public void SquareVandermondeMatrixLUDecomposition()
{
// fails now for d = 8 because determinant slightly off
for (int d = 1; d <= 7; d++)
{
Console.WriteLine("d={0}", d);
double[] x = new double[d];
for (int i = 0; i < d; i++)
{
x[i] = i;
}
double det = 1.0;
for (int i = 0; i < d; i++)
{
for (int j = 0; j < i; j++)
{
det = det * (x[i] - x[j]);
}
}
// LU decompose the matrix
SquareMatrix V = CreateVandermondeMatrix(d);
LUDecomposition LU = V.LUDecomposition();
// test that the decomposition works
SquareMatrix P = LU.PMatrix();
SquareMatrix L = LU.LMatrix();
SquareMatrix U = LU.UMatrix();
Assert.IsTrue(TestUtilities.IsNearlyEqual(P * V, L * U));
// check that the determinant agrees with the analytic expression
Console.WriteLine("det {0} {1}", LU.Determinant(), det);
Assert.IsTrue(TestUtilities.IsNearlyEqual(LU.Determinant(), det));
// check that the inverse works
SquareMatrix VI = LU.Inverse();
//PrintMatrix(VI);
//PrintMatrix(V * VI);
SquareMatrix I = TestUtilities.CreateSquareUnitMatrix(d);
Assert.IsTrue(TestUtilities.IsNearlyEqual(V * VI, I));
// test that a solution works
ColumnVector t = new ColumnVector(d);
for (int i = 0; i < d; i++)
{
t[i] = 1.0;
}
ColumnVector s = LU.Solve(t);
Assert.IsTrue(TestUtilities.IsNearlyEqual(V * s, t));
}
}
public static void LUDecomposition()
{
SquareMatrix A = new SquareMatrix(new double[, ] {
{ 1, -2, 3 },
{ 2, -5, 12 },
{ 0, 2, -10 }
});
ColumnVector b = new ColumnVector(2, 8, -4);
LUDecomposition lud = A.LUDecomposition();
ColumnVector x = lud.Solve(b);
PrintMatrix("x", x);
PrintMatrix("Ax", A * x);
SquareMatrix L = lud.LMatrix();
SquareMatrix U = lud.UMatrix();
SquareMatrix P = lud.PMatrix();
PrintMatrix("LU", L * U);
PrintMatrix("PA", P * A);
SquareMatrix AI = lud.Inverse();
PrintMatrix("A * AI", A * AI);
Console.WriteLine($"det(a) = {lud.Determinant()}");
}
public static Double Det(Double[,] matrix)
{
var rows = matrix.GetLength(0);
var columns = matrix.GetLength(1);
if (rows == columns)
{
switch (rows)
{
case 0: return(0.0);
case 1: return(matrix[0, 0]);
case 2: return(Helpers.Det2(matrix));
case 3: return(Helpers.Det3(matrix));
case 4: return(Helpers.Det4(matrix));
}
return(LUDecomposition.Determinant(matrix));
}
return(0.0);
}
public void Simple()
{
var matrix = new Matrix(3, 3);
// [ 3, 1, 8 ]
// [ 2, -5, 4 ]
// [-1, 6, -2 ]
// Determinant = 14
matrix[0, 0] = 3;
matrix[0, 1] = 1;
matrix[0, 2] = 8;
matrix[1, 0] = 2;
matrix[1, 1] = -5;
matrix[1, 2] = 4;
matrix[2, 0] = -1;
matrix[2, 1] = 6;
matrix[2, 2] = -2;
var decomposition = new LUDecomposition(matrix);
Assert.AreEqual(14.0d, decomposition.Determinant(), 0.00000001d);
}
public void SquareUnitMatrixLUDecomposition()
{
for (int d = 1; d <= 10; d++)
{
SquareMatrix I = UnitMatrix.OfDimension(d).ToSquareMatrix();
Assert.IsTrue(I.Trace() == d);
