/// <summary>
/// The main method that runs the TFQMR iterative solver.
/// </summary>
public void UseSolver()
{
// Create a sparse matrix. For now the size will be 10 x 10 elements
Matrix matrix = CreateMatrix(10);
// Create the right hand side vector. The size is the same as the matrix
// and all values will be 2.0.
Vector rightHandSideVector = new DenseVector(10, 2.0);
// Create a preconditioner. The possibilities are:
// 1) No preconditioner - Simply do not provide the solver with a preconditioner.
// 2) A simple diagonal preconditioner - Create an instance of the Diagonal class.
// 3) A ILU preconditioner - Create an instance of the IncompleteLu class.
// 4) A ILU preconditioner with pivoting and drop tolerances - Create an instance of the Ilutp class.
// Here we'll use the simple diagonal preconditioner.
// We need a link to the matrix so the pre-conditioner can do it's work.
IPreConditioner preconditioner = new Diagonal();
// Create a new iterator. This checks for convergence of the results of the
// iterative matrix solver.
// In this case we'll create the default iterator
IIterator iterator = Iterator.CreateDefault();
// Create the solver
TFQMR solver = new TFQMR(preconditioner, iterator);
// Now that all is set we can solve the matrix equation.
Vector solutionVector = solver.Solve(matrix, rightHandSideVector);
// Another way to get the values is by using the overloaded solve method
// In this case the solution vector needs to be of the correct size.
solver.Solve(matrix, rightHandSideVector, solutionVector);
// Finally you can check the reason the solver finished the iterative process
// by calling the SolutionStatus property on the iterator
ICalculationStatus status = iterator.Status;
if (status is CalculationCancelled)
Console.WriteLine("The user cancelled the calculation.");
if (status is CalculationIndetermined)
Console.WriteLine("Oh oh, something went wrong. The iterative process was never started.");
if (status is CalculationConverged)
Console.WriteLine("Yippee, the iterative process converged.");
if (status is CalculationDiverged)
Console.WriteLine("I'm sorry the iterative process diverged.");
if (status is CalculationFailure)
Console.WriteLine("Oh dear, the iterative process failed.");
if (status is CalculationStoppedWithoutConvergence)
Console.WriteLine("Oh dear, the iterative process did not converge.");
}
public void CanSolveForRandomMatrix(int order)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var monitor = new Iterator<double>(new IIterationStopCriterium<double>[]
{
new IterationCountStopCriterium<double>(1000),
new ResidualStopCriterium(1e-10)
});
var solver = new TFQMR(monitor);
var matrixX = solver.Solve(matrixA, matrixB);
// The solution X row dimension is equal to the column dimension of A
Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);
// The solution X has the same number of columns as B
Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);
var matrixBReconstruct = matrixA*matrixX;
// Check the reconstruction.
for (var i = 0; i < matrixB.RowCount; i++)
{
for (var j = 0; j < matrixB.ColumnCount; j++)
{
Assert.AreEqual(matrixB[i, j], matrixBReconstruct[i, j], 1.0e-7);
}
}
}
public void CanSolveForRandomVector(int order)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var vectorb = MatrixLoader.GenerateRandomDenseVector(order);
var monitor = new Iterator <Complex>(new IIterationStopCriterium <Complex>[]
{
new IterationCountStopCriterium <Complex>(1000),
new ResidualStopCriterium(1e-10),
});
var solver = new TFQMR(monitor);
var resultx = solver.Solve(matrixA, vectorb);
Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
var matrixBReconstruct = matrixA * resultx;
// Check the reconstruction.
for (var i = 0; i < order; i++)
{
Assert.AreEqual(vectorb[i].Real, matrixBReconstruct[i].Real, 1e-5);
Assert.AreEqual(vectorb[i].Imaginary, matrixBReconstruct[i].Imaginary, 1e-5);
}
}
public void CanSolveForRandomMatrix(int order)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var monitor = new Iterator(new IIterationStopCriterium <double>[]
{
new IterationCountStopCriterium <double>(1000),
new ResidualStopCriterium(1e-10)
});
var solver = new TFQMR(monitor);
var matrixX = solver.Solve(matrixA, matrixB);
// The solution X row dimension is equal to the column dimension of A
Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);
// The solution X has the same number of columns as B
Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);
var matrixBReconstruct = matrixA * matrixX;
// Check the reconstruction.
