public Vector computeExactSolution(Vector nodes, double time)
{
Vector exactSolution = new Vector(nodes.Length);
for (int i = 0; i < nodes.Length; i++)
exactSolution[i] = ExactSolution(nodes[i], time);
return exactSolution;
}
public static double evaluateLagrangePolynome(double x, Vector nodes, int j)
{
int idx;
//Gibt die Position zurück, wenn x einer Stützerstelle entspricht.
if ((idx = nodes.ContainsValue(x)) != -1)
{
if (idx == j)
return 1.0;
else
return 0.0;
}
Vector barycentricWeights = computeBarycentricWeights(nodes);
double tempProd = 1.0;
for(int i = 0; i < nodes.Length; i++)
{
if (i != j)
tempProd *= (x - nodes[i]);
}
tempProd *= barycentricWeights[j];
return tempProd;
}
public Vector computeSolutionVectorWithMultipleSteps(Vector initial, OrdinaryDifferentialEquation ode, double startTime, double endTime, double step)
{
double tempTime = startTime;
int N = Convert.ToInt32((endTime - startTime) / step) + 1;
Vector tempSolution = initial;
Vector solution = new Vector(N);
solution[0] = initial[0];
Vector temp = new Vector(1);
for (int i = 1; i < N; i++)
{
temp[0] = solution[i - 1];
solution[i] = computeSolutionForNextStep(temp, ode, tempTime, tempTime + step)[0];
tempTime += step;
}
if (!GeneralHelper.isXAlmostEqualToY(startTime + (N-1) * step, endTime))
{
tempTime = startTime + N * step;
tempSolution = computeSolutionForNextStep(tempSolution, ode, tempTime, endTime);
}
return solution;
}
private static double computeIntegralSummation(Func<double, double> myFunction, Vector nodes, Vector weights)
{
double evaluation = 0.0;
for (int i = 0; i <nodes.Length; i++)
{
evaluation += myFunction(nodes[i]) * weights[i];
}
return evaluation;
}
/// <summary>
/// Creates a diagonal matrix using values from input vector for diagonal entries.
/// </summary>
/// <param name="_diag">Vector, which should be mapped as a diagonal matrix.</param>
public Matrix(Vector _diag)
{
noRows = _diag.Length;
noColumns = noRows;
values = new double[noRows, noColumns];
for (int i = 0; i < noRows; i++)
values[i, i] = _diag[i];
}
public static String ConvertToMatLabPlotString(Vector nodes, Vector evaluation)
{
StringBuilder sb = new StringBuilder();
sb.Append(ConvertVectorToMatLabVector(nodes, "X")).AppendLine();
sb.Append(ConvertVectorToMatLabVector(evaluation, "Y")).AppendLine();
sb.Append("figure").AppendLine();
sb.Append("plot(X,Y)");
return sb.ToString();
}
private Matrix computeSBPMatrix(Vector nodes, Vector weights)
{
Matrix massMatrix = IntegrationToolbox.generateMassMatrix(weights);
Matrix diffMatrix = InterpolationToolbox.computeLagrangePolynomeDerivativeMatrix(nodes);
Matrix left = massMatrix * diffMatrix;
Matrix B = left + !left;
return B;
}
public Vector ComputeConstant()
{
Vector constant = new Vector((polynomOrder + 1) * elements.Length);
for(int i = 0; i < elements.Length; i++)
{
constant.InjectMatrixAtPosition(elements[i].GetConstantSolution(Ground), i * (polynomOrder+1), 0);
}
return constant;
}
public Vector evaluateOrdinaryDifferentialEquation(Vector startSolution, double time)
{
Vector evaluatedODE = new Vector(myOrdinaryFunctionList.Count);
for(int i = 0; i < myOrdinaryFunctionList.Count; i++)
{
evaluatedODE[i] = myOrdinaryFunctionList.ElementAt(i)(startSolution[i], time);
}
return evaluatedODE;
}
