public void InverseTest()
{
double[][] value =
{
new double[] { 2, -1, 0 },
new double[] { -1, 2, -1 },
new double[] { 0, -1, 2 }
};
double[][] expected =
{
new double[] { 0.7500, 0.5000, 0.2500 },
new double[] { 0.5000, 1.0000, 0.5000 },
new double[] { 0.2500, 0.5000, 0.7500 },
};
var target = new JaggedQrDecomposition(value);
double[][] actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-10));
Assert.IsTrue(Matrix.IsEqual(value, target.Reverse(), 1e-4));
target = new JaggedQrDecomposition(value.Transpose(), true);
actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-10));
Assert.IsTrue(Matrix.IsEqual(value.Transpose(), target.Reverse(), 1e-4));
}
public void full_decomposition()
{
double[][] value =
{
new double[] { 1 },
new double[] { 2 },
new double[] { 3 }
};
var target = new JaggedQrDecomposition(value, economy: false);
// Decomposition Identity
var Q = target.OrthogonalFactor;
var QtQ = Matrix.Dot(Q.Transpose(), Q);
Assert.IsTrue(Matrix.IsEqual(QtQ, Jagged.Identity(3), 1e-6));
double[][] reverse = target.Reverse();
Assert.IsTrue(Matrix.IsEqual(value, reverse, 1e-6));
// Linear system solving
double[][] B = Matrix.ColumnVector(new double[] { 4, 5, 6 }).ToJagged();
double[][] expected = Jagged.ColumnVector(new double[] { 2.285714285714286 });
double[][] actual = target.Solve(B);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
var QQt = Matrix.Dot(Q, Q.Transpose());
Assert.IsTrue(Matrix.IsEqual(QQt, Jagged.Identity(3), 1e-6));
}
public void SolveTransposeTest()
{
double[][] a =
{
new double[] { 2, 1, 4 },
new double[] { 6, 2, 2 },
new double[] { 0, 1, 6 },
};
double[][] b =
{
new double[] { 1, 0, 7 },
new double[] { 5, 2, 1 },
new double[] { 1, 5, 2 },
};
double[][] expected =
{
new double[] { 0.5062, 0.2813, 0.0875 },
new double[] { 0.1375, 1.1875, -0.0750 },
new double[] { 0.8063, -0.2188, 0.2875 },
};
double[][] actual = new JaggedQrDecomposition(b, true).SolveTranspose(a);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
}
public void QrDecompositionConstructorTest()
{
double[][] value =
{
new double[] { 2, -1, 0 },
new double[] { -1, 2, -1 },
new double[] { 0, -1, 2 }
};
var target = new JaggedQrDecomposition(value);
// Decomposition Identity
var Q = target.OrthogonalFactor;
var QQt = Matrix.Dot(Q, Q.Transpose());
Assert.IsTrue(Matrix.IsEqual(QQt, Jagged.Identity(3), 1e-6));
Assert.IsTrue(Matrix.IsEqual(value, target.Reverse(), 1e-6));
// Linear system solving
double[][] B = Matrix.ColumnVector(new double[] { 1, 2, 3 }).ToJagged();
double[][] expected = Matrix.ColumnVector(new double[] { 2.5, 4.0, 3.5 }).ToJagged();
double[][] actual = target.Solve(B);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
public void SolveTransposeTest()
{
double[][] a =
{
new double[] { 2, 1, 4 },
new double[] { 6, 2, 2 },
new double[] { 0, 1, 6 },
};
double[][] b =
{
new double[] { 1, 0, 7 },
new double[] { 5, 2, 1 },
new double[] { 1, 5, 2 },
};
double[][] expected =
{
new double[] { 0.5062, 0.2813, 0.0875 },
new double[] { 0.1375, 1.1875, -0.0750 },
new double[] { 0.8063, -0.2188, 0.2875 },
};
double[][] actual = new JaggedQrDecomposition(b, true).SolveTranspose(a);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
var bX = b.Dot(expected);
var Xb = expected.Dot(b);
Assert.IsTrue(Matrix.IsEqual(Xb, a, atol: 1e-3));
