/// <summary>Implementation of Runge-Kutta algoritm with per-point accurancy control from
/// Dormand and Prince article.
/// (J.R.Dormand, P.J.Prince, A family of embedded Runge-Kuttae formulae)</summary>
/// <param name="t0">Left end of current time span</param>
/// <param name="x0">Initial phase vector value</param>
/// <param name="f">System right parts vector function</param>
/// <param name="opts">Options used by solver</param>
/// <example>Let our problem will be:
/// dx/dt=y+1,
/// dy/dt=-x+2.
/// x(0)=0, y(0)=1.
/// To solve it, we just have to write
/// <code>
/// var sol=Ode.RK547M(0,new Vector(0,1),(t,x)=>new Vector(y+1,-x+2));
/// </code>
/// and then enumerate solution point from <see cref="System.IEnumerable"/> 'sol'.
/// </example>
/// <returns>Endless sequence of solution points</returns>
public static IEnumerable <SolPoint> RK547M(double t0, Vector x0, Func <double, Vector, Vector> f, Options opts)
{
// Safety factors for automatic step control
// See Solving Ordinary Differential Equations I Non stiff problems by E. Hairer, S.P. Norsett, G.Wanner
// 2nd edition, Springer.
// Safety factor is recommended on page 168
const double SafetyFactor = 0.8;
const double MaxFactor = 5.0d; // Maximum possible step increase
const double MinFactor = 0.2d; // Maximum possible step decrease
int Refine = 4;
Vector S = Vector.Zeros(Refine - 1);
for (int i = 0; i < Refine - 1; i++)
{
S[i] = (double)(i + 1) / Refine;
}
double t = t0;
Vector x = x0.Clone(); // Lower order approximation
Vector x1 = x0.Clone(); // Higher order approximation
int n = x0.Length; // Dimensions of the system
// Initial step value. During computational process, we modify it.
double dt = opts.InitialStep;
// Create formulae parameters(a's,b's and c's)
// In original article, a is a step array.
double[][] a = new double[6][];
a[0] = new double[] { 1 / 5d };
a[1] = new double[] { 3 / 40d, 9 / 40d };
a[2] = new double[] { 44 / 45d, -56 / 15d, 32 / 9d };
a[3] = new double[] { 19372 / 6561d, -25360 / 2187d, 64448 / 6561d, -212 / 729d };
a[4] = new double[] { 9017 / 3168d, -355 / 33d, 46732 / 5247d, 49 / 176d, -5103 / 18656d };
a[5] = new double[] { 35 / 384d, 0, 500 / 1113d, 125 / 192d, -2187 / 6784d, 11 / 84d };
Vector c = new Vector(0, 1 / 5d, 3 / 10d, 4 / 5d, 8 / 9d, 1, 1);
// Coeffs for higher order
Vector b1 = new Vector(5179 / 57600d, 0, 7571 / 16695d, 393 / 640d, -92097 / 339200d, 187 / 2100d, 1 / 40d);
// Coeffs for lower order
Vector b = new Vector(35 / 384d, 0, 500 / 1113d, 125 / 192d, -2187 / 6784d, 11 / 84d, 0);
const int s = 7; // == c.Length
double dt0;
// Compute initial step (see E. Hairer book)
if (opts.InitialStep == 0)
{
double d0 = 0.0d, d1 = 0.0d;
double[] sc = new double[n];
var f0 = f(t0, x0);
for (int i = 0; i < n; i++)
{
sc[i] = opts.AbsoluteTolerance + opts.RelativeTolerance * Math.Abs(x0[i]);
d0 = Math.Max(d0, Math.Abs(x0[i]) / sc[i]);
d1 = Math.Max(d1, Math.Abs(f0[i]) / sc[i]);
}
var h0 = Math.Min(d0, d1) < 1e-5 ? 1e-6 : 1e-2 * (d0 / d1);
var f1 = f(t0 + h0, x0 + h0 * f0);
double d2 = 0;
for (int i = 0; i < n; i++)
{
d2 = Math.Max(d2, Math.Abs(f0[i] - f1[i]) / sc[i] / h0);
}
dt = Math.Max(d1, d2) <= 1e-15 ? Math.Max(1e-6, h0 * 1e-3) : Math.Pow(1e-2 / Math.Max(d1, d2), 1 / 5d);
if (dt > 100 * h0)
{
dt = 100 * h0;
}
}
else
{
dt = opts.InitialStep;
}
dt0 = dt;
// Output initial point
double tout = t0;
Vector xout = x0.Clone();
if (opts.OutputStep > 0) // Store previous solution point if OutputStep is specified (non-zero)
{
tout += opts.OutputStep;
}
yield return(new SolPoint(t0, x0.Clone()));
// Pre-allocate arrays
Vector[] k = new Vector[s];
Vector[] x2 = new Vector[s - 1];
for (int i = 0; i < s - 1; i++)
{
x2[i] = Vector.Zeros(n);
}
Vector prevX = Vector.Zeros(n);
double prevErr = 1.0d; // Previous error, used for step PI-filtering
double prevDt;
while (true) // Main loop - produces numerical solution
{
Vector.Copy(x1, prevX);
double e = 0.0d; // error factor, should be < 1
//int ii = 0;
do
{
prevDt = dt;
Vector.Copy(prevX, x1);
// Compute internal method variables.