for (var i = 0; i < matrixB.RowCount; i++)
{
for (var j = 0; j < matrixB.ColumnCount; j++)
{
Assert.AreEqual(matrixB[i, j], matrixBReconstruct[i, j], 1.0e-7);
}
}
}
public void CanSolveForRandomVector(int order)
{
// Due to datatype "float" it can happen that solution will not converge for specific random matrix
// That's why we will do 3 tries and downgrade stop criterium each time
for (var iteration = 6; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var vectorb = MatrixLoader.GenerateRandomDenseVector(order);
var monitor = new Iterator(new IIterationStopCriterium[]
{
new IterationCountStopCriterium(MaximumIterations),
new ResidualStopCriterium((float)Math.Pow(1.0 / 10.0, iteration)),
});
var solver = new TFQMR(monitor);
var resultx = solver.Solve(matrixA, vectorb);
if (!(monitor.Status is CalculationConverged))
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
var matrixBReconstruct = matrixA * resultx;
// Check the reconstruction.
for (var i = 0; i < order; i++)
{
Assert.AreEqual(vectorb[i], matrixBReconstruct[i], (float)Math.Pow(1.0 / 10.0, iteration - 3));
}
return;
}
}
public void SolveLongMatrixThrowsArgumentException()
{
var matrix = new SparseMatrix(3, 2);
Vector input = new DenseVector(3);
var solver = new TFQMR();
Assert.Throws<ArgumentException>(() => solver.Solve(matrix, input));
}
public void SolveLongMatrix()
{
var matrix = new SparseMatrix(3, 2);
Vector <Complex32> input = new DenseVector(3);
var solver = new TFQMR();
solver.Solve(matrix, input);
}
public void SolveLongMatrixThrowsArgumentException()
{
var matrix = new SparseMatrix(3, 2);
var input = new DenseVector(3);
var solver = new TFQMR();
Assert.Throws <ArgumentException>(() => solver.Solve(matrix, input));
}
public void SolveWideMatrix()
{
var matrix = new SparseMatrix(2, 3);
Vector <float> input = new DenseVector(2);
var solver = new TFQMR();
solver.Solve(matrix, input);
}
public void CanSolveForRandomMatrix(int order)
{
for (var iteration = 5; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var monitor = new Iterator(new IIterationStopCriterium[]
{
new IterationCountStopCriterium(1000),
new ResidualStopCriterium((float)Math.Pow(1.0 / 10.0, iteration))
});
var solver = new TFQMR(monitor);
var matrixX = solver.Solve(matrixA, matrixB);
if (!(monitor.Status is CalculationConverged))
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
// The solution X row dimension is equal to the column dimension of A
Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);
// The solution X has the same number of columns as B
Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);
var matrixBReconstruct = matrixA * matrixX;
// Check the reconstruction.
for (var i = 0; i < matrixB.RowCount; i++)
{
for (var j = 0; j < matrixB.ColumnCount; j++)
{
Assert.AreEqual(matrixB[i, j].Real, matrixBReconstruct[i, j].Real, (float)Math.Pow(1.0 / 10.0, iteration - 3));
Assert.AreEqual(matrixB[i, j].Imaginary, matrixBReconstruct[i, j].Imaginary, (float)Math.Pow(1.0 / 10.0, iteration - 3));
}
}
return;
}
Assert.Fail("Solution was not found in 3 tries");
}
public void CanSolveForRandomMatrix(int order)
{
for (var iteration = 5; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var monitor = new Iterator <Complex32>(new IIterationStopCriterium <Complex32>[]
{
new IterationCountStopCriterium <Complex32>(1000),
new ResidualStopCriterium((float)Math.Pow(1.0 / 10.0, iteration))
});
var solver = new TFQMR(monitor);
var matrixX = solver.Solve(matrixA, matrixB);
if (!monitor.HasConverged)
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
// The solution X row dimension is equal to the column dimension of A
Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);
// The solution X has the same number of columns as B
Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);
var matrixBReconstruct = matrixA * matrixX;
// Check the reconstruction.