Assert.IsFalse(Matrix.IsEqual(bX, a, atol: 1e-3));
double[,] actualMatrix = new QrDecomposition(b.ToMatrix(), true).SolveTranspose(a.ToMatrix());
Assert.IsTrue(Matrix.IsEqual(expected, actualMatrix, 0.001));
}
public void solve_for_diagonal()
{
double[][] value =
{
new double[] { 2, -1, 0 },
new double[] { -1, 2, -1 },
new double[] { 0, -1, 2 }
};
double[] b = { 1, 2, 3 };
double[][] expected =
{
new double[] { 0.75, 1, 0.75 },
new double[] { 0.50, 2, 1.50 },
new double[] { 0.25, 1, 2.25 }
};
var target = new JaggedQrDecomposition(value);
double[][] actual = target.SolveForDiagonal(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-10));
Assert.IsTrue(Matrix.IsEqual(value.Transpose(), target.Reverse(), 1e-6));
}
public void QrDecompositionConstructorTest2()
{
double[][] value =
{
new double[] { 1 },
new double[] { 2 },
new double[] { 3 }
};
var target = new JaggedQrDecomposition(value);
// Decomposition Identity
var Q = target.OrthogonalFactor;
var QQt = Matrix.Dot(Q.Transpose(), Q);
Assert.IsTrue(Matrix.IsEqual(QQt, Jagged.Identity(1), 1e-6));
double[][] reverse = target.Reverse();
Assert.IsTrue(Matrix.IsEqual(value, reverse, 1e-6));
// Linear system solving
double[][] B = Matrix.ColumnVector(new double[] { 4, 5, 6 }).ToJagged();
double[][] expected = Jagged.ColumnVector(new double[] { 2.285714285714286 });
double[][] actual = target.Solve(B);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
double[][] R = target.UpperTriangularFactor;
actual = value.Dot(R.Inverse());
Assert.IsTrue(Matrix.IsEqual(Q, actual, 0.0000000000001));
}
public void QrDecompositionConstructorTest()
{
double[][] value =
{
new double[] { 2, -1, 0 },
new double[] { -1, 2, -1 },
new double[] { 0, -1, 2 }
};
var target = new JaggedQrDecomposition(value);
// Decomposition Identity
var Q = target.OrthogonalFactor;
var QQt = Q.Multiply(Q.Transpose());
Assert.IsTrue(Matrix.IsEqual(QQt, Matrix.JaggedIdentity(3), 0.0000001));
// Linear system solving
double[][] B = Matrix.ColumnVector(new double[] { 1, 2, 3 }).ToArray();
double[][] expected = Matrix.ColumnVector(new double[] { 2.5, 4.0, 3.5 }).ToArray();
double[][] actual = target.Solve(B);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
public void SolveTest2()
{
// Example from Lecture notes for MATHS 370: Advanced Numerical Methods
// http://www.math.auckland.ac.nz/~sharp/370/qr-solving.pdf
double[,] value =
{
{ 1, 0, 0 },
{ 1, 7, 49 },
{ 1, 14, 196 },
{ 1, 21, 441 },
{ 1, 28, 784 },
{ 1, 35, 1225 },
};
// Matrices
{
double[,] b =
{
{ 4 },
{ 1 },
{ 0 },
{ 0 },
{ 2 },
{ 5 },
};
double[,] expected =
{
{ 3.9286 },
{ -0.5031 },
{ 0.0153 },
};
var target = new QrDecomposition(value);
double[,] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, atol: 1e-4));
Assert.IsTrue(Matrix.IsEqual(value, target.Reverse(), 1e-6));
var target2 = new JaggedQrDecomposition(value.ToJagged());
double[][] actual2 = target2.Solve(b.ToJagged());
Assert.IsTrue(Matrix.IsEqual(expected, actual2, atol: 1e-4));
Assert.IsTrue(Matrix.IsEqual(value, target2.Reverse(), 1e-6));
}
// Vectors
{
double[] b = { 4, 1, 0, 0, 2, 5 };
double[] expected = { 3.9286, -0.5031, 0.0153 };
var target = new QrDecomposition(value);
double[] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, atol: 1e-4));
}
}
/// <summary>
/// Gets the null-space of a matrix.