k[0] = dt * f(t, x1);
for (int i = 1; i < s; i++)
{
Vector.Copy(x1, x2[i - 1]);
for (int j = 0; j < i; j++)
{
x2[i - 1].MulAdd(k[j], a[i - 1][j]);
}
k[i] = dt * f(t + dt * c[i], x2[i - 1]);
}
//Try to compute X in the next time point.
Vector.Copy(prevX, x);
for (int l = 0; l < s; l++)
{
x.MulAdd(k[l], b[l]);
x1.MulAdd(k[l], b1[l]);
}
// Compute error (see p. 168 of book indicated above)
// error compulation in L-infinity norm is commented
e = Math.Abs(x[0] - x1[0]) / Math.Max(opts.AbsoluteTolerance, opts.RelativeTolerance * Math.Max(Math.Abs(prevX[0]), Math.Abs(x1[0])));
for (int i = 1; i < n; i++)
{
e = Math.Max(e, Math.Abs(x[i] - x1[i]) / Math.Max(opts.AbsoluteTolerance, opts.RelativeTolerance * Math.Max(Math.Abs(prevX[i]), Math.Abs(x1[i]))));
}
// PI-filter. Beta = 0.08
dt = e == 0 ? dt : dt *Math.Min(MaxFactor, Math.Max(MinFactor, SafetyFactor *Math.Pow(1.0d / e, 1.0d / 5.0d) * Math.Pow(prevErr, 0.08d)));
if (opts.MaxStep < Double.MaxValue)
{
dt = Math.Min(dt, opts.MaxStep);
}
//if (opts.MinStep > 0) dt = Math.Max(dt, opts.MinStep);
if (Double.IsNaN(dt))
{
throw new ArgumentException("Derivatives function returned NaN");
}
if (dt < 1e-12)
{
throw new ArgumentException("Cannot generate numerical solution");
}
} while(e > 1.0d); // Repeat until solving vector euclidean norm doesn't satisfy break condition
prevErr = e;
// Output data
if (opts.OutputStep > 0) // Output points with specified step
{
while (t <= tout && tout <= t + prevDt)
{
yield return(new SolPoint(tout, Vector.Lerp(tout, t, xout, t + prevDt, x1)));
tout += opts.OutputStep;
}
}
else
if (Refine > 1) // Output interpolated points set by Refine property
{
Vector ts = Vector.Zeros(S.Length);
for (int i = 0; i < S.Length; i++)
{
ts[i] = t + prevDt * S[i];
}
var ys = RKinterp(S, xout, k);
for (int i = 0; i < S.Length; i++)
{
yield return(new SolPoint(ts[i], ys[i]));
}
}
yield return(new SolPoint(t + prevDt, x1.Clone()));
// Update current time and state
t = t + prevDt;
Vector.Copy(x1, xout);
}
}
/// <summary>
/// Implementation of Gear's BDF method with dynamically changed step size and order. Order changes between 1 and 3.
/// </summary>
/// <param name="t0">Initial time point</param>
/// <param name="x0">Initial phase vector</param>
/// <param name="f">Right parts of the system</param>
/// <param name="opts">Options for accuracy control and initial step size</param>
/// <returns>Sequence of infinite number of solution points.</returns>
public static IEnumerable <SolPoint> GearBDF(double t0, Vector x0, Func <double, Vector, Vector> f, Options opts)
{
double t = t0;
Vector x = x0.Clone();
int n = x0.Length;
double tout = t0;
Vector xout = new Vector();
if (opts.OutputStep > 0) // Store previous solution point if OutputStep is specified (non-zero)
{
xout = x0.Clone();
tout += opts.OutputStep;
}
// Firstly, return initial point
yield return(new SolPoint(t0, x0.Clone()));
//Initial step size.
