public void QrDecompositionConstructorTest()
{
double[,] value =
{
{ 2, -1, 0 },
{ -1, 2, -1 },
{ 0, -1, 2 }
};
QrDecomposition target = new QrDecomposition(value);
// Decomposition Identity
var Q = target.OrthogonalFactor;
var QQt = Q.Multiply(Q.Transpose());
Assert.IsTrue(Matrix.IsEqual(QQt, Matrix.Identity(3), 0.0000001));
// Linear system solving
double[,] B = Matrix.ColumnVector(new double[] { 1, 2, 3 });
double[,] expected = Matrix.ColumnVector(new double[] { 2.5, 4.0, 3.5 });
double[,] actual = target.Solve(B);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
public void InverseTestNaN()
{
int n = 5;
var I = Matrix.Identity(n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
double[,] value = Matrix.Magic(n);
value[i, j] = double.NaN;
var target = new QrDecomposition(value);
var solution = target.Solve(I);
var inverse = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(solution, inverse));
}
}
}
/// <summary>QR decomposition.</summary>
protected static void qr(double[,] m, out double[,] Q, out double[,] R, out double[] d)
{
var qr = new QrDecomposition(m);
Q = qr.OrthogonalFactor;
R = qr.UpperTriangularFactor;
d = qr.Diagonal;
}
public void SolveTransposeTest()
{
double[,] a =
{
{ 2, 1, 4 },
{ 6, 2, 2 },
{ 0, 1, 6 },
};
double[,] b =
{
{ 1, 0, 7 },
{ 5, 2, 1 },
{ 1, 5, 2 },
};
double[,] expected =
{
{ 0.5062, 0.2813, 0.0875 },
{ 0.1375, 1.1875, -0.0750 },
{ 0.8063, -0.2188, 0.2875 },
};
double[,] actual = new QrDecomposition(b, true).SolveTranspose(a);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
}
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 },
};
QrDecomposition target = new QrDecomposition(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 };
QrDecomposition target = new QrDecomposition(value);
double[] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-4));
}
}
public void SolveTest()
{
double[,] value =
{
{ 2, -1, 0 },
{ -1, 2, -1 },
{ 0, -1, 2 }
};
double[] b = { 1, 2, 3 };
double[] expected = { 2.5000, 4.0000, 3.5000 };
QrDecomposition target = new QrDecomposition(value);
double[] actual = target.Solve(b);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
public void InverseTest()
{
double[,] value =
{
{ 2, -1, 0 },
{ -1, 2, -1 },
{ 0, -1, 2 }
};
double[,] expected =
{
{ 0.7500, 0.5000, 0.2500},
{ 0.5000, 1.0000, 0.5000},
{ 0.2500, 0.5000, 0.7500},
};
QrDecomposition target = new QrDecomposition(value);
double[,] actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
target = new QrDecomposition(value.Transpose(), true);
actual = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.0000000000001));
}
static void Main(string[] args)
{
Console.WriteLine("Please take a look at the source for examples!");
Console.ReadKey();
#region 1. Declaring matrices
// 1.1 Using standard .NET declaration
double[,] A =
{
{1, 2, 3},
{6, 2, 0},
{0, 0, 1}
};
double[,] B =
{
{2, 0, 0},
{0, 2, 0},
{0, 0, 2}
};
{
// 1.2 Using Accord extension methods
double[,] Bi = Matrix.Identity(3).Multiply(2);
double[,] Bj = Matrix.Diagonal(3, 2.0); // both are equal to B
// 1.2 Using Accord extension methods with implicit typing
var I = Matrix.Identity(3);
}
#endregion
#region 2. Matrix Operations
{
// 2.1 Addition
var C = A.Add(B);
// 2.2 Subtraction
var D = A.Subtract(B);
// 2.3 Multiplication
{
// 2.3.1 By a scalar
var halfM = A.Multiply(0.5);
// 2.3.2 By a vector
double[] m = A.Multiply(new double[] { 1, 2, 3 });
// 2.3.3 By a matrix
var M = A.Multiply(B);
// 2.4 Transposing
var At = A.Transpose();
}
}
// 2.5 Elementwise operations
// 2.5.1 Elementwise multiplication
A.ElementwiseMultiply(B); // A.*B
// 2.5.1 Elementwise division
A.ElementwiseDivide(B); // A./B
#endregion
#region 3. Matrix characteristics
{
// 3.1 Calculating the determinant
double det = A.Determinant();
// 3.2 Calculating the trace
double tr = A.Trace();
// 3.3 Computing the sum vector
{
double[] sumVector = A.Sum();
// 3.3.1 Computing the total sum of elements
double sum = sumVector.Sum();
// 3.3.2 Computing the sum along the rows
sumVector = A.Sum(0); // Equivalent to Octave's sum(A, 1)
// 3.3.2 Computing the sum along the columns
sumVector = A.Sum(1); // Equivalent to Octave's sum(A, 2)
}
}
#endregion
#region 4. Linear Algebra
{
// 4.1 Computing the inverse
var invA = A.Inverse();
// 4.2 Computing the pseudo-inverse
var pinvA = A.PseudoInverse();
// 4.3 Solving a linear system (Ax = B)
var x = A.Solve(B);
}
#endregion
#region 5. Special operators
{
// 5.1 Finding the indices of elements
double[] v = { 5, 2, 2, 7, 1, 0 };
int[] idx = v.Find(e => e > 2); // finding the index of every element in v higher than 2.
