public void Create()
{
var criterion = new ResidualStopCriterion<double>(1e-8, 50);
Assert.AreEqual(1e-8, criterion.Maximum, "Incorrect maximum");
Assert.AreEqual(50, criterion.MinimumIterationsBelowMaximum, "Incorrect iteration count");
}
public void DetermineStatusWithNonMatchingResidualVectorThrowsArgumentException()
{
var criterion = new ResidualStopCriterion <float>(1e-6f, 50);
Assert.That(() => criterion.DetermineStatus(
1,
Vector <float> .Build.Dense(3, 4),
Vector <float> .Build.Dense(3, 4),
Vector <float> .Build.Dense(4, 4)), Throws.ArgumentException);
}
public void DetermineStatusWithIllegalIterationNumberThrowsArgumentOutOfRangeException()
{
var criterion = new ResidualStopCriterion <float>(1e-6f, 50);
Assert.That(() => criterion.DetermineStatus(
-1,
Vector <float> .Build.Dense(3, 4),
Vector <float> .Build.Dense(3, 5),
Vector <float> .Build.Dense(3, 6)), Throws.TypeOf <ArgumentOutOfRangeException>());
}
public void DetermineStatusWithIllegalIterationNumberThrowsArgumentOutOfRangeException()
{
var criterion = new ResidualStopCriterion<double>(1e-8, 50);
Assert.That(() => criterion.DetermineStatus(
-1,
Vector<double>.Build.Dense(3, 4),
Vector<double>.Build.Dense(3, 5),
Vector<double>.Build.Dense(3, 6)), Throws.TypeOf<ArgumentOutOfRangeException>());
}
private Complex[] CalculateSystemOfLinearEquations(Complex[,] matrixSystem, Complex[] matrixRightSide)
{
var matrixA = DenseMatrix.OfArray(matrixSystem);
var vectorB = new DenseVector(matrixRightSide);
var iterationCountStopCriterion = new IterationCountStopCriterion <Complex>(1000);
var residualStopCriterion = new ResidualStopCriterion <Complex>(1e-10);
var monitor = new Iterator <Complex>(iterationCountStopCriterion, residualStopCriterion);
var solver = new BiCgStab();
return(matrixA.SolveIterative(vectorB, solver, monitor).ToArray());
}
public void DetermineStatusWithSourceNaN()
{
var criterion = new ResidualStopCriterion <float>(1e-3f, 10);
var solution = new DenseVector(new[] { 1.0f, 1.0f, 2.0f });
var source = new DenseVector(new[] { 1.0f, 1.0f, float.NaN });
var residual = new DenseVector(new[] { 1000.0f, 1000.0f, 2001.0f });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Diverged, status, "Should be diverged");
}
public void DetermineStatusWithConvergenceAtFirstIteration()
{
var criterion = new ResidualStopCriterion <float>(1e-6);
var solution = new DenseVector(new[] { 1.0f, 1.0f, 1.0f });
var source = new DenseVector(new[] { 1.0f, 1.0f, 1.0f });
var residual = new DenseVector(new[] { 0.0f, 0.0f, 0.0f });
var status = criterion.DetermineStatus(0, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status, "Should be done");
}
public void DetermineStatusWithResidualNaN()
{
var criterion = new ResidualStopCriterion <Complex32>(1e-3f, 10);
var solution = new DenseVector(new[] { new Complex32(1.0f, 1), new Complex32(1.0f, 1), new Complex32(2.0f, 1) });
var source = new DenseVector(new[] { new Complex32(1.0f, 1), new Complex32(1.0f, 1), new Complex32(2.0f, 1) });
var residual = new DenseVector(new[] { new Complex32(1000.0f, 1), new Complex32(float.NaN, 1), new Complex32(2001.0f, 1) });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Diverged, status, "Should be diverged");
}
public void DetermineStatusWithConvergenceAtFirstIteration()
{
var criterion = new ResidualStopCriterion <Complex32>(1e-6);
var solution = new DenseVector(new[] { Complex32.One, Complex32.One, Complex32.One });
var source = new DenseVector(new[] { Complex32.One, Complex32.One, Complex32.One });
var residual = new DenseVector(new[] { Complex32.Zero, Complex32.Zero, Complex32.Zero });
var status = criterion.DetermineStatus(0, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status, "Should be done");
}
// Method to set up parameters for iterative solver
// Code for this method taken from the lecture slides (Lectures 5 and 6, solving 1D Heat Equation)
private void SetUpSolver()
{
// Stop after 1000 iterations
IterationCountStopCriterion <double> iterationCountStopCriterion
= new IterationCountStopCriterion <double>(1000);
// Stop after the error is less than 1e-8
ResidualStopCriterion <double> residualStopCriterion
= new ResidualStopCriterion <double>(1e-8);
monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
solver = new BiCgStab();
