public void TestWarmStarting()
{
MatrixF A = new MatrixF(new float[, ] {
{ -21, 2, -4, 0 },
{ 2, 3, 0.1f, -1 },
{ 2, 10, 111.1f, -11 },
{ 23, 112, 111.1f, -143 }
});
VectorF b = new VectorF(new float[] { 20, 28, -12, 0.1f });
SorMethodF solver = new SorMethodF();
VectorF x = solver.Solve(A, null, b);
int fullIterationCount = solver.NumberOfIterations;
VectorF solution = MatrixF.SolveLinearEquations(A, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(solution, x));
// Now test make separate solve calls with warm-starting
solver.MaxNumberOfIterations = 3;
x = solver.Solve(A, null, b);
Assert.AreEqual(3, solver.NumberOfIterations);
solver.MaxNumberOfIterations = 100;
x = solver.Solve(A, x, b);
Assert.AreEqual(fullIterationCount, solver.NumberOfIterations + 3);
}
public void Test6()
{
MatrixF A = new MatrixF(new float[, ] {
{ -21, 2, -4, 0 },
{ 2, 3, 0.1f, -1 },
{ 2, 10, 111.1f, -11 },
{ 23, 112, 111.1f, -143 }
});
VectorF b = new VectorF(new float[] { 20, 28, -12, 0.1f });
SorMethodF solver = new SorMethodF();
solver.MaxNumberOfIterations = 4;
VectorF x = solver.Solve(A, null, b);
VectorF solution = MatrixF.SolveLinearEquations(A, b);
Assert.IsFalse(VectorF.AreNumericallyEqual(solution, x));
Assert.AreEqual(4, solver.NumberOfIterations);
// Compare with Gauss-Seidel. Must be equal.
GaussSeidelMethodF gsSolver = new GaussSeidelMethodF();
gsSolver.MaxNumberOfIterations = 4;
VectorF gsSolution = gsSolver.Solve(A, null, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(gsSolution, x));
}
public void Test()
{
// Make a random list.
RandomHelper.Random = new Random(77);
List <VectorF> points = new List <VectorF>();
for (int i = 0; i < 10; i++)
{
var vector = new VectorF(4);
RandomHelper.Random.NextVectorF(vector, -1, 10);
points.Add(vector);
}
PrincipalComponentAnalysisF pca = new PrincipalComponentAnalysisF(points);
Assert.Greater(pca.Variances[0], pca.Variances[1]);
Assert.Greater(pca.Variances[1], pca.Variances[2]);
Assert.Greater(pca.Variances[2], pca.Variances[3]);
Assert.Greater(pca.Variances[3], 0);
Assert.IsTrue(pca.V.GetColumn(0).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(1).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(2).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(3).IsNumericallyNormalized);
// Compute covariance matrix and check if it is diagonal in the transformed space.
MatrixF cov = StatisticsHelper.ComputeCovarianceMatrix(points);
MatrixF transformedCov = pca.V.Transposed * cov * pca.V;
for (int row = 0; row < transformedCov.NumberOfRows; row++)
{
for (int column = 0; column < transformedCov.NumberOfColumns; column++)
{
if (row != column)
{
Assert.IsTrue(Numeric.IsZero(transformedCov[row, column]));
}
}
}
// The principal components must be Eigenvectors which means that multiplying with the covariance
// matrix does only change the length!
VectorF v0 = pca.V.GetColumn(0);
VectorF v0Result = cov * v0;
Assert.IsTrue(VectorF.AreNumericallyEqual(v0.Normalized, v0Result.Normalized));
VectorF v1 = pca.V.GetColumn(1);
VectorF v1Result = cov * v1;
Assert.IsTrue(VectorF.AreNumericallyEqual(v1.Normalized, v1Result.Normalized));
VectorF v2 = pca.V.GetColumn(2);
VectorF v2Result = cov * v2;
Assert.IsTrue(VectorF.AreNumericallyEqual(v2.Normalized, v2Result.Normalized));
VectorF v3 = pca.V.GetColumn(3);
VectorF v3Result = cov * v3;
Assert.IsTrue(VectorF.AreNumericallyEqual(v3.Normalized, v3Result.Normalized));
}
/// <summary>
/// Solves the specified linear system of equations <i>Ax=b</i>.
