public Vector <double>[] bvpsolver_zzz(Func <double, Vector <double>, Vector <double> > system,
double ya, double yb,
double xa, double xb,
int N)
{
Vector <double>[] res;
Vector <double> y0;
double s_guess = (yb - ya) / (xb - xa), s_guess_pre = 0;
double fais = yb, fais_pre = 0, dfaids = 0;
do
{
y0 = Vector <double> .Build.DenseOfArray(new[] { ya, s_guess });
// observer_pr ob_pr = this->write_targetFunc_end;
res = RungeKutta.FourthOrder(y0, xa, xb, N, system);
fais_pre = fais;
fais = res[N - 1][0] - yb;
dfaids = (fais - fais_pre) / (s_guess - s_guess_pre);
s_guess_pre = s_guess;
s_guess = s_guess - fais / dfaids;
//count++;
} while (fais > 0.001 * yb + 1e-6 || fais < -0.001 * yb - 1e-6);
return(res);
}
static void Main(string[] args)
{
int N = 1000000;
Vector <double> y0 = Vector <double> .Build.Dense(new[] { -7.0 / 4.0, 55.0 / 8.0 });
Func <double, Vector <double>, Vector <double> > der = DerivativeMaker();
Vector <double>[] res = RungeKutta.FourthOrder(y0, 0, 10, N, der);
double[] x = new double[N];
double[] y = new double[N];
for (int i = 0; i < N; i++)
{
double[] temp = res[i].ToArray();
x[i] = temp[0];
y[i] = temp[1];
}
//Test
Console.WriteLine(y[N / 10]); // gives 164,537981852489
Console.WriteLine(Math.Exp(-1) + 3 * Math.Exp(4) - 5.0 / 2 + 23.0 / 8); //gives 164,537329540604, which is y(1)
Console.ReadKey();
}
static Vector <double>[] function_RungeKutta(double P, double R, double p, double c, double a, double k,
double t_first, double t_end, int N)
{
Vector <double> variables = Vector <double> .Build.Dense(new[] { P, R });
Func <double, Vector <double>, Vector <double> > der = DerivativeMaker(p, c, a, k);
Vector <double>[] res = RungeKutta.FourthOrder(variables, t_first, t_end, N, der);
return(res);
}
public double[,] SolveRK4(Vector <double> x0, double start, double end, int n, Func <double, Vector <double>, Vector <double> > func)
{
var answer = new double[n, 2];
var res = RungeKutta.FourthOrder(x0, start, end, n, func);
for (var i = 0; i < n; i++)
{
var temp = res[i].ToArray();
answer[i, 0] = temp[0];
answer[i, 1] = temp[1];
}
return(answer);
}
public List <double[]> SolveODE(double[] y0, double startTime, double endTime, int n, Func <double, double[], double[]> func)
{
var y0Vector = Vector <double> .Build.DenseOfArray(y0);
var result = RungeKutta.FourthOrder(y0Vector, startTime, endTime, n, (t, y) =>
{
var res = func(t, y.AsArray());
return(Vector <double> .Build.DenseOfArray(res));
});
var resultList = new List <double[]>();
for (var i = 0; i < n; i++)
{
var vector = result[i];
resultList.Add(vector.AsArray());
}
return(resultList);
}
public void RK4Test()
{
Func <double, double, double> ode = (t, y) => t + 2 * y * t;
Func <double, double> sol = (t) => 0.5 * (Math.Exp(t * t) - 1);
double ratio = double.NaN;
double error = 0;
double oldError = 0;
for (int k = 0; k < 4; k++)
{
double y0 = 0;
double[] y_t = RungeKutta.FourthOrder(y0, 0, 2, Convert.ToInt32(Math.Pow(2, k + 6)), ode);
error = Math.Abs(sol(2) - y_t.Last());
if (oldError != 0)
{
ratio = Math.Log(oldError / error, 2);
}
oldError = error;
Console.WriteLine(string.Format("{0}, {1}", error, ratio));
}
Assert.AreEqual(4, ratio, 0.01);// Check error convergence order
}
static void Main(string[] args)
{
int N = 10000;
Vector <double> y0 = Vector <double> .Build.Dense(new[] { -7.0 / 4.0, 55.0 / 8.0 });
//Func<double, Vector<double>, Vector<double>> der = DerivativeMaker();
Vector <double>[] res = RungeKutta.FourthOrder(y0, 0, 10, N, DerivativeMaker_zzz2);