/// </summary>
///
public static Double[][] Null(this Double[][] matrix)
{
var qr = new JaggedQrDecomposition(matrix, economy: false);
var Q = qr.OrthogonalFactor;
var threshold = matrix.GetLength().Max() * Constants.DoubleEpsilon;
int[] idx = qr.Diagonal.Find(x => (Double)Math.Abs(x) < threshold);
return(Q.GetColumns(idx));
}
public void SolveTest()
{
double[][] value =
{
new double[] { 2, -1, 0 },
new double[] { -1, 2, -1 },
new double[] { 0, -1, 2 }
};
double[] b = { 1, 2, 3 };
double[] expected = { 2.5000, 4.0000, 3.5000 };
var target = new JaggedQrDecomposition(value);
double[] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
public void JaggedQrDecompositionCachingTest()
{
double[,] value =
{
{ 2, -1, 0 },
{ -1, 2, -1 },
{ 0, -1, 2 }
};
var target = new JaggedQrDecomposition(value.ToJagged());
// Decomposition Identity
double[][] Q1 = target.OrthogonalFactor;
double[][] Q2 = target.OrthogonalFactor;
double[][] utf1 = target.UpperTriangularFactor;
double[][] utf2 = target.UpperTriangularFactor;
Assert.AreSame(Q1, Q2);
Assert.AreSame(utf1, utf2);
}
public void InverseTest1()
{
double[][] value =
{
new double[] { 41.9, 29.1, 1 },
new double[] { 43.4, 29.3, 1 },
new double[] { 43.9, 29.5, 0 },
new double[] { 44.5, 29.7, 0 },
new double[] { 47.3, 29.9, 0 },
new double[] { 47.5, 30.3, 0 },
new double[] { 47.9, 30.5, 0 },
new double[] { 50.2, 30.7, 0 },
new double[] { 52.8, 30.8, 0 },
new double[] { 53.2, 30.9, 0 },
new double[] { 56.7, 31.5, 0 },
new double[] { 57.0, 31.7, 0 },
new double[] { 63.5, 31.9, 0 },
new double[] { 65.3, 32.0, 0 },
new double[] { 71.1, 32.1, 0 },
new double[] { 77.0, 32.5, 0 },
new double[] { 77.8, 32.9, 0 }
};
double[][] expected = new JaggedSingularValueDecomposition(value,
computeLeftSingularVectors: true,
computeRightSingularVectors: true,
autoTranspose: true).Inverse();
var target = new JaggedQrDecomposition(value);
double[][] actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, atol: 1e-4));
var targetMat = new QrDecomposition(value.ToMatrix());
double[][] actualMat = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actualMat, atol: 1e-4));
}
public void InverseTestNaN()
{
int n = 5;
var I = Jagged.Identity(n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
double[][] value = Jagged.Magic(n);
value[i][j] = double.NaN;
var target = new JaggedQrDecomposition(value);
var solution = target.Solve(I);
var inverse = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(solution, inverse));
}
}
}
public void InverseTestNaN()
{
int n = 5;
var I = Matrix.JaggedIdentity(n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
double[][] value = Matrix.JaggedMagic(n);
value[i][j] = double.NaN;
var target = new JaggedQrDecomposition(value);
var solution = target.Solve(I);
var inverse = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(solution, inverse));
}
}
}
public void InverseTest()
{
double[][] value =
{
new double[] { 2, -1, 0 },
new double[] { -1, 2, -1 },
new double[] { 0, -1, 2 }
};
double[][] expected =
{
new double[] { 0.7500, 0.5000, 0.2500},
new double[] { 0.5000, 1.0000, 0.5000},
new double[] { 0.2500, 0.5000, 0.7500},
};
var target = new JaggedQrDecomposition(value);
double[][] actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
target = new JaggedQrDecomposition(value.Transpose(), true);
actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
public void InverseTest1()
{
double[][] value =
{
new double[] { 41.9, 29.1, 1 },
new double[] { 43.4, 29.3, 1 },
new double[] { 43.9, 29.5, 0 },
new double[] { 44.5, 29.7, 0 },
new double[] { 47.3, 29.9, 0 },
new double[] { 47.5, 30.3, 0 },
new double[] { 47.9, 30.5, 0 },
new double[] { 50.2, 30.7, 0 },
new double[] { 52.8, 30.8, 0 },
new double[] { 53.2, 30.9, 0 },
new double[] { 56.7, 31.5, 0 },
new double[] { 57.0, 31.7, 0 },