/// </summary>
/// <param name="matrixA">The matrix A.</param>
/// <param name="initialX">
/// The initial guess for x. If this value is <see langword="null"/>, a zero vector will be used
/// as initial guess.
/// </param>
/// <param name="vectorB">The vector b.</param>
/// <returns>The solution vector x.</returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentNullException">
/// <paramref name="vectorB"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// <paramref name="matrixA"/> is not a square matrix.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of elements of <paramref name="initialX"/> does not match.
/// </exception>
public override VectorF Solve(MatrixF matrixA, VectorF initialX, VectorF vectorB)
{
NumberOfIterations = 0;
if (matrixA == null)
{
throw new ArgumentNullException("matrixA");
}
if (vectorB == null)
{
throw new ArgumentNullException("vectorB");
}
if (matrixA.IsSquare == false)
{
throw new ArgumentException("Matrix A must be a square matrix.", "matrixA");
}
if (matrixA.NumberOfRows != vectorB.NumberOfElements)
{
throw new ArgumentException("The number of rows of A and b do not match.");
}
if (initialX != null && initialX.NumberOfElements != vectorB.NumberOfElements)
{
throw new ArgumentException("The number of elements of the initial guess for x and b do not match.");
}
VectorF xOld = initialX ?? new VectorF(vectorB.NumberOfElements);
VectorF xNew = new VectorF(vectorB.NumberOfElements);
bool isConverged = false;
// Make iterations until max iteration count or the result has converged.
for (int i = 0; i < MaxNumberOfIterations && !isConverged; i++)
{
for (int j = 0; j < vectorB.NumberOfElements; j++)
{
float delta = 0;
for (int k = 0; k < j; k++)
{
delta += matrixA[j, k] * xNew[k];
}
for (int k = j + 1; k < vectorB.NumberOfElements; k++)
{
delta += matrixA[j, k] * xOld[k];
}
delta = (vectorB[j] - delta) / matrixA[j, j];
xNew[j] = xOld[j] + RelaxationFactor * (delta - xOld[j]);
}
// Test convergence
isConverged = VectorF.AreNumericallyEqual(xOld, xNew, Epsilon);
xOld = xNew.Clone();
NumberOfIterations = i + 1;
}
return(xNew);
}
public void Test1()
{
MatrixF A = new MatrixF(new float[, ] {
{ 4 }
});
VectorF b = new VectorF(new float[] { 20 });
SorMethodF solver = new SorMethodF();
VectorF x = solver.Solve(A, null, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(new VectorF(1, 5), x));
Assert.AreEqual(2, solver.NumberOfIterations);
}
public void SolveWithDefaultInitialGuess()
{
MatrixF A = new MatrixF(new float[, ] {
{ 4 }
});
VectorF b = new VectorF(new float[] { 20 });
JacobiMethodF solver = new JacobiMethodF();
VectorF x = solver.Solve(A, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(new VectorF(1, 5), x));
Assert.AreEqual(2, solver.NumberOfIterations);
}
public void Test3()
{
MatrixF A = new MatrixF(new float[, ] {
{ 2, 0 },
{ 0, 2 }
});
VectorF b = new VectorF(new float[] { 20, 28 });
JacobiMethodF solver = new JacobiMethodF();
VectorF x = solver.Solve(A, null, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(b / 2, x));
Assert.AreEqual(2, solver.NumberOfIterations);
}
public void Test2()
{
MatrixF A = new MatrixF(new float[, ] {
{ 1, 0 },
{ 0, 1 }
});
VectorF b = new VectorF(new float[] { 20, 28 });
GaussSeidelMethodF solver = new GaussSeidelMethodF();
VectorF x = solver.Solve(A, null, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(b, x));
Assert.AreEqual(2, solver.NumberOfIterations);
}
public void Test4()
{
MatrixF A = new MatrixF(new float[, ] {
{ -12, 2 },
{ 2, 3 }
});
VectorF b = new VectorF(new float[] { 20, 28 });
SorMethodF solver = new SorMethodF();
VectorF x = solver.Solve(A, null, b);
VectorF solution = MatrixF.SolveLinearEquations(A, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(solution, x));
}
public void Test5()
{
