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namespace Mapack
{
using System;
/// <summary>LU decomposition of a rectangular matrix.</summary>
/// <remarks>
/// For an m-by-n matrix <c>A</c> with m >= n, the LU decomposition is an m-by-n
/// unit lower triangular matrix <c>L</c>, an n-by-n upper triangular matrix <c>U</c>,
/// and a permutation vector <c>piv</c> of length m so that <c>A(piv)=L*U</c>.
/// If m < n, then <c>L</c> is m-by-m and <c>U</c> is m-by-n.
/// The LU decompostion with pivoting always exists, even if the matrix is
/// singular, so the constructor will never fail. The primary use of the
/// LU decomposition is in the solution of square systems of simultaneous
/// linear equations. This will fail if <see cref="NonSingular"/> returns <see langword="false"/>.
/// </remarks>
public class LuDecomposition
{
private Matrix LU;
private int pivotSign;
private int[] pivotVector;
/// <summary>Construct a LU decomposition.</summary>
public LuDecomposition(Matrix value)
{
if (value == null)
{
throw new ArgumentNullException("value");
}
this.LU = (Matrix) value.Clone();
double[][] lu = LU.Array;
int rows = value.Rows;
int columns = value.Columns;
pivotVector = new int[rows];
for (int i = 0; i < rows; i++)
{
pivotVector[i] = i;
}
pivotSign = 1;
double[] LUrowi;
double[] LUcolj = new double[rows];
// Outer loop.
for (int j = 0; j < columns; j++)
{
// Make a copy of the j-th column to localize references.
for (int i = 0; i < rows; i++)
{
LUcolj[i] = lu[i][j];
}
// Apply previous transformations.
for (int i = 0; i < rows; i++)
{
LUrowi = lu[i];
// Most of the time is spent in the following dot product.
int kmax = Math.Min(i,j);
double s = 0.0;
for (int k = 0; k < kmax; k++)
{
s += LUrowi[k]*LUcolj[k];
}
LUrowi[j] = LUcolj[i] -= s;
}
// Find pivot and exchange if necessary.
int p = j;
for (int i = j+1; i < rows; i++)
{
if (Math.Abs(LUcolj[i]) > Math.Abs(LUcolj[p]))
{
p = i;
}
}
if (p != j)
{
for (int k = 0; k < columns; k++)
{
double t = lu[p][k];
lu[p][k] = lu[j][k];
lu[j][k] = t;
}
int v = pivotVector[p];
pivotVector[p] = pivotVector[j];
pivotVector[j] = v;
pivotSign = - pivotSign;
}
// Compute multipliers.
if (j < rows & lu[j][j] != 0.0)
{
for (int i = j+1; i < rows; i++)
{
lu[i][j] /= lu[j][j];
}
}
}
}
/// <summary>Returns if the matrix is non-singular.</summary>
public bool NonSingular
{
get
{
for (int j = 0; j < LU.Columns; j++)
if (LU[j, j] == 0)
return false;
return true;
}
}
/// <summary>Returns the determinant of the matrix.</summary>
public double Determinant
{
get
{
if (LU.Rows != LU.Columns) throw new ArgumentException("Matrix must be square.");
double determinant = (double) pivotSign;
for (int j = 0; j < LU.Columns; j++)
determinant *= LU[j, j];
return determinant;
}
}
/// <summary>Returns the lower triangular factor <c>L</c> with <c>A=LU</c>.</summary>
public Matrix LowerTriangularFactor
{
get
{
int rows = LU.Rows;
int columns = LU.Columns;
Matrix X = new Matrix(rows, columns);
for (int i = 0; i < rows; i++)
for (int j = 0; j < columns; j++)
if (i > j)
X[i,j] = LU[i,j];
else if (i == j)
X[i,j] = 1.0;
else
X[i,j] = 0.0;
return X;
}
}
/// <summary>Returns the lower triangular factor <c>L</c> with <c>A=LU</c>.</summary>
public Matrix UpperTriangularFactor
{
get
{
int rows = LU.Rows;
int columns = LU.Columns;
Matrix X = new Matrix(rows, columns);
for (int i = 0; i < rows; i++)
for (int j = 0; j < columns; j++)
if (i <= j)
X[i,j] = LU[i,j];
else
X[i,j] = 0.0;
return X;
}
}
/// <summary>Returns the pivot permuation vector.</summary>
public double[] PivotPermutationVector
{
get
{
int rows = LU.Rows;
double[] p = new double[rows];
for (int i = 0; i < rows; i++)
{
p[i] = (double) this.pivotVector[i];
}
return p;
}
}
/// <summary>Solves a set of equation systems of type <c>A * X = B</c>.</summary>
/// <param name="value">Right hand side matrix with as many rows as <c>A</c> and any number of columns.</param>
/// <returns>Matrix <c>X</c> so that <c>L * U * X = B</c>.</returns>
public Matrix Solve(Matrix value)
{
if (value == null)
{
throw new ArgumentNullException("value");
}
if (value.Rows != this.LU.Rows)
{
throw new ArgumentException("Invalid matrix dimensions.", "value");
}
if (!this.NonSingular)
{
throw new InvalidOperationException("Matrix is singular");
}
// Copy right hand side with pivoting
int count = value.Columns;
Matrix X = value.Submatrix(pivotVector, 0, count-1);
int rows = LU.Rows;
int columns = LU.Columns;
double[][] lu = LU.Array;
// Solve L*Y = B(piv,:)
for (int k = 0; k < columns; k++)
{
for (int i = k + 1; i < columns; i++)
{
for (int j = 0; j < count; j++)
{
X[i,j] -= X[k,j] * lu[i][k];
}
}
}
// Solve U*X = Y;
for (int k = columns - 1; k >= 0; k--)
{
for (int j = 0; j < count; j++)
{
X[k,j] /= lu[k][k];
}
for (int i = 0; i < k; i++)
{
for (int j = 0; j < count; j++)
{
X[i,j] -= X[k,j] * lu[i][k];
}
}
}
return X;
}
}
}
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