for (var i = 0; i < matrixB.RowCount; i++)
{
for (var j = 0; j < matrixB.ColumnCount; j++)
{
Assert.AreEqual(matrixB[i, j].Real, matrixBReconstruct[i, j].Real, (float)Math.Pow(1.0 / 10.0, iteration - 3));
Assert.AreEqual(matrixB[i, j].Imaginary, matrixBReconstruct[i, j].Imaginary, (float)Math.Pow(1.0 / 10.0, iteration - 3));
}
}
return;
}
Assert.Fail("Solution was not found in 3 tries");
}
public void CanSolveForRandomMatrix(int order)
{
// Due to datatype "float" it can happen that solution will not converge for specific random matrix
// That's why we will do 4 tries and downgrade stop criterium each time
for (var iteration = 6; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var monitor = new Iterator(new IIterationStopCriterium <float>[]
{
new IterationCountStopCriterium(MaximumIterations),
new ResidualStopCriterium((float)Math.Pow(1.0 / 10.0, iteration))
});
var solver = new TFQMR(monitor);
var matrixX = solver.Solve(matrixA, matrixB);
if (!(monitor.Status is CalculationConverged))
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
// The solution X row dimension is equal to the column dimension of A
Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);
// The solution X has the same number of columns as B
Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);
var matrixBReconstruct = matrixA * matrixX;
// Check the reconstruction.
for (var i = 0; i < matrixB.RowCount; i++)
{
for (var j = 0; j < matrixB.ColumnCount; j++)
{
Assert.AreApproximatelyEqual(matrixB[i, j], matrixBReconstruct[i, j], (float)Math.Pow(1.0 / 10.0, iteration - 3));
}
}
return;
}
Assert.Fail("Solution was not found in 3 tries");
}
public void CanSolveForRandomMatrix(int order)
{
// Due to datatype "float" it can happen that solution will not converge for specific random matrix
// That's why we will do 4 tries and downgrade stop criterium each time
for (var iteration = 6; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var monitor = new Iterator(new IIterationStopCriterium<float>[]
{
new IterationCountStopCriterium(MaximumIterations),
new ResidualStopCriterium((float)Math.Pow(1.0/10.0, iteration))
});
var solver = new TFQMR(monitor);
var matrixX = solver.Solve(matrixA, matrixB);
if (!(monitor.Status is CalculationConverged))
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
// The solution X row dimension is equal to the column dimension of A
Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);
// The solution X has the same number of columns as B
Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);
var matrixBReconstruct = matrixA * matrixX;
// Check the reconstruction.
for (var i = 0; i < matrixB.RowCount; i++)
{
for (var j = 0; j < matrixB.ColumnCount; j++)
{
Assert.AreApproximatelyEqual(matrixB[i, j], matrixBReconstruct[i, j], (float)Math.Pow(1.0 / 10.0, iteration - 3));
}
}
return;
}
Assert.Fail("Solution was not found in 3 tries");
}
public void SolveScaledUnitMatrixAndBackMultiply()
{
// Create the identity matrix
var matrix = SparseMatrix.Identity(100);
// Scale it with a funny number
matrix.Multiply(Math.PI, matrix);
// Create the y vector
var y = DenseVector.Create(matrix.RowCount, i => 1);
// Create an iteration monitor which will keep track of iterative convergence
var monitor = new Iterator(new IIterationStopCriterium <double>[]
{
new IterationCountStopCriterium <double>(MaximumIterations),
new ResidualStopCriterium(ConvergenceBoundary),
new DivergenceStopCriterium(),
new FailureStopCriterium()
});
var solver = new TFQMR(monitor);
// Solve equation Ax = y
var x = solver.Solve(matrix, y);
// Now compare the results
Assert.IsNotNull(x, "#02");
Assert.AreEqual(y.Count, x.Count, "#03");
// Back multiply the vector
var z = matrix.Multiply(x);
// Check that the solution converged
Assert.IsTrue(monitor.Status is CalculationConverged, "#04");
// Now compare the vectors
for (var i = 0; i < y.Count; i++)
{
Assert.IsTrue((y[i] - z[i]).IsSmaller(ConvergenceBoundary, 1), "#05-" + i);
}
}
public void CanSolveForRandomVector(int order)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var vectorb = MatrixLoader.GenerateRandomDenseVector(order);
var monitor = new Iterator(new IIterationStopCriterium<double>[]
{
new IterationCountStopCriterium(1000),
new ResidualStopCriterium(1e-10),
});
var solver = new TFQMR(monitor);
var resultx = solver.Solve(matrixA, vectorb);
Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
var bReconstruct = matrixA * resultx;
// Check the reconstruction.