public static String ConvertToMatLabPlotStringWithAxisLabelAndTitle(Vector nodes, Vector evaluation, String xAxisLabel, String yAxisName, String title)
{
StringBuilder sb = new StringBuilder(ConvertToMatLabPlotString(nodes, evaluation));
sb.AppendLine();
sb.Append("title('").Append(title).Append("');").AppendLine();
sb.Append("xlabel('").Append(xAxisLabel).Append("');").AppendLine();
sb.Append("ylabel('").Append(yAxisName).Append("');").AppendLine();
return sb.ToString();
}
/// <summary>
/// Aquidistante Stützstellen berechnen
/// </summary>
/// <param name="leftBorder">Linker Rand</param>
/// <param name="rightBorder">Rechter Rand</param>
/// <param name="N">Zerlegungsparameter</param>
/// <returns>Vektor mit aquidistanten Stützstellen</returns>
private Vector computeAquiDistantNodes(double leftBorder, double rightBorder, int N)
{
Vector aquiDistantNodes = new Vector(N + 1);
for(int i = 0; i <= N; i++)
{
aquiDistantNodes[i] = leftBorder + ((rightBorder - leftBorder) / (double)N) * (double)i;
}
return aquiDistantNodes;
}
/// <summary>
/// Berechnet die Gauß Lobatto Stützstellen und deren zugehörige Gewichte.
/// </summary>
/// <param name="N">Grad N des Legendre Stützpolynoms</param>
/// <param name="nodes">Ausgabe der GL-Stützstellen</param>
/// <param name="weights">Ausgabe der GL-Gewichte</param>
public static void computeGaussLobattoNodesAndWeights(int N, out Vector nodes, out Vector weights)
{
if( N == 0)
{
throw new Exception("Gauß Lobatto Nodes and Weights not defined for N = 0.");
}
Vector gaussNodes = new Vector(N + 1);
Vector gaussWeights = new Vector(N + 1);
if (N == 1)
{
gaussNodes[0] = -1.0;
gaussWeights[0] = 1.0;
gaussNodes[1] = 1.0;
gaussWeights[1] = 1.0;
}
else
{
int upperBound = (N + 1) / 2 - 1;
double q, qDerivative, evaPolynomial, delta;
gaussNodes[0] = -1;
gaussWeights[0] = 2.0 / (N * (N + 1));
gaussNodes[N] = 1;
gaussWeights[N] = gaussWeights[0];
for(int j=1;j<= upperBound; j++)
{
gaussNodes[j] = -Math.Cos( ((j + 0.25) * Math.PI / (double)N) - ((3.0 / (8.0 * N * Math.PI)) * (1.0 / (j + 0.25))) );
for(int k = 0; k <= NEWTON_ITERATOR; k++)
{
evaluateQandL(N, gaussNodes[j], out q,out qDerivative,out evaPolynomial);
delta = -q / qDerivative;
gaussNodes[j] += delta;
if (Math.Abs(delta) <= 2 * GeneralHelper.EPSILON * Math.Abs(gaussNodes[j]))
k = NEWTON_ITERATOR + 1;
}
evaluateQandL(N, gaussNodes[j], out q, out qDerivative, out evaPolynomial);
gaussNodes[N - j] = -gaussNodes[j];
gaussWeights[j] = 2.0 / (N * (N + 1) * evaPolynomial * evaPolynomial);
gaussWeights[N - j] = gaussWeights[j];
}
if(N%2 == 0)
{
evaluateQandL(N, 0.0, out q, out qDerivative, out evaPolynomial);
gaussNodes[N / 2] = 0.0;
gaussWeights[N / 2] = 2.0 / (N * (N + 1) * evaPolynomial * evaPolynomial);
}
}
nodes = gaussNodes;
weights = gaussWeights;
}
private Vector GetZValues(int x, int y)
{
Vector zValues = new Vector(DimZ);
for(int i = 0; i < DimZ; i++)
{
zValues[i] = matrices[i][x, y];
}
return zValues;
}
private static Vector computeSpaceDiscretationVector(double leftSpaceBoundary, double rightSpaceBoundary, int N)
{
Vector spaceDiscretization = new Vector(N - 1);
double spaceStep = (Math.Abs(leftSpaceBoundary) + Math.Abs(leftSpaceBoundary)) / (double)N;
for (int i = 0; i < N-1; i++)
{
double spaceLocation = leftSpaceBoundary + (i + 1) * spaceStep;
spaceDiscretization[i] = spaceLocation;
}