double[] t = Vector <double> .Build.Dense(N, i => i * 10.0 / N).ToArray();
double[] x = new double[N];
double[] y = new double[N];
for (int i = 0; i < N; i++)
{
double[] temp = res[i].ToArray();
x[i] = temp[0];
y[i] = temp[1];
//Console.WriteLine(t[i]);
}
var plt = new ScottPlot.Plot(800, 600);
plt.PlotScatter(t, x, label: "x", markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
plt.PlotScatter(t, y, label: "y", markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
plt.Grid(false);
plt.Legend(fontSize: 10);
//plt.PlotSignal(new[,] { x, y });
plt.SaveFig("quickstart.png");
//Test
Console.WriteLine(y[N / 10]); // gives 164,537981852489
Console.WriteLine(Math.Exp(-1) + 3 * Math.Exp(4) - 5.0 / 2 + 23.0 / 8); //gives 164,537329540604, which is y(1)
Console.ReadKey();
}
private void button_calc_Click_3(object sender, EventArgs e)
{
DataTable dt = (DataTable)dataGridView1.DataSource;
bendStiffener.max_k = (double)dt.Rows[0][1];
bendStiffener.D1 = (double)dt.Rows[1][1];
bendStiffener.d1 = (double)dt.Rows[2][1];
bendStiffener.d2 = (double)dt.Rows[3][1];
bendStiffener.L1 = (double)dt.Rows[4][1];
bendStiffener.L2 = (double)dt.Rows[5][1];
bendStiffener.L3 = (double)dt.Rows[6][1];
bendStiffener.EIp = (double)dt.Rows[7][1];
bendStiffener.Es = (double)dt.Rows[8][1];
bendStiffener.F = (double)dt.Rows[9][1];
bendStiffener.q = (double)dt.Rows[10][1];
int N = 10000;
//Vector<double>[] res = bendStiffener.bvpsolver_zzz(bendStiffener.Ode_to_solve, 0.0, bendStiffener.q, 0.0, 5.0, N);
double[] x = Vector <double> .Build.Dense(N, i => i * 5.0 / N).ToArray();
double[] y = new double[N];
double[] dydx = new double[N];
//initialisation of plot
formsPlot1.Reset();
var plt = new ScottPlot.Plot(800, 600);
//plt.PlotScatter(x, y, label: "y", markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
//plt.PlotSignal(new[,] { x, y });
//plt.SaveFig("quickstart.png");
double[] lineX = new double[] { 0, 5 };
double[] lineY = new double[] { bendStiffener.max_k, bendStiffener.max_k };
//formsPlot1.plt.PlotScatter(lineX, lineY, label: "Pass", lineWidth: 5, markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
//double[] max_pnt = new double[] { x[Array.IndexOf(dydx, dydx.Max())], dydx.Max() };
//formsPlot1.plt.PlotPoint(max_pnt[0], max_pnt[1], color: Color.Magenta, markerSize: 15);
//initialisation of plot
//--------
double ya = 0;
double yb = bendStiffener.q;
double xa = 0;
double xb = 5;
Func <double, Vector <double>, Vector <double> > system = bendStiffener.Ode_to_solve;
Vector <double>[] res;
Vector <double> y0;
//double s_guess = (yb - ya) / (xb - xa), s_guess_pre = 0;
double fais = yb, fais_pre = 0, dfaids = 0, dfai, ds;
//bool changeDirectionFlag = false;
int countTimes = 1000;
double[] faisLst = new double[countTimes];
double[] sLst = new double[countTimes];
sLst = Vector <double> .Build.Dense(N, i => - 0.05 + i * 0.3 / countTimes).ToArray();
for (int count = 0; count < countTimes; count++)
{
y0 = Vector <double> .Build.DenseOfArray(new[] { ya, sLst[count] });
// observer_pr ob_pr = this->write_targetFunc_end;
res = RungeKutta.FourthOrder(y0, xa, xb, N, system);
fais_pre = fais;
fais = res[N - 1][0] - yb;
////dfaids = (fais - fais_pre) / (s_guess - s_guess_pre);
//dfai = fais - fais_pre;
//ds = s_guess - s_guess_pre;
//s_guess_pre = s_guess;
////s_guess = s_guess - fais / dfaids;
//s_guess = s_guess - fais * ds / dfai;
////s_guess = s_guess - fais / (fais - fais_pre) * (s_guess - s_guess_pre);
faisLst[count] = fais;
//sLst[count] = s_guess;
formsPlot1.plt.PlotPoint(sLst[count], fais, color: Color.Magenta, markerSize: 15);