Vector dx = f(t0, x0).Clone();
double dt;
if (opts.InitialStep != 0)
{
dt = opts.InitialStep;
}
else
{
var tol = opts.RelativeTolerance;
var ewt = Vector.Zeros(n);
var ywt = Vector.Zeros(n);
var sum = 0.0;
for (int i = 0; i < n; i++)
{
ewt[i] = opts.RelativeTolerance * Math.Abs(x[i]) + opts.AbsoluteTolerance;
ywt[i] = ewt[i] / tol;
sum = sum + (double)dx[i] * dx[i] / (ywt[i] * ywt[i]);
}
dt = Math.Sqrt(tol / ((double)1.0d / (ywt[0] * ywt[0]) + sum / n));
}
dt = Math.Min(dt, opts.MaxStep);
var resdt = dt;
int qmax = 5;
int qcurr = 2;
//Compute Nordstieck's history matrix at t=t0;
Matrix zn = new Matrix(n, qmax + 1);
for (int i = 0; i < n; i++)
{
zn[i, 0] = x[i];
zn[i, 1] = dt * dx[i];
for (int j = qcurr; j < qmax + 1; j++)
{
zn[i, j] = 0.0d;
}
}
var eold = Vector.Zeros(n);
NordsieckState currstate = new NordsieckState();
currstate.delta = 0.0d;
currstate.Dq = 0.0d;
currstate.dt = dt;
currstate.en = eold;
currstate.tn = t;
currstate.xn = x0;
currstate.qn = qcurr;
currstate.qmax = qmax;
currstate.nsuccess = 0;
currstate.zn = zn;
currstate.rFactor = 1.0d;
bool isIterationFailed = false;
//Can produce any number of solution points
while (true)
{
// Reset fail flag
isIterationFailed = false;
// Predictor step
var z0 = currstate.zn.Clone();
currstate.zn = NordsieckState.ZNew(currstate.zn);
currstate.en = Vector.Zeros(n);
currstate.xn = currstate.zn.CloneColumn(0);
// Corrector step
currstate = PredictorCorrectorScheme(currstate, ref isIterationFailed, f, opts);
if (isIterationFailed) // If iterations are not finished - bad convergence
{
currstate.zn = z0;
currstate.nsuccess = 0;
currstate.ChangeStep();
}
else // Iterations finished
{
var r = Math.Min(1.1d, Math.Max(0.2d, currstate.rFactor));
if (currstate.delta >= 1.0d)
{
if (opts.MaxStep < Double.MaxValue)
{
r = Math.Min(r, opts.MaxStep / currstate.dt);
}
if (opts.MinStep > 0)
{
r = Math.Max(r, opts.MinStep / currstate.dt);
}
r = Math.Min(r, opts.MaxScale);
r = Math.Max(r, opts.MinScale);
currstate.dt = currstate.dt * r; // Decrease step
currstate.zn = NordsieckState.Rescale(currstate.zn, r);
}
else
{
// Output data
if (opts.OutputStep > 0) // Output points with specified step
{
while (currstate.tn <= tout && tout <= currstate.tn + currstate.dt)
{
yield return(new SolPoint(tout, Vector.Lerp(tout, currstate.tn,
xout, currstate.tn + currstate.dt, currstate.xn)));
tout += opts.OutputStep;
}
Vector.Copy(currstate.xn, xout);
}
else // Output each point
{
yield return(new SolPoint(currstate.tn + currstate.dt, currstate.xn));
}
currstate.tn = currstate.tn + currstate.dt;
if (opts.MaxStep < Double.MaxValue)
{
r = Math.Min(r, opts.MaxStep / currstate.dt);
}
if (opts.MinStep > 0)
{
r = Math.Max(r, opts.MinStep / currstate.dt);
}
r = Math.Min(r, opts.MaxScale);
r = Math.Max(r, opts.MinScale);
currstate.dt = currstate.dt * r;
currstate.zn = NordsieckState.Rescale(currstate.zn, r);
}
}
}
}