return spaceDiscretization;
}
/// <summary>
/// Berechnet die Visualisierungsmatrix
/// </summary>
/// <param name="lagrangeEvaluation">Auswertung der Funktion an den erzeugenden Stützstellen</param>
/// <param name="lagrangeNodes">Lagrange Polynome erzeugende Stützstellen </param>
/// <param name="nodes">Auswertungsstellen der Interpolation</param>
/// <returns></returns>
private Matrix computeVisualizationMatrix(Vector lagrangeEvaluation, Vector lagrangeNodes, Vector nodes)
{
Matrix visualMatrix = new Matrix(nodes.Length, lagrangeNodes.Length);
for(int i = 0; i < nodes.Length; i++)
{
for(int k = 0; k < lagrangeNodes.Length; k++)
{
visualMatrix[i, k] = InterpolationToolbox.evaluateLagrangePolynome(nodes[i], lagrangeNodes, k);
}
}
return visualMatrix;
}
private static Vector evaluateExactSolution(double time, double leftSpaceBoundary, double rightSpaceBoundary, int N)
{
Vector evaluation = new Vector(N - 1);
double spaceStep = (Math.Abs(leftSpaceBoundary) + Math.Abs(leftSpaceBoundary)) / (double)N;
for(int i = 0; i< N-1; i++)
{
double spaceLocation = leftSpaceBoundary + (i + 1) * spaceStep;
evaluation[i] = exactSolution(time, spaceLocation);
}
return evaluation;
}
private static String ConvertVectorToMatLabVector(Vector array, String arrayName)
{
StringBuilder sb = new StringBuilder();
sb.Append(arrayName).Append(" = ");
sb.Append("[");
for(int i = 0; i < array.Length - 1; i++)
{
sb.Append(array[i] + ";");
}
sb.Append(array[array.Length-1] + "];");
return sb.ToString().Replace(",",".") ;
}
public Vector computeSolutionForNextStep(Vector initial,OrdinaryDifferentialEquation ode, double startTime, double endTime)
{
Vector tempSolution = initial;
Vector evaluatedODE = ode.evaluateOrdinaryDifferentialEquation(tempSolution, startTime);
double deltaTime = endTime - startTime;
for(int i = 0; i < 5; i++)
{
double nextTimeStep = startTime + B[i] * deltaTime;
evaluatedODE = (Vector) (A[i] * evaluatedODE) + ode.evaluateOrdinaryDifferentialEquation(tempSolution, nextTimeStep);
tempSolution = tempSolution + (Vector)(C[i] * deltaTime * evaluatedODE);
}
return tempSolution;
}
/// <summary>
/// Solves the equation Ax = b by using Gauß Elimination Algorithm for a square matrix A
/// </summary>
/// <param name="A">Matrix A</param>
/// <param name="b"></param>
/// <returns></returns>
public static Vector Solve(Matrix A, Vector b)
{
int N = A.NoRows;
if (A.NoRows != A.NoColumns)
throw new SolverException("Matrix A is not square.");
//Transform to lower triangular matrix
for (int i = 0; i < N; i++)
{
int pivotIdx = A.GetPivotRow(i, i);
if (A[pivotIdx,i] == 0.00)
throw new SolverException("Matrix A is not singular");
A.SwapRow(i, pivotIdx);
b.SwapRow(i, pivotIdx);
for (int row = i + 1; row < N; row++)
{
for (int col = i + 1; col < N; col++)
{
A[row, col] = A[row, col] - A[i, col] * ((A[row, i] / A[i, i]));
}
//Update vecctor b
b[row, 0] = b[row, 0] - b[i, 0] * ((A[row, i] / A[i, i]));
//Set lower Triangular Matrix with zeros
A[row, i] = 0.0;
}
}
//Transform lower triangular matrix to diagnol matrix
for (int i = N-1; i >= 0; i--)
{
//Normalize
b[i,0] = b[i,0]/A[i,i];
A[i, i] = 1.0;
for (int row = i - 1; row >= 0; row--)
{
b[row, 0] = b[row, 0] - b[i, 0] * (A[row, i]);
A[row, i] = 0.0;
}
}
return b;
}
public static Vector computeBarycentricWeights(Vector nodes)