LUDecomposition LU = I.LUDecomposition();
Assert.IsTrue(LU.Determinant() == 1.0);
SquareMatrix II = LU.Inverse();
Assert.IsTrue(TestUtilities.IsNearlyEqual(II, I));
}
}
public void MatrixLUDecomposition2()
{
/*
* MATLAB:
* mc2x2 = [1 2;3 4]
* [L_mc, U_mc, P_mc] = lu(mc2x2)
* P_mcv = P_mc * [0:1:length(P_mc)-1]'
* det(mc2x2)
* [L_mch, U_mch, P_mch] = lu(mc2x2')
* P_mchv = P_mch * [0:1:length(P_mch)-1]'
* det(mc2x2')
*/
Converter <int, double> int2Double = delegate(int i) { return(i); };
Matrix mc2X2 = new Matrix(new double[][]
{
new double[] { 1, 2 },
new double[] { 3, 4 }
});
LUDecomposition mcLU = mc2X2.LUDecomposition;
Matrix mcL = new Matrix(new double[][] {
new double[] { 1, 0 },
new double[] { 1d / 3d, 1 }
});
Matrix mcU = new Matrix(new double[][] {
new double[] { 3, 4 },
new double[] { 0, 2d / 3d }
});
Matrix mcP = new Matrix(new double[][] {
new double[] { 0, 1 },
new double[] { 1, 0 }
});
Vector mcPv = new Vector(new double[] { 1, 0 });
Assert.That(mcLU.L, NumericIs.AlmostEqualTo(mcL), "real LU L-matrix");
Assert.That(mcLU.U, NumericIs.AlmostEqualTo(mcU), "real LU U-matrix");
Assert.That(mcLU.PermutationMatrix, NumericIs.AlmostEqualTo(mcP), "real LU permutation matrix");
Assert.That(mcLU.PivotVector, NumericIs.AlmostEqualTo(mcPv), "real LU pivot");
Assert.That((Vector)Array.ConvertAll(mcLU.Pivot, int2Double), NumericIs.AlmostEqualTo(mcPv), "real LU pivot II");
Assert.That(mcLU.Determinant(), NumericIs.AlmostEqualTo((double)(-2)), "real LU determinant");
Assert.That(mc2X2.Determinant(), NumericIs.AlmostEqualTo((double)(-2)), "real LU determinant II");
Assert.That(mcLU.IsNonSingular, "real LU non-singular");
Assert.That(mcLU.L * mcLU.U, NumericIs.AlmostEqualTo(mcLU.PermutationMatrix * mc2X2), "real LU product");
Matrix mc2X2H = Matrix.Transpose(mc2X2);
LUDecomposition mchLU = mc2X2H.LUDecomposition;
Matrix mchL = new Matrix(new double[][] {
new double[] { 1, 0 },
new double[] { 0.5, 1 }
});
Matrix mchU = new Matrix(new double[][] {
new double[] { 2, 4 },
new double[] { 0, 1 }
});
Matrix mchP = new Matrix(new double[][] {
new double[] { 0, 1 },
new double[] { 1, 0 }
});
Vector mchPv = new Vector(new double[] { 1, 0 });
Assert.That(mchLU.L, NumericIs.AlmostEqualTo(mchL), "real LU L-matrix (H)");
Assert.That(mchLU.U, NumericIs.AlmostEqualTo(mchU), "real LU U-matrix (H)");
Assert.That(mchLU.PermutationMatrix, NumericIs.AlmostEqualTo(mchP), "real LU permutation matrix (H)");
Assert.That(mchLU.PivotVector, NumericIs.AlmostEqualTo(mchPv), "real LU pivot (H)");
Assert.That((Vector)Array.ConvertAll(mchLU.Pivot, int2Double), NumericIs.AlmostEqualTo(mchPv), "real LU pivot II (H)");
Assert.That(mchLU.Determinant(), NumericIs.AlmostEqualTo((double)(-2)), "real LU determinant (H)");
