sin(x[0])) * ((1.0 / 3.0) + Math.Sin(x[0])) * x[2]);
break;
}
odesol1.Add(new Vector(xt, yNew[0], 0));
for (int j = 0; j < yNew.Length; j++)
{
yOld[j] = yNew[j];
}
yNew = RK4Step(udash, u2dash, u3dash, xt, h, yOld);
xt = xt + h;
i++;
}
return(new List <Vector>[] { odesol1 });
}
/// <summary>
/// A method for computing a solution to a third order IVP ODE using RKF
/// </summary>
/// <param name="f1">The first function to be solved</param>
/// <param name="f2">The second function to be solved</param>
/// <param name="f3">The third function to be solved</param>
/// <param name="y0">The initial values for the system</param>
/// <param name="a">The lower bound to the range</param>
/// <param name="b">The upper bound to the range</param>
/// <param name="NumSteps">The number of steps to take in the range</param>
/// <returns>A list of points calculated by the function</returns>
public List <Vector>[] RKF(double[] y0, double a, double b, int NumSteps, int choice)
{
List <Vector> odesol1 = new List <Vector>();
double h = (Math.Max(a, b) - Math.Min(a, b)) / (NumSteps - 1.0);
double xt = a, x0 = 0;
double[] yNew = y0, yOld = new double[yNew.Length];
int i = 0;
double deriv = 0;
List <double[]> resultStep = new List <double[]>();
//double tolerance = 1E-9;
double err = 0, tol = 1E-4, /*maxErr = 0, delta = 0,*/ s = 0;
double hmin = 1E-9;
double hmax = 5;
Function udash = x => x[2];
Function u2dash = x => x[3];
Function u3dash = x => x[0];
Function kappa1 = x => (1.0 / 3.0) + Math.Sin(x[0]);
Function kappa2 = x => (1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0);
Function kappa3 = x => (1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0);
Function kappa4 = x => (1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0);
Function kappa5 = x => (1.0 / 10.0) + Math.Sin(x[0]);
Function kappa6 = x => 1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3);
Function kappa7 = x => 1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4);
while (i < NumSteps)
{
/*if (i >= NumSteps)
* {
* break;
* }*/
switch (choice)
{
case (1):
deriv = DrawCurve.RichardsonWithError(kappa1, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 3.0) + Math.Sin(x[0]))) * x[3] - (((1.0 / 3.0) + Math.Sin(x[0])) * ((1.0 / 3.0) + Math.Sin(x[0])) * x[2]);
break;
case (2):
deriv = DrawCurve.RichardsonWithError(kappa2, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0))) * x[3] - (((1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0)) * ((1.0 / 5.0) + 2 * Math.Sin(x[0]) + 3 * Math.Cos(x[0] / 3.0)) * x[2]);;
break;
case (3):
deriv = DrawCurve.RichardsonWithError(kappa3, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0))) * x[3] - (((1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0)) * ((1.0 / 5.0) + Math.Sin(x[0] / 2.0) - Math.Cos(x[0] / 3.0)) * x[2]);
break;
case (4):
deriv = DrawCurve.RichardsonWithError(kappa4, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0))) * x[3] - (((1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0)) * ((1.0 / 5.0) + Math.Sin(x[0] / 3.0) + 2 * Math.Cos(x[0] / 2.0)) * x[2]);
break;
case (5):
deriv = DrawCurve.RichardsonWithError(kappa5, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 10.0) + Math.Sin(x[0]))) * x[3] - (((1.0 / 10.0) + Math.Sin(x[0])) * ((1.0 / 10.0) + Math.Sin(x[0])) * x[2]);
break;
case (6):
deriv = DrawCurve.RichardsonWithError(kappa6, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3))) * x[3] - ((1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3)) * (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (3)) * x[2]);
break;
case (7):
deriv = DrawCurve.RichardsonWithError(kappa7, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4))) * x[3] - ((1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4)) * (1 + (3.0 / (5 + 4 * Math.Cos(x[0]))) * (4)) * x[2]);
break;
default:
deriv = DrawCurve.RichardsonWithError(kappa5, xt, 1E-6, 10000, 1E-9);
u3dash = x => (deriv / ((1.0 / 10.0) + Math.Sin(x[0]))) * x[3] - (((1.0 / 10.0) + Math.Sin(x[0])) * ((1.0 / 10.0) + Math.Sin(x[0])) * x[2]);
break;
}
odesol1.Add(new Vector(xt, yNew[0], 0));
for (int j = 0; j < yNew.Length; j++)
{
yOld[j] = yNew[j];
}
resultStep = RKFStep(udash, u2dash, u3dash, xt, h, yOld);
/*for (int j = 0; j < 3; j++)
* {
* err = Math.Max(err, resultStep[2][j]);
* }*/
err = resultStep[2][0];
s = Math.Pow(0.5 * tol / err, 0.25);
if (err < tol)
{
yNew = resultStep[0];
xt = x0 + h;
x0 = xt;
}
if (s < 0.1)
{
s = 0.1;
}
else if (s > 4.0)
{
s = 4.0;
}
h *= s;
if (h > b - x0)
{
h = b - x0;
}
if (h > hmax)
{
h = hmax;
}
else if (h < hmin)
{
h = hmin;
}
i++;
err = 0;
}
deriv = 0;
return(new List <Vector>[] { odesol1 });
}
