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);
}
/// <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);
// 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:
var solver = new CompositeSolver(SolverSetup <double> .LoadFromAssembly(typeof(CompositeSolver).Assembly));
// 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 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();
}