public void T01_BasicLinearEquations()
{
const double Accuracy = 2.2205e-15, SvdAccuracy = 4.46e-15, DeterminantAccuracy = 1.2e-13;
// given w=-1, x=2, y=-3, and z=4...
// w - 2x + 3y - 5z = -1 - 4 - 9 - 20 = -34
// w + 2x + 3y + 5z = -1 + 4 - 9 + 20 = 14
// 5w - 3x + 2y - z = -5 - 6 - 6 - 4 = -21
// 5w + 3x + 2y + z = -5 + 6 - 6 + 4 = -1
double[] coefficientArray = new double[16]
{
1, -2, 3, -5,
1, 2, 3, 5,
5, -3, 2, -1,
5, 3, 2, 1,
};
double[] valueArray = new double[4] {
-34, 14, -21, -1
};
// first solve using Gauss-Jordan elimination
Matrix4 coefficients = new Matrix4(coefficientArray);
Matrix gjInverse, values = new Matrix(valueArray, 1);
Matrix solution = GaussJordan.Solve(coefficients.ToMatrix(), values, out gjInverse);
CheckSolution(solution, Accuracy);
CheckSolution(gjInverse * values, Accuracy); // make sure multiplying by the matrix inverse also gives the right answer
// then solve using LU decomposition
LUDecomposition lud = new LUDecomposition(coefficients.ToMatrix());
solution = lud.Solve(values);
CheckSolution(solution, Accuracy);
CheckSolution(lud.GetInverse() * values, Accuracy); // check that the inverse can be multiplied to produce a good solution
solution.Multiply(1.1); // mess up the solution
lud.RefineSolution(values, solution); // test that refinement can fix it
CheckSolution(solution, Accuracy);
// check that the computed determinants are what we expect
bool negative;
Assert.AreEqual(coefficients.GetDeterminant(), lud.GetDeterminant(), DeterminantAccuracy);
Assert.AreEqual(Math.Log(Math.Abs(coefficients.GetDeterminant())), lud.GetLogDeterminant(out negative), Accuracy);
Assert.AreEqual(coefficients.GetDeterminant() < 0, negative);
// check that the inverses are about the same from both methods
Assert.IsTrue(gjInverse.Equals(lud.GetInverse(), Accuracy));
// then solve using QR decomposition
QRDecomposition qrd = new QRDecomposition(coefficients.ToMatrix());
solution = qrd.Solve(values);
CheckSolution(solution, Accuracy);
CheckSolution(qrd.GetInverse() * values, Accuracy); // check that the inverse can be multiplied to produce a good solution
// TODO: test qrd.Update()
// finally, try solving using singular value decomposition
SVDecomposition svd = new SVDecomposition(coefficients.ToMatrix());
solution = svd.Solve(values);
CheckSolution(solution, SvdAccuracy);
CheckSolution(svd.GetInverse() * values, SvdAccuracy);
Assert.IsTrue(gjInverse.Equals(svd.GetInverse(), Accuracy));
Assert.AreEqual(4, svd.GetRank());
Assert.AreEqual(0, svd.GetNullity());
}