MatrixF A = new MatrixF(new float[, ] {
{ -21, 2, -4, 0 },
{ 2, 3, 0.1f, -1 },
{ 2, 10, 111.1f, -11 },
{ 23, 112, 111.1f, -143 }
});
VectorF b = new VectorF(new float[] { 20, 28, -12, 0.1f });
SorMethodF solver = new SorMethodF();
VectorF x = solver.Solve(A, null, b);
VectorF solution = MatrixF.SolveLinearEquations(A, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(solution, x));
}
public void Test6()
{
MatrixF A = new MatrixF(new float[, ] {
{ -21, 2, -4, 0 },
{ 2, 3, 0.1f, -1 },
{ 2, 10, 111.1f, -11 },
{ 23, 112, 111.1f, -143 }
});
VectorF b = new VectorF(new float[] { 20, 28, -12, 0.1f });
JacobiMethodF solver = new JacobiMethodF();
solver.MaxNumberOfIterations = 10;
VectorF x = solver.Solve(A, null, b);
VectorF solution = MatrixF.SolveLinearEquations(A, b);
Assert.IsFalse(VectorF.AreNumericallyEqual(solution, x));
Assert.AreEqual(10, solver.NumberOfIterations);
}
public void Test7()
{
MatrixF A = new MatrixF(new float[, ] {
{ -21, 2, -4, 0 },
{ 2, 3, 0.1f, -1 },
{ 2, 10, 111.1f, -11 },
{ 23, 112, 111.1f, -143 }
});
VectorF b = new VectorF(new float[] { 20, 28, -12, 0.1f });
SorMethodF solver = new SorMethodF();
solver.Epsilon = 0.1f;
VectorF x = solver.Solve(A, null, b);
VectorF solution = MatrixF.SolveLinearEquations(A, b);
Assert.IsTrue(VectorF.AreNumericallyEqual(solution, x, 0.1f));
Assert.IsFalse(VectorF.AreNumericallyEqual(solution, x));
Assert.Greater(12, solver.NumberOfIterations); // For normal accuracy (EpsilonF) we need 12 iterations.
}
/// <summary>
/// Solves the specified linear system of equations <i>Ax=b</i>.
/// </summary>
/// <param name="matrixA">The matrix A.</param>
/// <param name="initialX">
/// The initial guess for x. If this value is <see langword="null"/>, a zero vector will be used
/// as initial guess.
/// </param>
/// <param name="vectorB">The vector b.</param>
/// <returns>The solution vector x.</returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentNullException">
/// <paramref name="vectorB"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// <paramref name="matrixA"/> is not a square matrix.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of elements of <paramref name="initialX"/> does not match.
/// </exception>
public override VectorF Solve(MatrixF matrixA, VectorF initialX, VectorF vectorB)
{
// TODO: We can possible improve the method by reordering after each step.
// This can be done randomly or we sort by the "convergence" of the elements.
// See book Physics-Based Animation.
NumberOfIterations = 0;
if (matrixA == null)
{
throw new ArgumentNullException("matrixA");
}
if (vectorB == null)
{
throw new ArgumentNullException("vectorB");
}
if (matrixA.IsSquare == false)
{
throw new ArgumentException("Matrix A must be a square matrix.", "matrixA");
}
if (matrixA.NumberOfRows != vectorB.NumberOfElements)
{
throw new ArgumentException("The number of rows of A and b do not match.");
}
if (initialX != null && initialX.NumberOfElements != vectorB.NumberOfElements)
{
throw new ArgumentException("The number of elements of the initial guess for x and b do not match.");
}
VectorF xOld = initialX ?? new VectorF(vectorB.NumberOfElements);
VectorF xNew = new VectorF(vectorB.NumberOfElements);
bool isConverged = false;
// Make iterations until max iteration count or the result has converged.
for (int i = 0; i < MaxNumberOfIterations && !isConverged; i++)
{
for (int j = 0; j < vectorB.NumberOfElements; j++)
{
float delta = 0;
for (int k = 0; k < j; k++)
{
delta += matrixA[j, k] * xNew[k];
}
for (int k = j + 1; k < vectorB.NumberOfElements; k++)
{
delta += matrixA[j, k] * xOld[k];
}
xNew[j] = (vectorB[j] - delta) / matrixA[j, j];
}
// Test convergence
isConverged = VectorF.AreNumericallyEqual(xOld, xNew, Epsilon);
xOld = xNew.Clone();
NumberOfIterations = i + 1;
}
return(xNew);
}