Assert.That(mc2X2H.Determinant(), NumericIs.AlmostEqualTo((double)(-2)), "real LU determinant II (H)");
Assert.That(mchLU.IsNonSingular, "real LU non-singular (H)");
Assert.That(mchLU.L * mchLU.U, NumericIs.AlmostEqualTo(mchLU.PermutationMatrix * mc2X2H), "real LU product (H)");
}
public void TestMatrix_LUDecomposition()
{
/*
* MATLAB:
* [L_mc, U_mc, P_mc] = lu(mc2x2)
* P_mcv = P_mc * [0:1:length(P_mc)-1]'
* det(mc2x2)
* [L_mch, U_mch, P_mch] = lu(mc2x2')
* P_mchv = P_mch * [0:1:length(P_mch)-1]'
* det(mc2x2')
*/
Converter <int, double> int2double = delegate(int i) { return(i); };
LUDecomposition LU = mc2x2.LUDecomposition;
Matrix L_mc = new Matrix(new double[][] {
new double[] { 1, 0 },
new double[] { 1d / 3d, 1 }
});
Matrix U_mc = new Matrix(new double[][] {
new double[] { 3, 4 },
new double[] { 0, 2d / 3d }
});
Matrix P_mc = new Matrix(new double[][] {
new double[] { 0, 1 },
new double[] { 1, 0 }
});
Vector P_mcv = new Vector(new double[] { 1, 0 });
NumericAssert.AreAlmostEqual(L_mc, LU.L, "real LU L-matrix");
NumericAssert.AreAlmostEqual(U_mc, LU.U, "real LU U-matrix");
NumericAssert.AreAlmostEqual(P_mc, LU.PermutationMatrix, "real LU permutation matrix");
NumericAssert.AreAlmostEqual(P_mcv, LU.PivotVector, "real LU pivot");
NumericAssert.AreAlmostEqual(P_mcv, Array.ConvertAll(LU.Pivot, int2double), "real LU pivot II");
NumericAssert.AreAlmostEqual(-2, LU.Determinant(), "real LU determinant");
NumericAssert.AreAlmostEqual(-2, mc2x2.Determinant(), "real LU determinant II");
Assert.IsTrue(LU.IsNonSingular, "real LU non-singular");
NumericAssert.AreAlmostEqual(LU.PermutationMatrix * mc2x2, LU.L * LU.U, "real LU product");
Matrix mc2x2h = Matrix.Transpose(mc2x2);
LUDecomposition LUH = mc2x2h.LUDecomposition;
Matrix L_mch = new Matrix(new double[][] {
new double[] { 1, 0 },
new double[] { 0.5, 1 }
});
Matrix U_mch = new Matrix(new double[][] {
new double[] { 2, 4 },
new double[] { 0, 1 }
});
Matrix P_mch = new Matrix(new double[][] {
new double[] { 0, 1 },
new double[] { 1, 0 }
});
Vector P_mchv = new Vector(new double[] { 1, 0 });
NumericAssert.AreAlmostEqual(L_mch, LUH.L, "real LU L-matrix (H)");
NumericAssert.AreAlmostEqual(U_mch, LUH.U, "real LU U-matrix (H)");
NumericAssert.AreAlmostEqual(P_mch, LUH.PermutationMatrix, "real LU permutation matrix (H)");
NumericAssert.AreAlmostEqual(P_mchv, LUH.PivotVector, "real LU pivot (H)");
NumericAssert.AreAlmostEqual(P_mchv, Array.ConvertAll(LUH.Pivot, int2double), "real LU pivot II (H)");
NumericAssert.AreAlmostEqual(-2, LUH.Determinant(), "real LU determinant (H)");
NumericAssert.AreAlmostEqual(-2, mc2x2h.Determinant(), "real LU determinant II (H)");
Assert.IsTrue(LUH.IsNonSingular, "real LU non-singular (H)");
NumericAssert.AreAlmostEqual(LUH.PermutationMatrix * mc2x2h, LUH.L * LUH.U, "real LU product (H)");
}