new double[] { 63.5, 31.9, 0 },
new double[] { 65.3, 32.0, 0 },
new double[] { 71.1, 32.1, 0 },
new double[] { 77.0, 32.5, 0 },
new double[] { 77.8, 32.9, 0 }
};
double[][] expected = new JaggedSingularValueDecomposition(value,
computeLeftSingularVectors: true,
computeRightSingularVectors: true,
autoTranspose: true).Inverse();
var target = new JaggedQrDecomposition(value);
double[][] actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, atol: 1e-4));
var targetMat = new QrDecomposition(value.ToMatrix());
double[][] actualMat = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actualMat, atol: 1e-4));
}
public void SolveTransposeTest()
{
double[][] a =
{
new double[] { 2, 1, 4 },
new double[] { 6, 2, 2 },
new double[] { 0, 1, 6 },
};
double[][] b =
{
new double[] { 1, 0, 7 },
new double[] { 5, 2, 1 },
new double[] { 1, 5, 2 },
};
double[][] expected =
{
new double[] { 0.5062, 0.2813, 0.0875 },
new double[] { 0.1375, 1.1875, -0.0750 },
new double[] { 0.8063, -0.2188, 0.2875 },
};
double[][] actual = new JaggedQrDecomposition(b, true).SolveTranspose(a);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
var bX = b.Dot(expected);
var Xb = expected.Dot(b);
Assert.IsTrue(Matrix.IsEqual(Xb, a, atol: 1e-3));
Assert.IsFalse(Matrix.IsEqual(bX, a, atol: 1e-3));
double[,] actualMatrix = new QrDecomposition(b.ToMatrix(), true).SolveTranspose(a.ToMatrix());
Assert.IsTrue(Matrix.IsEqual(expected, actualMatrix, 0.001));
}
public void SolveTest()
{
double[][] value =
{
new double[] { 2, -1, 0 },
new double[] { -1, 2, -1 },
new double[] { 0, -1, 2 }
};
double[] b = { 1, 2, 3 };
double[] expected = { 2.5000, 4.0000, 3.5000 };
var target = new JaggedQrDecomposition(value);
double[] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
public void SolveTest3()
{
double[][] value =
{
new double[] { 41.9, 29.1, 1 },
new double[] { 43.4, 29.3, 1 },
new double[] { 43.9, 29.5, 0 },
new double[] { 44.5, 29.7, 0 },
new double[] { 47.3, 29.9, 0 },
new double[] { 47.5, 30.3, 0 },
new double[] { 47.9, 30.5, 0 },
new double[] { 50.2, 30.7, 0 },
new double[] { 52.8, 30.8, 0 },
new double[] { 53.2, 30.9, 0 },
new double[] { 56.7, 31.5, 0 },
new double[] { 57.0, 31.7, 0 },
new double[] { 63.5, 31.9, 0 },
new double[] { 65.3, 32.0, 0 },
new double[] { 71.1, 32.1, 0 },
new double[] { 77.0, 32.5, 0 },
new double[] { 77.8, 32.9, 0 }
};
double[][] b =
{
new double[] { 251.3 },
new double[] { 251.3 },
new double[] { 248.3 },
new double[] { 267.5 },
new double[] { 273.0 },
new double[] { 276.5 },
new double[] { 270.3 },
new double[] { 274.9 },
new double[] { 285.0 },
new double[] { 290.0 },
new double[] { 297.0 },
new double[] { 302.5 },
new double[] { 304.5 },
new double[] { 309.3 },
new double[] { 321.7 },
new double[] { 330.7 },
new double[] { 349.0 }
};
double[][] expected =
{
new double[] { 1.7315235669547167 },
new double[] { 6.25142110500275 },
new double[] { -5.0909763966987178 },
};
var target = new JaggedQrDecomposition(value);
double[][] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, atol: 1e-4));
var reconstruct = value.Dot(expected);
Assert.IsTrue(Matrix.IsEqual(reconstruct, b, rtol: 1e-1));
double[] b2 = b.GetColumn(0);
double[] expected2 = expected.GetColumn(0);
var target2 = new JaggedQrDecomposition(value);
double[] actual2 = target2.Solve(b2);
Assert.IsTrue(Matrix.IsEqual(expected2, actual2, atol: 1e-4));
var targetMat = new QrDecomposition(value.ToMatrix());
double[,] actualMat = targetMat.Solve(b.ToMatrix());
Assert.IsTrue(Matrix.IsEqual(expected, actualMat, atol: 1e-4));
var reconstructMat = value.ToMatrix().Dot(expected);
Assert.IsTrue(Matrix.IsEqual(reconstruct, b, rtol: 1e-1));
var targetMat2 = new QrDecomposition(value.ToMatrix());
Assert.IsTrue(Matrix.IsEqual(target2.Diagonal, targetMat2.Diagonal, atol: 1e-4));