for (var i = 0; i < order; i++)
{
Assert.AreApproximatelyEqual(vectorb[i], bReconstruct[i], 1e-7);
}
}
public void CanSolveForRandomVector(int order)
{
for (var iteration = 5; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var vectorb = MatrixLoader.GenerateRandomDenseVector(order);
var monitor = new Iterator <Complex32>(new IIterationStopCriterium <Complex32>[]
{
new IterationCountStopCriterium <Complex32>(1000),
new ResidualStopCriterium((float)Math.Pow(1.0 / 10.0, iteration))
});
var solver = new TFQMR(monitor);
var resultx = solver.Solve(matrixA, vectorb);
if (!monitor.HasConverged)
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
var matrixBReconstruct = matrixA * resultx;
// Check the reconstruction.
for (var i = 0; i < order; i++)
{
Assert.AreEqual(vectorb[i].Real, matrixBReconstruct[i].Real, (float)Math.Pow(1.0 / 10.0, iteration - 3));
Assert.AreEqual(vectorb[i].Imaginary, matrixBReconstruct[i].Imaginary, (float)Math.Pow(1.0 / 10.0, iteration - 3));
}
return;
}
Assert.Fail("Solution was not found in 3 tries");
}
/// <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 criteriums to monitor an iterative calculation. There are next available stop criteriums:
// - DivergenceStopCriterium: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterium: monitors residuals for NaN's;
// - IterationCountStopCriterium: monitors the numbers of iteration steps;
// - ResidualStopCriterium: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
var iterationCountStopCriterium = new IterationCountStopCriterium<double>(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
var residualStopCriterium = new ResidualStopCriterium(1e-10);
// Create monitor with defined stop criteriums
var monitor = new Iterator<double>(new IIterationStopCriterium<double>[] { iterationCountStopCriterium, residualStopCriterium });
// Create Transpose Free Quasi-Minimal Residual solver
var solver = new TFQMR(monitor);
// 1. Solve the matrix equation
var resultX = solver.Solve(matrixA, vectorB);
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(solver.IterationResult);
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 SolveUnitMatrixAndBackMultiply()
{
// Create the identity matrix
var matrix = SparseMatrix.Identity(100);
// Create the y vector
var y = DenseVector.Create(matrix.RowCount, i => 1);
// Create an iteration monitor which will keep track of iterative convergence
var monitor = new Iterator<Complex32>(new IIterationStopCriterium<Complex32>[]
{
new IterationCountStopCriterium<Complex32>(MaximumIterations),
new ResidualStopCriterium(ConvergenceBoundary),
new DivergenceStopCriterium(),
new FailureStopCriterium()
});
var solver = new TFQMR(monitor);
// Solve equation Ax = y
var x = solver.Solve(matrix, y);
// Now compare the results
Assert.IsNotNull(x, "#02");
Assert.AreEqual(y.Count, x.Count, "#03");
// Back multiply the vector
var z = matrix.Multiply(x);
// Check that the solution converged
Assert.IsTrue(monitor.HasConverged, "#04");
// Now compare the vectors
for (var i = 0; i < y.Count; i++)
{
Assert.IsTrue((y[i] - z[i]).Magnitude.IsSmaller(ConvergenceBoundary, 1), "#05-" + i);
}
}
public void SolvePoissonMatrixAndBackMultiply()
{
// Create the matrix
var matrix = new SparseMatrix(25);
// Assemble the matrix. We assume we're solving the Poisson equation
// on a rectangular 5 x 5 grid
const int GridSize = 5;
// The pattern is:
// 0 .... 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 ... 0
for (var i = 0; i < matrix.RowCount; i++)
{
// Insert the first set of -1's
if (i > (GridSize - 1))
{
matrix[i, i - GridSize] = -1;
}
// Insert the second set of -1's
if (i > 0)
{
matrix[i, i - 1] = -1;
}
// Insert the centerline values
matrix[i, i] = 4;
// Insert the first trailing set of -1's
if (i < matrix.RowCount - 1)
{
matrix[i, i + 1] = -1;
}
// Insert the second trailing set of -1's
if (i < matrix.RowCount - GridSize)
{
matrix[i, i + GridSize] = -1;
}
}
// Create the y vector
var y = DenseVector.Create(matrix.RowCount, i => 1);
// Create an iteration monitor which will keep track of iterative convergence