//---------
double[][] dydx2 = new double[countTimes][];
for (int i = 0; i < N; i++)
{
double[] temp = res[i].ToArray();
y[i] = temp[0];
dydx[i] = temp[1];
//Console.WriteLine(t[i]);
}
dydx2[count] = (double[])dydx.Clone();
//formsPlot1.plt.PlotScatter(x, dydx2[count], label: "dydx" + count.ToString(), lineWidth: 5, markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
}
//formsPlot1.plt.PlotScatter(x, dydx, label: "dydx", lineWidth: 5, markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
formsPlot1.plt.PlotScatter(sLst, faisLst, label: "fais-s", lineWidth: 5, markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
//plt.PlotScatter(x, dydx, label: "dydx", markerShape: (MarkerShape)Enum.Parse(typeof(MarkerShape), "none"));
formsPlot1.plt.Grid(true);
formsPlot1.plt.Legend(fontSize: 20, location: legendLocation.lowerLeft);
formsPlot1.Render();
}
public Vector <double>[] bvpsolver_zzz(Func <double, Vector <double>, Vector <double> > system,
double ya, double yb,
double xa, double xb,
int N)
{
//先求解方程得到初始斜率的估计值，再进行打靶法迭代。--zzz
Vector <double>[] res;
Vector <double> y0;
double s_guess, s_guess_pre;
double fais, fais_pre;
double dfai, ds;
int count = 0;
//计算20组s_guess和fais，然后样条插值得到连续函数，再通过解方程，得到使fais=0的初始s_guess
int M = 50;
double[] sLst = Vector <double> .Build.Dense(M, i => yb / M *i).ToArray();
double[] faisLst = new double[M];
count = 0;
while (count < M)
{
y0 = Vector <double> .Build.DenseOfArray(new[] { ya, sLst[count] });
// observer_pr ob_pr = this->write_targetFunc_end;
res = RungeKutta.FourthOrder(y0, xa, xb, N, system);
faisLst[count] = res[N - 1][0] - yb;
count++;
}
//样条插值得到连续函数
var cubicSpl = CubicSpline.InterpolateNatural(sLst, faisLst);
/*如果初始值离解太远，牛顿法会不收敛。故采用Mathnet包中的RobustNewtonRaphson
* double s_cur = 0, s_next;
* count = 0;
* while (count < 1000)
* {
* fais = cubicSpl.Interpolate(s_cur);
* dfaids = cubicSpl.Differentiate(s_cur);
* if (fais < 1e-5 && fais > -1e-5)
* {
* break;
* }
*
* s_next = s_cur - fais / dfaids;
* s_cur = s_next;
* count += 1;
* }*/
//解方程fais=0,得到初始斜率s_guess。该法先尝试牛顿法，如失败会采用二分法（bisection）。
s_guess = RobustNewtonRaphson.FindRoot(cubicSpl.Interpolate, cubicSpl.Differentiate, 0, yb, 1e-5);
//利用解得的s_guess，构造s_guess_pre，目的是求导数dfai/ds。
if (s_guess == 0)
{
s_guess_pre = 1e-2;
}
else
{
s_guess_pre = s_guess * 0.99;
}
//求s_guess_pre对应的fais_pre，目的是求导数dfai/ds。
y0 = Vector <double> .Build.DenseOfArray(new[] { ya, s_guess_pre });
res = RungeKutta.FourthOrder(y0, xa, xb, N, system);
fais_pre = res[N - 1][0] - yb;
count = 0;
while (count < 50)
{
y0 = Vector <double> .Build.DenseOfArray(new[] { ya, s_guess });
res = RungeKutta.FourthOrder(y0, xa, xb, N, system);
fais = res[N - 1][0] - yb;
dfai = fais - fais_pre;
ds = s_guess - s_guess_pre;
if (fais < 1e-5 && fais > -1e-5)
{
break;
}
fais_pre = fais;
s_guess_pre = s_guess;
s_guess = s_guess - fais * ds / dfai;
count++;
}
return(res);
}
/// <summary>
/// Método Runge Kutta de Quarta ordem para equações diferenciais
/// </summary>
/// <param name="y0">Valor Inicial</param>
/// <param name="start">Tempo Inicial</param>
/// <param name="end">Tempo Final</param>
/// <param name="dydx">Equação diferencial a ser calculada</param>
/// <returns></returns>
public double RungeKutta4(double y0, double start, double end, Func <double, double, double> dydx)
{
return(RungeKutta.FourthOrder(y0, start, end, 10, dydx)[9]);
}
public Vector <double>[] Integrate(IReadOnlyList <double> Control)
{
_params = Control;
return(RungeKutta.FourthOrder(_x0, 0, _tMax, _sizeOfRes, OdeFunction));
}