Assert.IsTrue(Matrix.IsEqual(target2.OrthogonalFactor, targetMat2.OrthogonalFactor, atol: 1e-4));
Assert.IsTrue(Matrix.IsEqual(target2.UpperTriangularFactor, targetMat2.UpperTriangularFactor, atol: 1e-4));
double[] actualMat2 = targetMat2.Solve(b2);
Assert.IsTrue(Matrix.IsEqual(expected2, actualMat2, atol: 1e-4));
}
public void SolveTest2()
{
// Example from Lecture notes for MATHS 370: Advanced Numerical Methods
// http://www.math.auckland.ac.nz/~sharp/370/qr-solving.pdf
double[][] value =
{
new double[] { 1, 0, 0 },
new double[] { 1, 7, 49 },
new double[] { 1, 14, 196 },
new double[] { 1, 21, 441 },
new double[] { 1, 28, 784 },
new double[] { 1, 35, 1225 },
};
// Matrices
{
double[][] b =
{
new double[] { 4 },
new double[] { 1 },
new double[] { 0 },
new double[] { 0 },
new double[] { 2 },
new double[] { 5 },
};
double[][] expected =
{
new double[] { 3.9286 },
new double[] { -0.5031 },
new double[] { 0.0153 },
};
var target = new JaggedQrDecomposition(value);
double[][] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-4));
}
// Vectors
{
double[] b = { 4, 1, 0, 0, 2, 5 };
double[] expected = { 3.9286, -0.5031, 0.0153 };
var target = new JaggedQrDecomposition(value);
double[] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-4));
}
}
public void SolveTransposeTest()
{
double[][] a =
{
new double[] { 2, 1, 4 },
new double[] { 6, 2, 2 },
new double[] { 0, 1, 6 },
};
double[][] b =
{
new double[] { 1, 0, 7 },
new double[] { 5, 2, 1 },
new double[] { 1, 5, 2 },
};
double[][] expected =
{
new double[] { 0.5062, 0.2813, 0.0875 },
new double[] { 0.1375, 1.1875, -0.0750 },
new double[] { 0.8063, -0.2188, 0.2875 },
};
double[][] actual = new JaggedQrDecomposition(b, true).SolveTranspose(a);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
}
public void SolveTest3()
{
double[][] value =
{
new double[] { 41.9, 29.1, 1 },
new double[] { 43.4, 29.3, 1 },
new double[] { 43.9, 29.5, 0 },
new double[] { 44.5, 29.7, 0 },
new double[] { 47.3, 29.9, 0 },
new double[] { 47.5, 30.3, 0 },
new double[] { 47.9, 30.5, 0 },
new double[] { 50.2, 30.7, 0 },
new double[] { 52.8, 30.8, 0 },
new double[] { 53.2, 30.9, 0 },
new double[] { 56.7, 31.5, 0 },
new double[] { 57.0, 31.7, 0 },
new double[] { 63.5, 31.9, 0 },
new double[] { 65.3, 32.0, 0 },
new double[] { 71.1, 32.1, 0 },
new double[] { 77.0, 32.5, 0 },
new double[] { 77.8, 32.9, 0 }
};
double[][] b =
{
new double[] { 251.3 },
new double[] { 251.3 },
new double[] { 248.3 },
new double[] { 267.5 },
new double[] { 273.0 },
new double[] { 276.5 },
new double[] { 270.3 },
new double[] { 274.9 },
new double[] { 285.0 },
new double[] { 290.0 },
new double[] { 297.0 },
new double[] { 302.5 },
new double[] { 304.5 },
new double[] { 309.3 },
new double[] { 321.7 },
new double[] { 330.7 },
new double[] { 349.0 }
};
double[][] expected =
{
new double[] { 1.7315235669547167 },
new double[] { 6.25142110500275 },
new double[] { -5.0909763966987178 },
};
var target = new JaggedQrDecomposition(value);
double[][] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, atol: 1e-4));
var reconstruct = value.Dot(expected);
Assert.IsTrue(Matrix.IsEqual(reconstruct, b, rtol: 1e-1));
double[] b2 = b.GetColumn(0);
double[] expected2 = expected.GetColumn(0);
var target2 = new JaggedQrDecomposition(value);
double[] actual2 = target2.Solve(b2);
Assert.IsTrue(Matrix.IsEqual(expected2, actual2, atol: 1e-4));
var targetMat = new QrDecomposition(value.ToMatrix());
double[,] actualMat = targetMat.Solve(b.ToMatrix());
Assert.IsTrue(Matrix.IsEqual(expected, actualMat, atol: 1e-4));
var reconstructMat = value.ToMatrix().Dot(expected);
Assert.IsTrue(Matrix.IsEqual(reconstruct, b, rtol: 1e-1));
var targetMat2 = new QrDecomposition(value.ToMatrix());
double[] actualMat2 = targetMat2.Solve(b2);
Assert.IsTrue(Matrix.IsEqual(expected2, actualMat2, atol: 1e-4));
}