{
double tempProd = 0.0;
int N = nodes.Length;
Vector baryWeights = new Vector(N);
for(int i = 0; i < N; i++)
{
tempProd = 1.0;
for(int j = 0; j < N; j++)
{
if(i != j)
tempProd *= (nodes[i] - nodes[j]);
}
baryWeights[i] = 1.0/tempProd;
}
return baryWeights;
}
public double computeSolution(double endTime, double timeStep)
{
double recentTime = 0.0;
Vector[] tempTimeDerivaties = new Vector[elements.Length];
while (recentTime < endTime)
{
if (recentTime + timeStep > endTime)
timeStep = endTime - recentTime;
//Vektoren zurüksetzen
for (int i = 0; i < elements.Length; i++)
tempTimeDerivaties[i] = new Vector(polynomOrder+1);
//Runga Kutta
for (int k = 0; k < 5; k++)
{
//Runga Kutta für jedes Element, damit Zeitsynchron berechnet wird
for (int i = 0; i < elements.Length; i++)
{
Vector solution = elements[i].GetSolution();
double nextTimeStep = recentTime + B[k] * timeStep;
tempTimeDerivaties[i] = (Vector)(A[k] * tempTimeDerivaties[i]) + elements[i].EvaluateTimeDerivative(nextTimeStep);
solution = solution + (Vector)(C[k] * timeStep * tempTimeDerivaties[i]);
elements[i].UpdateSolution(solution);
}
//Rand updaten
for (int i = 0; i < elements.Length; i++)
{
elements[i].UpdateBorderValues();
}
}
recentTime += timeStep;
}
//for (int k = 0; k < elements.Length; k++)
//{
// for (int i = 0; i < elements[k].GetSolution().Length; i++)
// Console.WriteLine(elements[k].GetSolution()[i]);
//}
return computeError(endTime);
}
public void evaluate()
{
//Anfangsbedingung
Vector initial = new Vector(1);
initial[0] = -1.0;
List<Func<double, double, double>> mySystem = new List<Func<double, double, double>>();
mySystem.Add(derivativeFunction);
OrdinaryDifferentialEquation testEquation = new OrdinaryDifferentialEquation(mySystem);
IODESolver odeSolver = new RungeKuttaSolver();
double[] errorList = new double[10];
for (int i = 1; i <= 10; i++)
{
double timeStep = Math.Pow(2.0, -i);
Vector res = odeSolver.computeSolutionVectorWithMultipleSteps(initial, testEquation, 1.0, 2.0, timeStep);
//Vector exact = null; //hier die exacte auswertung
//double error = (res - exact) * (res - exact);
}
}
public static Matrix computeLagrangePolynomeDerivativeMatrix(Vector nodes)
{
Matrix D = new Matrix(nodes.Length, nodes.Length);
Vector baryWeights = computeBarycentricWeights(nodes);
for (int i = 0; i < nodes.Length; i++)
{
D[i, i] = 0.0;
for(int j = 0; j < nodes.Length; j++)
{
if(i != j)
{
D[i, j] = baryWeights[j] / baryWeights[i] * (1.0/(nodes[i] - nodes[j])) ;
D[i, i] -= D[i, j];
}
}
}
return D;
}
public Vector computeSolutionWithMultipleSteps(Vector initial, OrdinaryDifferentialEquation ode, double startTime, double endTime, double step)
{
double tempTime = startTime;
int N = Convert.ToInt32((endTime - startTime)/ step);
Vector tempSolution = initial;
for(int i = 1; i <= N; i++)
{
tempSolution = computeSolutionForNextStep(tempSolution, ode, tempTime, tempTime + step);
tempTime += step;
}
if (!GeneralHelper.isXAlmostEqualToY(startTime + N * step, endTime))
{
tempTime = startTime + N * step;
tempSolution = computeSolutionForNextStep(tempSolution, ode, tempTime, endTime);
}
return tempSolution;
}
public static double evaluateLagrangeRepresentation(double x, Vector nodes, Vector functionValues)
{
int idx;
//Gibt die Position zurück, wenn x einer Stützerstelle entspricht.