}
public void DetermineStatusWithConvergenceAtFirstIteration()
{
var criterion = new ResidualStopCriterion <double>(1e-12);
var solution = Vector <double> .Build.Dense(new[] { 1.0, 1.0, 1.0 });
var source = Vector <double> .Build.Dense(new[] { 1.0, 1.0, 1.0 });
var residual = Vector <double> .Build.Dense(new[] { 0.0, 0.0, 0.0 });
var status = criterion.DetermineStatus(0, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status, "Should be done");
}
public void DetermineStatusWithResidualNaN()
{
var criterion = new ResidualStopCriterion <double>(1e-3, 10);
var solution = Vector <double> .Build.Dense(new[] { 1.0, 1.0, 2.0 });
var source = Vector <double> .Build.Dense(new[] { 1.0, 1.0, 2.0 });
var residual = Vector <double> .Build.Dense(new[] { 1000.0, double.NaN, 2001.0 });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Diverged, status, "Should be diverged");
}
public void Clone()
{
var criterion = new ResidualStopCriterion<Complex32>(1e-3f, 10);
var clone = criterion.Clone();
Assert.IsInstanceOf(typeof(ResidualStopCriterion<Complex32>), clone, "Wrong criterion type");
var clonedCriterion = clone as ResidualStopCriterion<Complex32>;
Assert.IsNotNull(clonedCriterion);
Assert.AreEqual(criterion.Maximum, clonedCriterion.Maximum, "Clone failed");
Assert.AreEqual(criterion.MinimumIterationsBelowMaximum, clonedCriterion.MinimumIterationsBelowMaximum, "Clone failed");
}
//Method to set up parameters for iterative solver
private void SetUpSolver()
{
// Stop calculation if 1000 iterations reached during calculation
IterationCountStopCriterion <double> iterationCountStopCriterion
= new IterationCountStopCriterion <double>(1000);
// Stop calculation if residuals are below 1e-8
// i.e. the calculation is considered converged
ResidualStopCriterion <double> residualStopCriterion
= new ResidualStopCriterion <double>(1e-8);
// Create monitor with defined stop criteria
monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
solver = new BiCgStab();
}
public void ResetCalculationState()
{
var criterion = new ResidualStopCriterion <Complex32>(1e-3f, 10);
var solution = new DenseVector(new[] { new Complex32(0.001f, 1), new Complex32(0.001f, 1), new Complex32(0.002f, 1) });
var source = new DenseVector(new[] { new Complex32(0.001f, 1), new Complex32(0.001f, 1), new Complex32(0.002f, 1) });
var residual = new DenseVector(new[] { new Complex32(1.000f, 0), new Complex32(1.000f, 0), new Complex32(2.001f, 0) });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should be running");
criterion.Reset();
Assert.AreEqual(IterationStatus.Continue, criterion.Status, "Should not have started");
}
public void Clone()
{
var criterion = new ResidualStopCriterion <Complex32>(1e-3f, 10);
var clone = criterion.Clone();
Assert.IsInstanceOf(typeof(ResidualStopCriterion <Complex32>), clone, "Wrong criterion type");
var clonedCriterion = clone as ResidualStopCriterion <Complex32>;
Assert.IsNotNull(clonedCriterion);
Assert.AreEqual(criterion.Maximum, clonedCriterion.Maximum, "Clone failed");
Assert.AreEqual(criterion.MinimumIterationsBelowMaximum, clonedCriterion.MinimumIterationsBelowMaximum, "Clone failed");
}
private void SetUpSolver()
{
// Stop calculation if 1000 iterations reached during calculation
IterationCountStopCriterion <double> iterationCountStopCriterion = new IterationCountStopCriterion <double>(100);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
ResidualStopCriterion <double> residualStopCriterion = new ResidualStopCriterion <double>(1e-8);
// Create monitor with defined stop criteria
monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
// Load all suitable solvers from current assembly. Below in this example, there is user-defined solver
// "class UserBiCgStab : IIterativeSolverSetup<double>" which uses regular BiCgStab solver. But user may create any other solver
// and solver setup classes which implement IIterativeSolverSetup<T> and pass assembly to next function:
solver = new BiCgStab();
}
public void Clone()
{
var criterion = new ResidualStopCriterion<float>(1e-3f, 10);
var clone = criterion.Clone();
Assert.IsInstanceOf(typeof (ResidualStopCriterion<float>), clone, "Wrong criterion type");
var clonedCriterion = clone as ResidualStopCriterion<float>;