var monitor = new Iterator<Complex32>(new IIterationStopCriterium<Complex32>[]
{
new IterationCountStopCriterium<Complex32>(MaximumIterations),
new ResidualStopCriterium(ConvergenceBoundary),
new DivergenceStopCriterium(),
new FailureStopCriterium()
});
var solver = new TFQMR(monitor);
// Solve equation Ax = y
var x = solver.Solve(matrix, y);
// Now compare the results
Assert.IsNotNull(x, "#02");
Assert.AreEqual(y.Count, x.Count, "#03");
// Back multiply the vector
var z = matrix.Multiply(x);
// Check that the solution converged
Assert.IsTrue(monitor.HasConverged, "#04");
// Now compare the vectors
for (var i = 0; i < y.Count; i++)
{
Assert.IsTrue((y[i] - z[i]).Magnitude.IsSmaller(1e-4f, 1), "#05-" + i);
}
}
public void CanSolveForRandomVector(int order)
{
for (var iteration = 5; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var vectorb = MatrixLoader.GenerateRandomDenseVector(order);
var monitor = new Iterator<Complex32>(new IIterationStopCriterium<Complex32>[]
{
new IterationCountStopCriterium<Complex32>(1000),
new ResidualStopCriterium((float) Math.Pow(1.0/10.0, iteration))
});
var solver = new TFQMR(monitor);
var resultx = solver.Solve(matrixA, vectorb);
if (!monitor.HasConverged)
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
var matrixBReconstruct = matrixA*resultx;
// Check the reconstruction.
for (var i = 0; i < order; i++)
{
Assert.AreEqual(vectorb[i].Real, matrixBReconstruct[i].Real, (float) Math.Pow(1.0/10.0, iteration - 3));
Assert.AreEqual(vectorb[i].Imaginary, matrixBReconstruct[i].Imaginary, (float) Math.Pow(1.0/10.0, iteration - 3));
}
return;
}
Assert.Fail("Solution was not found in 3 tries");
}
public void SolvePoissonMatrixAndBackMultiply()
{
// Create the matrix
var matrix = new SparseMatrix(100);
// Assemble the matrix. We assume we're solving the Poisson equation
// on a rectangular 10 x 10 grid
const int GridSize = 10;
// The pattern is:
// 0 .... 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 ... 0
for (var i = 0; i < matrix.RowCount; i++)
{
// Insert the first set of -1's
if (i > (GridSize - 1))
{
matrix[i, i - GridSize] = -1;
}
// Insert the second set of -1's
if (i > 0)
{
matrix[i, i - 1] = -1;
}
// Insert the centerline values
matrix[i, i] = 4;
// Insert the first trailing set of -1's
if (i < matrix.RowCount - 1)
{
matrix[i, i + 1] = -1;
}
// Insert the second trailing set of -1's
if (i < matrix.RowCount - GridSize)
{
matrix[i, i + GridSize] = -1;
}
}
// Create the y vector
var y = DenseVector.Create(matrix.RowCount, i => 1);
// Create an iteration monitor which will keep track of iterative convergence
var monitor = new Iterator(new IIterationStopCriterium <double>[]
{
new IterationCountStopCriterium <double>(MaximumIterations),
new ResidualStopCriterium(ConvergenceBoundary),
new DivergenceStopCriterium(),
new FailureStopCriterium()
});
var solver = new TFQMR(monitor);
// Solve equation Ax = y
var x = solver.Solve(matrix, y);
// Now compare the results
Assert.IsNotNull(x, "#02");
Assert.AreEqual(y.Count, x.Count, "#03");
// Back multiply the vector
var z = matrix.Multiply(x);
// Check that the solution converged
Assert.IsTrue(monitor.Status is CalculationConverged, "#04");
// Now compare the vectors
for (var i = 0; i < y.Count; i++)
{
Assert.IsTrue(Math.Abs(y[i] - z[i]).IsSmaller(ConvergenceBoundary, 1), "#05-" + i);
}
}
public void SolveLongMatrix()
{
var matrix = new SparseMatrix(3, 2);
Vector<float> input = new DenseVector(3);
var solver = new TFQMR();
solver.Solve(matrix, input);
}
public void CanSolveForRandomVector([Values(4, 8, 10)] int order)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var vectorb = MatrixLoader.GenerateRandomDenseVector(order);
var monitor = new Iterator(new IIterationStopCriterium[]
{
new IterationCountStopCriterium(1000),
new ResidualStopCriterium(1e-10),
});
var solver = new TFQMR(monitor);
var resultx = solver.Solve(matrixA, vectorb);
Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
var matrixBReconstruct = matrixA * resultx;
// Check the reconstruction.