if( (idx = nodes.ContainsValue(x)) != -1)
return functionValues[idx];
Vector barycentricWeights = computeBarycentricWeights(nodes);
double tempQuot = 0.0;
double denominator = 0.0;
double numerator = 0.0;
for(int i = 0; i < nodes.Length; i++)
{
tempQuot = barycentricWeights[i] / (x - nodes[i]);
numerator += functionValues[i] * tempQuot;
denominator += tempQuot;
}
return numerator / denominator;
}
public void TaskOneEnergy()
{
DGSystemController controller = new DGSystemController();
controller.createDGElements(101, 3, 0.0, 20.0, 2);
double test = controller.ComputeEnergy();
Console.WriteLine(test);
Dictionary<double, double> Energy = controller.ComputeSolution(1.0);
Vector time = new Vector(Energy.Count);
Vector energyVector = new Vector(Energy.Count);
int counter = 0;
foreach (KeyValuePair<double, double> entry in Energy)
{
time[counter] = entry.Key;
energyVector[counter] = entry.Value;
counter++;
Console.WriteLine("Time:" + entry.Key + " Energy:" + entry.Value);
}
string matlabString = MatLabConverter.ConvertToMatLabPlotStringWithAxisLabelAndTitle(time, energyVector, "Zeit t", "Energie e", "Energieverlauf");
}
private Vector visualizeFunctionDerivative(Matrix visualizationMatrix, Vector evaluation, Vector nodes)
{
Matrix diffMatrix = InterpolationToolbox.computeLagrangePolynomeDerivativeMatrix(nodes);
Vector differentialFunctionEvaluation = diffMatrix * evaluation;
Vector visualizedDerivativeEvaluation = visualizationMatrix * differentialFunctionEvaluation;
return visualizedDerivativeEvaluation;
}
private Vector visualizeFunction(Matrix visualizationMatrix, Vector evaluation)
{
Vector visualizedEvaluation = visualizationMatrix * evaluation;
return visualizedEvaluation;
}
/// <summary>
/// Evaluiert eine Funktion an gegebenen Stützstellen.
/// </summary>
/// <param name="function">Delegate an die Funktion</param>
/// <param name="nodes">Die Stützstellen</param>
/// <returns>Funktionsauswertung an den Stützstellen</returns>
private Vector evaluateFunctionAtNodes(Func<double, double> function, Vector nodes)
{
Vector evaluation = new Vector(nodes.Length);
for (int i = 0; i < nodes.Length; i++)
evaluation[i] = function(nodes[i]);
return evaluation;
}
private Vector ComputeV(Matrix Solution)
{
Vector v = new Vector(Solution.NoRows);
Vector h = Solution.GetColumn(0);
Vector hv = Solution.GetColumn(1);
for(int i = 0; i < v.Length; i++){
v[i] = hv[i] / h[i];
}
return v;
}