/// <summary>
/// Solves a linear equation system.
/// </summary>
/// <param name="decomposition">The LU decomposition.</param>
/// <param name="b">The vector.</param>
/// <returns>The vector containing the unknown variables of the equation.</returns>
private static Vector SolveEquation(LUDecomposition decomposition, Vector b)
{
// P*b
Vector pb = decomposition.P * b;
// L*y = P*b with forward substitution
Vector y = new Vector(pb.Size);
for (Int32 rowIndex = 0; rowIndex < decomposition.L.NumberOfRows; ++rowIndex)
{
y[rowIndex] = pb[rowIndex];
for (Int32 columnIndex = 0; columnIndex < rowIndex; ++columnIndex)
{
y[rowIndex] -= decomposition.L[rowIndex, columnIndex] * y[columnIndex];
}
y[rowIndex] /= decomposition.L[rowIndex, rowIndex] == 0 ? 1 : decomposition.L[rowIndex, rowIndex];
}
// U*x = y with back substitution
Vector x = new Vector(y.Size);
for (Int32 rowIndex = x.Size - 1; rowIndex >= 0; --rowIndex)
{
x[rowIndex] = y[rowIndex];
for (Int32 columnIndex = rowIndex + 1; columnIndex < decomposition.U.NumberOfRows; ++columnIndex)
{
x[rowIndex] -= decomposition.U[rowIndex, columnIndex] * x[columnIndex];
}
x[rowIndex] /= decomposition.U[rowIndex, rowIndex] == 0 ? 1 : decomposition.U[rowIndex, rowIndex];
}
return(x);
}
/// <summary>
/// Computes the determinant of the matrix.
/// </summary>
/// <param name="matrix">The matrix.</param>
/// <returns>The determinant of the matrix.</returns>
/// <exception cref="System.ArgumentNullException">The matrix is null.</exception>
/// <exception cref="System.ArgumentException">The matrix is not square.</exception>
public static Double Determinant(this Matrix matrix)
{
if (matrix == null)
{
throw new ArgumentNullException(nameof(matrix));
}
if (matrix.NumberOfRows != matrix.NumberOfColumns)
{
throw new ArgumentException(NumericsMessages.MatrixIsNotSquare, nameof(matrix));
}
if (matrix.NumberOfRows == 0)
{
return(0);
}
if (matrix.NumberOfRows == 1)
{
return(matrix[0, 0]);
}
if (matrix.NumberOfRows == 2)
{
return(matrix[0, 0] * matrix[1, 1] - matrix[0, 1] * matrix[1, 0]);
}
return(LUDecomposition.ComputeDeterminant(matrix));
}
/// <summary>
/// Computes the determinant of the specified matrix.
/// </summary>
/// <param name="matrix">The matrix.</param>
/// <returns>The determinant of the specified matrix.</returns>
/// <exception cref="System.ArgumentNullException">The matrix is null.</exception>
/// <exception cref="System.ArgumentException">The matrix is not square.</exception>
public static Double ComputeDeterminant(Matrix matrix)
{
LUDecomposition decomposition = new LUDecomposition(matrix);
decomposition.Compute();
return(decomposition.Determinant);
}
/// <summary>
/// Decomposes the specified matrix.
/// </summary>
/// <param name="matrix">The matrix.</param>
/// <returns>The decomposed (LU) matrix.</returns>
/// <exception cref="System.ArgumentNullException">The matrix is null.</exception>
/// <exception cref="System.ArgumentException">The matrix is not square.</exception>
public static Matrix Decompose(Matrix matrix)
{
LUDecomposition decomposition = new LUDecomposition(matrix);
decomposition.Compute();
return(decomposition.LU);
}
/// <summary>
/// Decomposes the specified matrix.
/// </summary>
/// <param name="matrix">The matrix to decompose.</param>
/// <param name="l">The L (lower triangular) matrix.</param>
/// <param name="u">The U (upper triangular) matrix.</param>
/// <exception cref="System.ArgumentNullException">The matrix is null.</exception>
/// <exception cref="System.ArgumentException">The matrix is not square.</exception>
public static void Decompose(Matrix matrix, out Matrix l, out Matrix u)
{
LUDecomposition decomposition = new LUDecomposition(matrix);
decomposition.Compute();
l = decomposition.L;
u = decomposition.U;
}
/// <summary>
/// Solves a linear equation system.
/// </summary>
/// <param name="a">The left side of the equation represented by a matrix.</param>
/// <param name="b">The right side of the equation represented by a vector.</param>
/// <returns>The vector containing the unknown variables of the equation.</returns>
/// <exception cref="System.ArgumentNullException">
/// The matrix is null.
/// or
/// The vector is null.
/// </exception>
/// <exception cref="System.ArgumentException">
/// The matrix is not square.
/// or
/// The size of the matrix does not match the size of the vector.
/// </exception>
public static Vector SolveEquation(Matrix a, Vector b)
{
if (a == null)
{
throw new ArgumentNullException(nameof(a));
}
if (b == null)
{
throw new ArgumentNullException(nameof(b));
}
if (!a.IsSquare)
{
throw new ArgumentException(NumericsMessages.MatrixIsNotSquare, nameof(a));
}
if (a.NumberOfRows != b.Size)
{
throw new ArgumentException(NumericsMessages.MatrixSizeDoesNotMatchVector, nameof(b));
}
LUDecomposition decomposition = new LUDecomposition(a);
decomposition.Compute();
return(SolveEquation(decomposition, b));
}
/// <summary>
/// Inverts the specified matrix.
/// </summary>
/// <param name="matrix">The matrix.</param>
/// <returns>The inverted matrix.</returns>
/// <exception cref="System.ArgumentNullException">The matrix is null.</exception>
/// <exception cref="System.ArgumentException">The matrix is not invertible.</exception>
public static Matrix Invert(Matrix matrix)
{
if (matrix == null)
{
throw new ArgumentNullException(nameof(matrix));
}
if (matrix.All(value => value == 0))
{
throw new ArgumentException(NumericsMessages.MatrixIsNotInvertible, nameof(matrix));
}
LUDecomposition decomposition = new LUDecomposition(matrix);
decomposition.Compute();
if (decomposition.Determinant == 0)
{
throw new ArgumentException(NumericsMessages.MatrixIsNotInvertible, nameof(matrix));
}
Matrix inverse = new Matrix(matrix.NumberOfRows, matrix.NumberOfColumns);
for (Int32 columnIndex = 0; columnIndex < matrix.NumberOfColumns; columnIndex++)
{
Vector b = VectorFactory.CreateUnitVector(matrix.NumberOfRows, columnIndex);
Vector y = SolveEquation(decomposition, b);
for (Int32 rowIndex = 0; rowIndex < inverse.NumberOfRows; ++rowIndex)
{
inverse[rowIndex, columnIndex] = y[rowIndex];
}
}
return(inverse);
}
/// <summary>
/// Inverts the specified matrix.
/// </summary>
/// <param name="matrix">The matrix.</param>
/// <returns>The inverse matrix.</returns>
/// <exception cref="System.ArgumentNullException">The matrix is null.</exception>
/// <exception cref="System.ArgumentException">The matrix is not invertible.</exception>
public static Matrix Invert(this Matrix matrix)
{
return(LUDecomposition.Invert(matrix));
}