for (var i = 0; i < order; i++)
{
Assert.AreEqual(vectorb[i].Real, matrixBReconstruct[i].Real, 1e-5);
Assert.AreEqual(vectorb[i].Imaginary, matrixBReconstruct[i].Imaginary, 1e-5);
}
}
/// <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 = new DenseMatrix(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 criteriums to monitor an iterative calculation. There are next available stop criteriums:
// - DivergenceStopCriterium: monitors an iterative calculation for signs of divergence;
// - FailureStopCriterium: monitors residuals for NaN's;
// - IterationCountStopCriterium: monitors the numbers of iteration steps;
// - ResidualStopCriterium: monitors residuals if calculation is considered converged;
// Stop calculation if 1000 iterations reached during calculation
var iterationCountStopCriterium = new IterationCountStopCriterium(1000);
// Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
var residualStopCriterium = new ResidualStopCriterium(1e-10);
// Create monitor with defined stop criteriums
var monitor = new Iterator(new IIterationStopCriterium[] { iterationCountStopCriterium, residualStopCriterium });
// Create Transpose Free Quasi-Minimal Residual solver
var solver = new TFQMR(monitor);
// 1. Solve the matrix equation
var resultX = solver.Solve(matrixA, vectorB);
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(solver.IterationResult);
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 SolveWideMatrix()
{
var matrix = new SparseMatrix(2, 3);
Vector<Complex32> input = new DenseVector(2);
var solver = new TFQMR();
solver.Solve(matrix, input);
}
public void CanSolveForRandomVector(int order)
{
// Due to datatype "float" it can happen that solution will not converge for specific random matrix
// That's why we will do 3 tries and downgrade stop criterium each time
for (var iteration = 6; iteration > 3; iteration--)
{
var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
var vectorb = MatrixLoader.GenerateRandomDenseVector(order);
var monitor = new Iterator<float>(new IIterationStopCriterium<float>[]
{
new IterationCountStopCriterium<float>(MaximumIterations),
new ResidualStopCriterium((float) Math.Pow(1.0/10.0, iteration))
});
var solver = new TFQMR(monitor);
var resultx = solver.Solve(matrixA, vectorb);
if (!monitor.HasConverged)
{
// Solution was not found, try again downgrading convergence boundary
continue;
}
Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
var matrixBReconstruct = matrixA*resultx;
// Check the reconstruction.
for (var i = 0; i < order; i++)
{
Assert.AreEqual(vectorb[i], matrixBReconstruct[i], (float) Math.Pow(1.0/10.0, iteration - 3));
}
return;
}
}
public void SolveScaledUnitMatrixAndBackMultiply()
{
// Create the identity matrix
Matrix<float> matrix = SparseMatrix.Identity(100);
// Scale it with a funny number
matrix.Multiply((float)Math.PI);
// Create the y vector
Vector<float> y = new DenseVector(matrix.RowCount, 1);
// Create an iteration monitor which will keep track of iterative convergence
var monitor = new Iterator(new IIterationStopCriterium<float>[]
{
new IterationCountStopCriterium(MaximumIterations),
new ResidualStopCriterium(ConvergenceBoundary),
new DivergenceStopCriterium(),
new FailureStopCriterium()
});
var solver = new TFQMR(monitor);
// Solve equation Ax = y
var x = solver.Solve(matrix, y);
// Now compare the results
Assert.IsNotNull(x, "#02");
Assert.AreEqual(y.Count, x.Count, "#03");
// Back multiply the vector
var z = matrix.Multiply(x);
// Check that the solution converged
Assert.IsTrue(monitor.Status is CalculationConverged, "#04");
// Now compare the vectors
for (var i = 0; i < y.Count; i++)
{
Assert.IsTrue((y[i] - z[i]).IsSmaller(ConvergenceBoundary, 1), "#05-" + i);
}
}