// 5.2 Selecting elements by index
double[] u = v.Submatrix(idx); // u is { 5, 7 }
// 5.3 Converting between different matrix representations
double[][] jaggedA = A.ToArray(); // from multidimensional to jagged array
// 5.4 Extracting a column or row from the matrix
double[] a = A.GetColumn(0); // retrieves the first column
double[] b = B.GetRow(1); // retrieves the second row
// 5.5 Taking the absolute of a matrix
var absA = A.Abs();
// 5.6 Applying some function to every element
var newv = v.Apply(e => e + 1);
}
#endregion
#region 7. Vector operations
{
double[] u = { 1, 2, 3 };
double[] v = { 4, 5, 6 };
var w1 = u.InnerProduct(v);
var w2 = u.OuterProduct(v);
var w3 = u.CartesianProduct(v);
double[] m = { 1, 2, 3, 4 };
double[,] M = Matrix.Reshape(m, 2, 2);
}
#endregion
#region Decompositions
{
// Singular value decomposition
{
SingularValueDecomposition svd = new SingularValueDecomposition(A);
var U = svd.LeftSingularVectors;
var S = svd.Diagonal;
var V = svd.RightSingularVectors;
}
// or (please see documentation for details)
{
SingularValueDecomposition svd = new SingularValueDecomposition(A.Transpose());
var U = svd.RightSingularVectors;
var S = svd.Diagonal;
var V = svd.LeftSingularVectors;
}
// Eigenvalue decomposition
{
EigenvalueDecomposition eig = new EigenvalueDecomposition(A);
var V = eig.Eigenvectors;
var D = eig.DiagonalMatrix;
}
// QR decomposition
{
QrDecomposition qr = new QrDecomposition(A);
var Q = qr.OrthogonalFactor;
var R = qr.UpperTriangularFactor;
}
// Cholesky decomposition
{
CholeskyDecomposition chol = new CholeskyDecomposition(A);
var R = chol.LeftTriangularFactor;
}
// LU decomposition
{
LuDecomposition lu = new LuDecomposition(A);
var L = lu.LowerTriangularFactor;
var U = lu.UpperTriangularFactor;
}
}
#endregion
}
/// <summary>
/// Creates a new object that is a copy of the current instance.
/// </summary>
/// <returns>
/// A new object that is a copy of this instance.
/// </returns>
public object Clone()
{
var clone = new QrDecomposition();
clone.qr = (double[,])qr.Clone();
clone.Rdiag = (double[])Rdiag.Clone();
return clone;
}
private double regress(double[][] inputs, double[] outputs, out double[,] designMatrix, bool robust)
{
if (inputs.Length != outputs.Length)
throw new ArgumentException("Number of input and output samples does not match", "outputs");
int parameters = Inputs;
int rows = inputs.Length; // number of instance points
int cols = Inputs; // dimension of each point
for (int i = 0; i < inputs.Length; i++)
{
if (inputs[i].Length != parameters)
{
throw new DimensionMismatchException("inputs", String.Format(
"Input vectors should have length {0}. The row at index {1} of the" +
" inputs matrix has length {2}.", parameters, i, inputs[i].Length));
}
}
ISolverMatrixDecomposition<double> solver;
// Create the problem's design matrix. If we
// have to add an intercept term, add a new
// extra column at the end and fill with 1s.
if (!addIntercept)
{
// Just copy values over
designMatrix = new double[rows, cols];
for (int i = 0; i < inputs.Length; i++)
for (int j = 0; j < inputs[i].Length; j++)
designMatrix[i, j] = inputs[i][j];
}
else
{
// Add an intercept term
designMatrix = new double[rows, cols + 1];
for (int i = 0; i < inputs.Length; i++)
{
for (int j = 0; j < inputs[i].Length; j++)
designMatrix[i, j] = inputs[i][j];
designMatrix[i, cols] = 1;
}
}
// Check if we have an overdetermined or underdetermined
// system to select an appropriate matrix solver method.
if (robust || cols >= rows)
{
// We have more variables than equations, an
// underdetermined system. Solve using a SVD:
solver = new SingularValueDecomposition(designMatrix,
computeLeftSingularVectors: true,
computeRightSingularVectors: true,
autoTranspose: true);
}
else
{
// We have more equations than variables, an
// overdetermined system. Solve using the QR:
solver = new QrDecomposition(designMatrix);
}
// Solve V*C = B to find C (the coefficients)
coefficients = solver.Solve(outputs);
// Calculate Sum-Of-Squares error
double error = 0.0;
double e;
for (int i = 0; i < outputs.Length; i++)
{
e = outputs[i] - Compute(inputs[i]);
error += e * e;
}
return error;
}
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 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 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));
}