Assert.IsNotNull(clonedCriterion);
// ReSharper disable PossibleNullReferenceException
Assert.AreEqual(criterion.Maximum, clonedCriterion.Maximum, "Clone failed");
Assert.AreEqual(criterion.MinimumIterationsBelowMaximum, clonedCriterion.MinimumIterationsBelowMaximum, "Clone failed");
// ReSharper restore PossibleNullReferenceException
}
public void ResetCalculationState()
{
var criterion = new ResidualStopCriterion <double>(1e-3, 10);
var solution = Vector <double> .Build.Dense(new[] { 0.001, 0.001, 0.002 });
var source = Vector <double> .Build.Dense(new[] { 0.001, 0.001, 0.002 });
var residual = Vector <double> .Build.Dense(new[] { 1.000, 1.000, 2.001 });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should be running");
criterion.Reset();
Assert.AreEqual(IterationStatus.Continue, criterion.Status, "Should not have started");
}
public void Clone()
{
var criterion = new ResidualStopCriterion <float>(1e-3f, 10);
var clone = criterion.Clone();
Assert.IsInstanceOf(typeof(ResidualStopCriterion <float>), clone, "Wrong criterion type");
var clonedCriterion = clone as ResidualStopCriterion <float>;
Assert.IsNotNull(clonedCriterion);
// ReSharper disable PossibleNullReferenceException
Assert.AreEqual(criterion.Maximum, clonedCriterion.Maximum, "Clone failed");
Assert.AreEqual(criterion.MinimumIterationsBelowMaximum, clonedCriterion.MinimumIterationsBelowMaximum, "Clone failed");
// ReSharper restore PossibleNullReferenceException
}
public void solve()
{
Matrix <float> A = null;
Vector <float> b = null;
Vector <float> x = null;
IIterationStopCriterion <float> count_stop_criterion = new IterationCountStopCriterion <float>(5000);
IIterationStopCriterion <float> residual_stop_criterion = new ResidualStopCriterion <float>(1e-10f);
Iterator <float> iterator = new Iterator <float>(count_stop_criterion, residual_stop_criterion);
IPreconditioner <float> preconditioner = null;
BiCgStab solver = new BiCgStab();
solver.Solve(A, b, x, iterator, preconditioner);
}
static int Main(string[] args)
{
if (args.Length < 1 || args.Length > 2)
{
Console.WriteLine("Use: LinearEquationSolver.exe <matrix.csv> [<number-of-iterations>]");
return(2);
}
Table table = Table.Load(args[0], new ReadSettings(Delimiter.Comma, false, false, FSharpOption <int> .None, ReadSettings.Default.ColumnTypes));
if (table.Count <= 1)
{
throw new Exception("Expecting 2 or more columns");
}
var n = args.Length >= 2 ? int.Parse(args[1], CultureInfo.InvariantCulture) : 1;
var columns = table.Select(column => column.Rows.AsReal as IEnumerable <double>);
var A = Matrix <double> .Build.DenseOfColumns(columns.Where((_, i) => i < table.Count - 1));
var b = Vector <double> .Build.DenseOfEnumerable(columns.Last());
var iterationCountStopCriterion = new IterationCountStopCriterion <double>(1000);
var residualStopCriterion = new ResidualStopCriterion <double>(1e-10);
var monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
var solver = new CompositeSolver(SolverSetup <double> .LoadFromAssembly(System.Reflection.Assembly.GetExecutingAssembly()));
Vector <double> x = null;
for (var i = 0; i < n; i++)
{
//x = A.Solve(b);
x = A.SolveIterative(b, solver, monitor);
}
Table result = Table.OfColumns(new[] { Column.Create("x", x.ToArray(), FSharpOption <int> .None) });
Table.Save(result, "result.csv");
return(0);
}
public void DetermineStatus()
{
var criterion = new ResidualStopCriterion <Complex32>(1e-3f, 10);
// the solution vector isn't actually being used so ...
var solution = new DenseVector(new[] { new Complex32(float.NaN, float.NaN), new Complex32(float.NaN, float.NaN), new Complex32(float.NaN, float.NaN) });
// Set the source values
var source = new DenseVector(new[] { new Complex32(1.000f, 1), new Complex32(1.000f, 1), new Complex32(2.001f, 1) });
// Set the residual values
var residual = new DenseVector(new[] { new Complex32(0.001f, 0), new Complex32(0.001f, 0), new Complex32(0.002f, 0) });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should still be running");
var status2 = criterion.DetermineStatus(16, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status2, "Should be done");
}
public void DetermineStatus()
{
var criterion = new ResidualStopCriterion <double>(1e-3, 10);
// the solution vector isn't actually being used so ...
var solution = Vector <double> .Build.Dense(new[] { double.NaN, double.NaN, double.NaN });
// Set the source values
var source = Vector <double> .Build.Dense(new[] { 1.000, 1.000, 2.001 });
// Set the residual values
var residual = Vector <double> .Build.Dense(new[] { 0.001, 0.001, 0.002 });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should still be running");
var status2 = criterion.DetermineStatus(16, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status2, "Should be done");
}
public void DetermineStatusWithResidualNaN()
{
var criterion = new ResidualStopCriterion<double>(1e-3, 10);
var solution = Vector<double>.Build.Dense(new[] { 1.0, 1.0, 2.0 });
var source = Vector<double>.Build.Dense(new[] { 1.0, 1.0, 2.0 });
var residual = Vector<double>.Build.Dense(new[] { 1000.0, double.NaN, 2001.0 });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Diverged, status, "Should be diverged");
}
public void ResetCalculationState()
{
var criterion = new ResidualStopCriterion<float>(1e-3f, 10);
var solution = new DenseVector(new[] { 0.001f, 0.001f, 0.002f });
var source = new DenseVector(new[] { 0.001f, 0.001f, 0.002f });
var residual = new DenseVector(new[] { 1.000f, 1.000f, 2.001f });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should be running");
criterion.Reset();
Assert.AreEqual(IterationStatus.Continue, criterion.Status, "Should not have started");
}
public void DetermineStatusWithSourceNaN()
{
var criterion = new ResidualStopCriterion<float>(1e-3f, 10);
var solution = new DenseVector(new[] { 1.0f, 1.0f, 2.0f });
var source = new DenseVector(new[] { 1.0f, 1.0f, float.NaN });
var residual = new DenseVector(new[] { 1000.0f, 1000.0f, 2001.0f });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Diverged, status, "Should be diverged");
}
public void DetermineStatusWithConvergenceAtFirstIteration()
{
var criterion = new ResidualStopCriterion<float>(1e-6);
var solution = new DenseVector(new[] { 1.0f, 1.0f, 1.0f });
var source = new DenseVector(new[] { 1.0f, 1.0f, 1.0f });
var residual = new DenseVector(new[] { 0.0f, 0.0f, 0.0f });
var status = criterion.DetermineStatus(0, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status, "Should be done");
}
public void ResetCalculationState()
{
var criterion = new ResidualStopCriterion<double>(1e-3, 10);
var solution = Vector<double>.Build.Dense(new[] { 0.001, 0.001, 0.002 });
var source = Vector<double>.Build.Dense(new[] { 0.001, 0.001, 0.002 });
var residual = Vector<double>.Build.Dense(new[] { 1.000, 1.000, 2.001 });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should be running");
criterion.Reset();
Assert.AreEqual(IterationStatus.Continue, criterion.Status, "Should not have started");
}
public void ResetCalculationState()
{
var criterion = new ResidualStopCriterion<Complex>(1e-3, 10);
var solution = new DenseVector(new[] { new Complex(0.001, 1), new Complex(0.001, 1), new Complex(0.002, 1) });
var source = new DenseVector(new[] { new Complex(0.001, 1), new Complex(0.001, 1), new Complex(0.002, 1) });
var residual = new DenseVector(new[] { new Complex(1.000, 0), new Complex(1.000, 0), new Complex(2.001, 0) });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should be running");
criterion.Reset();
Assert.AreEqual(IterationStatus.Continue, criterion.Status, "Should not have started");
}
/// <summary>
/// Run example
/// </summary>
/// <seealso cref="http://en.wikipedia.org/wiki/Biconjugate_gradient_stabilized_method">Biconjugate gradient stabilized method</seealso>
public void Run()
{
// Format matrix output to console
var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
formatProvider.TextInfo.ListSeparator = " ";
// Solve next system of linear equations (Ax=b):
// 5*x + 2*y - 4*z = -7
// 3*x - 7*y + 6*z = 38
// 4*x + 1*y + 5*z = 43
// Create matrix "A" with coefficients
var matrixA = DenseMatrix.OfArray(new[, ] {
{ 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 }
});
Console.WriteLine(@"Matrix 'A' with coefficients");
Console.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
Console.WriteLine();
// Create vector "b" with the constant terms.
var vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
Console.WriteLine(@"Vector 'b' with the constant terms");
Console.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
Console.WriteLine();
// Create stop criteria to monitor an iterative calculation. There are next available stop criteria:
// - DivergenceStopCriterion: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterion: monitors residuals for NaN's;
// - IterationCountStopCriterion: monitors the numbers of iteration steps;
// - ResidualStopCriterion: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
var iterationCountStopCriterion = new IterationCountStopCriterion <double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
var residualStopCriterion = new ResidualStopCriterion <double>(1e-10);
// Create monitor with defined stop criteria
var monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
// Create Bi-Conjugate Gradient Stabilized solver
var solver = new BiCgStab();
// 1. Solve the matrix equation
var resultX = matrixA.SolveIterative(vectorB, solver, monitor);
Console.WriteLine(@"1. Solve the matrix equation");
Console.WriteLine();
// 2. Check solver status of the iterations.
// Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
// Possible values are:
// - CalculationCancelled: calculation was cancelled by the user;
// - CalculationConverged: calculation has converged to the desired convergence levels;
// - CalculationDiverged: calculation diverged;
// - CalculationFailure: calculation has failed for some reason;
// - CalculationIndetermined: calculation is indetermined, not started or stopped;
// - CalculationRunning: calculation is running and no results are yet known;
// - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
Console.WriteLine(@"2. Solver status of the iterations");
Console.WriteLine(monitor.Status);
Console.WriteLine();
// 3. Solution result vector of the matrix equation
Console.WriteLine(@"3. Solution result vector of the matrix equation");
Console.WriteLine(resultX.ToString("#0.00\t", formatProvider));
Console.WriteLine();
// 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
var reconstructVecorB = matrixA * resultX;
Console.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
Console.WriteLine();
}
public void DetermineStatusWithConvergenceAtFirstIteration()
{
var criterion = new ResidualStopCriterion<Complex32>(1e-6);
var solution = new DenseVector(new[] {Complex32.One, Complex32.One, Complex32.One});
var source = new DenseVector(new[] {Complex32.One, Complex32.One, Complex32.One});
var residual = new DenseVector(new[] {Complex32.Zero, Complex32.Zero, Complex32.Zero});
var status = criterion.DetermineStatus(0, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status, "Should be done");
}
public void DetermineStatusWithNonMatchingSourceVectorThrowsArgumentException()
{
var criterion = new ResidualStopCriterion<Complex32>(1e-6f, 50);
Assert.That(() => criterion.DetermineStatus(
1,
Vector<Complex32>.Build.Dense(3, 4),
Vector<Complex32>.Build.Dense(4, 4),
Vector<Complex32>.Build.Dense(3, 4)), Throws.ArgumentException);
}
private void InitializeSolver()
{
result = Vector<double>.Build.Dense(n * m);
Control.LinearAlgebraProvider = new OpenBlasLinearAlgebraProvider();
//Control.UseNativeMKL();
//Control.LinearAlgebraProvider = new MklLinearAlgebraProvider();
//Control.UseManaged();
var iterationCountStopCriterion = new IterationCountStopCriterion<double>(1000);
var residualStopCriterion = new ResidualStopCriterion<double>(1e-7);
monitor = new Iterator<double>(iterationCountStopCriterion, residualStopCriterion);
solver = new BiCgStab();
preconditioner = new MILU0Preconditioner();
}
public void DetermineStatusWithConvergenceAtFirstIteration()
{
var criterion = new ResidualStopCriterion<double>(1e-12);
var solution = Vector<double>.Build.Dense(new[] { 1.0, 1.0, 1.0 });
var source = Vector<double>.Build.Dense(new[] { 1.0, 1.0, 1.0 });
var residual = Vector<double>.Build.Dense(new[] { 0.0, 0.0, 0.0 });
var status = criterion.DetermineStatus(0, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status, "Should be done");
}
public void DetermineStatus()
{
var criterion = new ResidualStopCriterion<Complex>(1e-3, 10);
// the solution vector isn't actually being used so ...
var solution = new DenseVector(new[] { new Complex(double.NaN, double.NaN), new Complex(double.NaN, double.NaN), new Complex(double.NaN, double.NaN) });
// Set the source values
var source = new DenseVector(new[] { new Complex(1.000, 1), new Complex(1.000, 1), new Complex(2.001, 1) });
// Set the residual values
var residual = new DenseVector(new[] { new Complex(0.001, 0), new Complex(0.001, 0), new Complex(0.002, 0) });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should still be running");
var status2 = criterion.DetermineStatus(16, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status2, "Should be done");
}
public void DetermineStatus()
{
var criterion = new ResidualStopCriterion<double>(1e-3, 10);
// the solution vector isn't actually being used so ...
var solution = Vector<double>.Build.Dense(new[] { double.NaN, double.NaN, double.NaN });
// Set the source values
var source = Vector<double>.Build.Dense(new[] { 1.000, 1.000, 2.001 });
// Set the residual values
var residual = Vector<double>.Build.Dense(new[] { 0.001, 0.001, 0.002 });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should still be running");
var status2 = criterion.DetermineStatus(16, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status2, "Should be done");
}
public void DetermineStatus()
{
var criterion = new ResidualStopCriterion<float>(1e-3f, 10);
// the solution vector isn't actually being used so ...
var solution = new DenseVector(new[] { float.NaN, float.NaN, float.NaN });
// Set the source values
var source = new DenseVector(new[] { 1.000f, 1.000f, 2.001f });
// Set the residual values
var residual = new DenseVector(new[] { 0.001f, 0.001f, 0.002f });
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Continue, status, "Should still be running");
var status2 = criterion.DetermineStatus(16, solution, source, residual);
Assert.AreEqual(IterationStatus.Converged, status2, "Should be done");
}
/// <summary>
/// Run example
/// </summary>
public void Run()
{
// Format matrix output to console
var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
formatProvider.TextInfo.ListSeparator = " ";
// Solve next system of linear equations (Ax=b):
// 5*x + 2*y - 4*z = -7
// 3*x - 7*y + 6*z = 38
// 4*x + 1*y + 5*z = 43
// Create matrix "A" with coefficients
var matrixA = DenseMatrix.OfArray(new[,] { { 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 } });
Console.WriteLine(@"Matrix 'A' with coefficients");
Console.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
Console.WriteLine();
// Create vector "b" with the constant terms.
var vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
Console.WriteLine(@"Vector 'b' with the constant terms");
Console.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
Console.WriteLine();
// Create stop criteria to monitor an iterative calculation. There are next available stop criteria:
// - DivergenceStopCriterion: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterion: monitors residuals for NaN's;
// - IterationCountStopCriterion: monitors the numbers of iteration steps;
// - ResidualStopCriterion: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
var iterationCountStopCriterion = new IterationCountStopCriterion<double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
var residualStopCriterion = new ResidualStopCriterion<double>(1e-10);
// Create monitor with defined stop criteria
var monitor = new Iterator<double>(iterationCountStopCriterion, residualStopCriterion);
// Create Multiple-Lanczos Bi-Conjugate Gradient Stabilized solver
var solver = new MlkBiCgStab();
// 1. Solve the matrix equation
var resultX = matrixA.SolveIterative(vectorB, solver, monitor);
Console.WriteLine(@"1. Solve the matrix equation");
Console.WriteLine();
// 2. Check solver status of the iterations.
// Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
// Possible values are:
// - CalculationCancelled: calculation was cancelled by the user;
// - CalculationConverged: calculation has converged to the desired convergence levels;
// - CalculationDiverged: calculation diverged;
// - CalculationFailure: calculation has failed for some reason;
// - CalculationIndetermined: calculation is indetermined, not started or stopped;
// - CalculationRunning: calculation is running and no results are yet known;
// - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
Console.WriteLine(@"2. Solver status of the iterations");
Console.WriteLine(monitor.Status);
Console.WriteLine();
// 3. Solution result vector of the matrix equation
Console.WriteLine(@"3. Solution result vector of the matrix equation");
Console.WriteLine(resultX.ToString("#0.00\t", formatProvider));
Console.WriteLine();
// 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
var reconstructVecorB = matrixA*resultX;
Console.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
Console.WriteLine();
}
public void DetermineStatusWithResidualNaN()
{
var criterion = new ResidualStopCriterion<Complex32>(1e-3f, 10);
var solution = new DenseVector(new[] {new Complex32(1.0f, 1), new Complex32(1.0f, 1), new Complex32(2.0f, 1)});
var source = new DenseVector(new[] {new Complex32(1.0f, 1), new Complex32(1.0f, 1), new Complex32(2.0f, 1)});
var residual = new DenseVector(new[] {new Complex32(1000.0f, 1), new Complex32(float.NaN, 1), new Complex32(2001.0f, 1)});
var status = criterion.DetermineStatus(5, solution, source, residual);
Assert.AreEqual(IterationStatus.Diverged, status, "Should be diverged");
}
public void DetermineStatusWithNonMatchingSolutionVectorThrowsArgumentException()
{
var criterion = new ResidualStopCriterion<double>(1e-8, 50);
Assert.That(() => criterion.DetermineStatus(
1,
Vector<double>.Build.Dense(4, 4),
Vector<double>.Build.Dense(3, 4),
Vector<double>.Build.Dense(3, 4)), Throws.ArgumentException);
}
public override void ExecuteExample()
{
// <seealso cref="http://en.wikipedia.org/wiki/Biconjugate_gradient_stabilized_method">Biconjugate gradient stabilized method</seealso>
MathDisplay.WriteLine("<b>Biconjugate gradient stabilised iterative solver</b>");
// Format matrix output to console
var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
formatProvider.TextInfo.ListSeparator = " ";
// Solve next system of linear equations (Ax=b):
// 5*x + 2*y - 4*z = -7
// 3*x - 7*y + 6*z = 38
// 4*x + 1*y + 5*z = 43
// Create matrix "A" with coefficients
var matrixA = DenseMatrix.OfArray(new[, ] {
{ 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 }
});
MathDisplay.WriteLine(@"Matrix 'A' with coefficients");
MathDisplay.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create vector "b" with the constant terms.
var vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
MathDisplay.WriteLine(@"Vector 'b' with the constant terms");
MathDisplay.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create stop criteria to monitor an iterative calculation. There are next available stop criteria:
// - DivergenceStopCriterion: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterion: monitors residuals for NaN's;
// - IterationCountStopCriterion: monitors the numbers of iteration steps;
// - ResidualStopCriterion: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
var iterationCountStopCriterion = new IterationCountStopCriterion <double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
var residualStopCriterion = new ResidualStopCriterion <double>(1e-10);
// Create monitor with defined stop criteria
var monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
// Create Bi-Conjugate Gradient Stabilized solver
var solverBiCgStab = new BiCgStab();
// 1. Solve the matrix equation
var resultX = matrixA.SolveIterative(vectorB, solverBiCgStab, monitor);
MathDisplay.WriteLine(@"1. Solve the matrix equation");
MathDisplay.WriteLine();
// 2. Check solver status of the iterations.
// Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
// Possible values are:
// - CalculationCancelled: calculation was cancelled by the user;
// - CalculationConverged: calculation has converged to the desired convergence levels;
// - CalculationDiverged: calculation diverged;
// - CalculationFailure: calculation has failed for some reason;
// - CalculationIndetermined: calculation is indetermined, not started or stopped;
// - CalculationRunning: calculation is running and no results are yet known;
// - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
MathDisplay.WriteLine(@"2. Solver status of the iterations");
MathDisplay.WriteLine(monitor.Status.ToString());
MathDisplay.WriteLine();
// 3. Solution result vector of the matrix equation
MathDisplay.WriteLine(@"3. Solution result vector of the matrix equation");
MathDisplay.WriteLine(resultX.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
var reconstructVecorB = matrixA * resultX;
MathDisplay.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
MathDisplay.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
MathDisplay.WriteLine("<b>Generalized product biconjugate gradient solver</b>");
// Solve next system of linear equations (Ax=b):
// 5*x + 2*y - 4*z = -7
// 3*x - 7*y + 6*z = 38
// 4*x + 1*y + 5*z = 43
// Create matrix "A" with coefficients
matrixA = DenseMatrix.OfArray(new[, ] {
{ 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 }
});
MathDisplay.WriteLine(@"Matrix 'A' with coefficients");
MathDisplay.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create vector "b" with the constant terms.
vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
MathDisplay.WriteLine(@"Vector 'b' with the constant terms");
MathDisplay.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create stop criteria to monitor an iterative calculation. There are next available stop criteria:
// - DivergenceStopCriterion: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterion: monitors residuals for NaN's;
// - IterationCountStopCriterion: monitors the numbers of iteration steps;
// - ResidualStopCriterion: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
iterationCountStopCriterion = new IterationCountStopCriterion <double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
residualStopCriterion = new ResidualStopCriterion <double>(1e-10);
// Create monitor with defined stop criteria
monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
// Create Generalized Product Bi-Conjugate Gradient solver
var solverGpBiCg = new GpBiCg();
// 1. Solve the matrix equation
resultX = matrixA.SolveIterative(vectorB, solverGpBiCg, monitor);
MathDisplay.WriteLine(@"1. Solve the matrix equation");
MathDisplay.WriteLine();
// 2. Check solver status of the iterations.
// Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
// Possible values are:
// - CalculationCancelled: calculation was cancelled by the user;
// - CalculationConverged: calculation has converged to the desired convergence levels;
// - CalculationDiverged: calculation diverged;
// - CalculationFailure: calculation has failed for some reason;
// - CalculationIndetermined: calculation is indetermined, not started or stopped;
// - CalculationRunning: calculation is running and no results are yet known;
// - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
MathDisplay.WriteLine(@"2. Solver status of the iterations");
MathDisplay.WriteLine(monitor.Status.ToString());
MathDisplay.WriteLine();
// 3. Solution result vector of the matrix equation
MathDisplay.WriteLine(@"3. Solution result vector of the matrix equation");
MathDisplay.WriteLine(resultX.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
reconstructVecorB = matrixA * resultX;
MathDisplay.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
MathDisplay.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
MathDisplay.WriteLine("<b>Composite linear equation solver</b>");
// Solve next system of linear equations (Ax=b):
// 5*x + 2*y - 4*z = -7
// 3*x - 7*y + 6*z = 38
// 4*x + 1*y + 5*z = 43
// Create matrix "A" with coefficients
matrixA = DenseMatrix.OfArray(new[, ] {
{ 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 }
});
MathDisplay.WriteLine(@"Matrix 'A' with coefficients");
MathDisplay.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create vector "b" with the constant terms.
vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
MathDisplay.WriteLine(@"Vector 'b' with the constant terms");
MathDisplay.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create stop criteria to monitor an iterative calculation. There are next available stop criteria:
// - DivergenceStopCriterion: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterion: monitors residuals for NaN's;
// - IterationCountStopCriterion: monitors the numbers of iteration steps;
// - ResidualStopCriterion: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
iterationCountStopCriterion = new IterationCountStopCriterion <double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
residualStopCriterion = new ResidualStopCriterion <double>(1e-10);
// Create monitor with defined stop criteria
monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
// Load all suitable solvers from current assembly. Below (see UserBiCgStab) there is a custom user-defined solver
// "class UserBiCgStab : IIterativeSolverSetup<double>" which derives from the regular BiCgStab solver. However users can
// create any other solver and solver setup classes that implement IIterativeSolverSetup<T> and load the assembly that
// contains them using the following function:
var solverComp = new CompositeSolver(SolverSetup <double> .LoadFromAssembly(Assembly.GetExecutingAssembly()));
// 1. Solve the linear system
resultX = matrixA.SolveIterative(vectorB, solverComp, monitor);
MathDisplay.WriteLine(@"1. Solve the matrix equation");
MathDisplay.WriteLine();
// 2. Check solver status of the iterations.
// Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
// Possible values are:
// - CalculationCancelled: calculation was cancelled by the user;
// - CalculationConverged: calculation has converged to the desired convergence levels;
// - CalculationDiverged: calculation diverged;
// - CalculationFailure: calculation has failed for some reason;
// - CalculationIndetermined: calculation is indetermined, not started or stopped;
// - CalculationRunning: calculation is running and no results are yet known;
// - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
MathDisplay.WriteLine(@"2. Solver status of the iterations");
MathDisplay.WriteLine(monitor.Status.ToString());
MathDisplay.WriteLine();
// 3. Solution result vector of the matrix equation
MathDisplay.WriteLine(@"3. Solution result vector of the matrix equation");
MathDisplay.WriteLine(resultX.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
reconstructVecorB = matrixA * resultX;
MathDisplay.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
MathDisplay.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
MathDisplay.WriteLine("<b>Multiple-Lanczos biconjugate gradient stabilised iterative solver</b>");
// Solve next system of linear equations (Ax=b):
// 5*x + 2*y - 4*z = -7
// 3*x - 7*y + 6*z = 38
// 4*x + 1*y + 5*z = 43
// Create matrix "A" with coefficients
matrixA = DenseMatrix.OfArray(new[, ] {
{ 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 }
});
MathDisplay.WriteLine(@"Matrix 'A' with coefficients");
MathDisplay.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create vector "b" with the constant terms.
vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
MathDisplay.WriteLine(@"Vector 'b' with the constant terms");
MathDisplay.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create stop criteria to monitor an iterative calculation. There are next available stop criteria:
// - DivergenceStopCriterion: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterion: monitors residuals for NaN's;
// - IterationCountStopCriterion: monitors the numbers of iteration steps;
// - ResidualStopCriterion: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
iterationCountStopCriterion = new IterationCountStopCriterion <double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
residualStopCriterion = new ResidualStopCriterion <double>(1e-10);
// Create monitor with defined stop criteria
monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
// Create Multiple-Lanczos Bi-Conjugate Gradient Stabilized solver
var solverLanczos = new MlkBiCgStab();
// 1. Solve the matrix equation
resultX = matrixA.SolveIterative(vectorB, solverLanczos, monitor);
MathDisplay.WriteLine(@"1. Solve the matrix equation");
MathDisplay.WriteLine();
// 2. Check solver status of the iterations.
// Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
// Possible values are:
// - CalculationCancelled: calculation was cancelled by the user;
// - CalculationConverged: calculation has converged to the desired convergence levels;
// - CalculationDiverged: calculation diverged;
// - CalculationFailure: calculation has failed for some reason;
// - CalculationIndetermined: calculation is indetermined, not started or stopped;
// - CalculationRunning: calculation is running and no results are yet known;
// - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
MathDisplay.WriteLine(@"2. Solver status of the iterations");
MathDisplay.WriteLine(monitor.Status.ToString());
MathDisplay.WriteLine();
// 3. Solution result vector of the matrix equation
MathDisplay.WriteLine(@"3. Solution result vector of the matrix equation");
MathDisplay.WriteLine(resultX.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
reconstructVecorB = matrixA * resultX;
MathDisplay.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
MathDisplay.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
MathDisplay.WriteLine("<b>Transpose freee quasi-minimal residual iterative solver</b>");
// Solve next system of linear equations (Ax=b):
// 5*x + 2*y - 4*z = -7
// 3*x - 7*y + 6*z = 38
// 4*x + 1*y + 5*z = 43
// Create matrix "A" with coefficients
matrixA = DenseMatrix.OfArray(new[, ] {
{ 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 }
});
MathDisplay.WriteLine(@"Matrix 'A' with coefficients");
MathDisplay.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create vector "b" with the constant terms.
vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
MathDisplay.WriteLine(@"Vector 'b' with the constant terms");
MathDisplay.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// Create stop criteria to monitor an iterative calculation. There are next available stop criteria:
// - DivergenceStopCriterion: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterion: monitors residuals for NaN's;
// - IterationCountStopCriterion: monitors the numbers of iteration steps;
// - ResidualStopCriterion: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
iterationCountStopCriterion = new IterationCountStopCriterion <double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
residualStopCriterion = new ResidualStopCriterion <double>(1e-10);
// Create monitor with defined stop criteria
monitor = new Iterator <double>(iterationCountStopCriterion, residualStopCriterion);
// Create Transpose Free Quasi-Minimal Residual solver
var solverTFQMR = new TFQMR();
// 1. Solve the matrix equation
resultX = matrixA.SolveIterative(vectorB, solverTFQMR, monitor);
MathDisplay.WriteLine(@"1. Solve the matrix equation");
MathDisplay.WriteLine();
// 2. Check solver status of the iterations.
// Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
// Possible values are:
// - CalculationCancelled: calculation was cancelled by the user;
// - CalculationConverged: calculation has converged to the desired convergence levels;
// - CalculationDiverged: calculation diverged;
// - CalculationFailure: calculation has failed for some reason;
// - CalculationIndetermined: calculation is indetermined, not started or stopped;
// - CalculationRunning: calculation is running and no results are yet known;
// - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
MathDisplay.WriteLine(@"2. Solver status of the iterations");
MathDisplay.WriteLine(monitor.Status.ToString());
MathDisplay.WriteLine();
// 3. Solution result vector of the matrix equation
MathDisplay.WriteLine(@"3. Solution result vector of the matrix equation");
MathDisplay.WriteLine(resultX.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
// 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
reconstructVecorB = matrixA * resultX;
MathDisplay.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
MathDisplay.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
MathDisplay.WriteLine();
}
