// Accord Math Library
// Accord.NET framework
// http://www.crsouza.com
//
// Modifications copyright � C�sar Souza, 2009-2010
// cesarsouza at gmail.com
//
// Original work copyright � Lutz Roeder, 2000
// Adapted from Mapack for COM and Jama routines
//
namespace Accord.Math.Decompositions
{
using System;
using Accord.Math;
/// <summary>
/// QR decomposition for a rectangular matrix.
/// </summary>
/// <remarks>
/// <para>
/// For an m-by-n matrix <c>A</c> with <c>m >= n</c>, the QR decomposition
/// is an m-by-n orthogonal matrix <c>Q</c> and an n-by-n upper triangular
/// matrix <c>R</c> so that <c>A = Q * R</c>.</para>
/// <para>
/// The QR decomposition always exists, even if the matrix does not have
/// full rank, so the constructor will never fail. The primary use of the
/// QR decomposition is in the least squares solution of nonsquare systems
/// of simultaneous linear equations.
/// This will fail if <see cref="FullRank"/> returns <see langword="false"/>.</para>
/// </remarks>
///
public sealed class QrDecomposition
{
private double[,] qr;
private double[] Rdiag;
/// <summary>Constructs a QR decomposition.</summary>
/// <param name="value">The matrix A to be decomposed.</param>
public QrDecomposition(double[,] value)
: this(value, false)
{
}
/// <summary>Constructs a QR decomposition.</summary>
/// <param name="value">The matrix A to be decomposed.</param>
/// <param name="transpose">True if the decomposition should be performed on
/// the transpose of A rather than A itself, false otherwise. Default is false.</param>
public QrDecomposition(double[,] value, bool transpose)
{
if (value == null)
{
throw new ArgumentNullException("value", "Matrix cannot be null.");
}
if ((!transpose && value.GetLength(0) < value.GetLength(1)) ||
(transpose && value.GetLength(1) < value.GetLength(0)))
{
throw new ArgumentException("Matrix has more columns than rows.", "value");
}
this.qr = transpose ? value.Transpose() : (double[,])value.Clone();
int rows = qr.GetLength(0);
int cols = qr.GetLength(1);
this.Rdiag = new double[cols];
for (int k = 0; k < cols; k++)
{
// Compute 2-norm of k-th column without under/overflow.
double nrm = 0;
for (int i = k; i < rows; i++)
{
nrm = Tools.Hypotenuse(nrm, qr[i, k]);
}
if (nrm != 0.0)
{
// Form k-th Householder vector.
if (qr[k, k] < 0)
{
nrm = -nrm;
}
for (int i = k; i < rows; i++)
{
qr[i, k] /= nrm;
}
qr[k, k] += 1.0;
// Apply transformation to remaining columns.
for (int j = k + 1; j < cols; j++)
{
double s = 0.0;
for (int i = k; i < rows; i++)
{
s += qr[i, k] * qr[i, j];
}
s = -s / qr[k, k];
for (int i = k; i < rows; i++)
{
qr[i, j] += s * qr[i, k];
}
}
}
this.Rdiag[k] = -nrm;
}
}
/// <summary>Least squares solution of <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>A matrix that minimized the two norm of <c>Q * R * X - B</c>.</returns>
/// <exception cref="T:System.ArgumentException">Matrix row dimensions must be the same.</exception>
/// <exception cref="T:System.InvalidOperationException">Matrix is rank deficient.</exception>
public double[,] Solve(double[,] value)
{
if (value == null)
throw new ArgumentNullException("value", "Matrix cannot be null.");
if (value.GetLength(0) != qr.GetLength(0))
throw new ArgumentException("Matrix row dimensions must agree.");
if (!this.FullRank)
throw new InvalidOperationException("Matrix is rank deficient.");
// Copy right hand side
int count = value.GetLength(1);
double[,] X = (double[,])value.Clone();
int m = qr.GetLength(0);
int n = qr.GetLength(1);
// Compute Y = transpose(Q)*B
for (int k = 0; k < n; k++)
{
for (int j = 0; j < count; j++)
{
double s = 0.0;
for (int i = k; i < m; i++)
s += qr[i, k] * X[i, j];
s = -s / qr[k, k];
for (int i = k; i < m; i++)
X[i, j] += s * qr[i, k];
}
}
// Solve R*X = Y;
for (int k = n - 1; k >= 0; k--)
{
for (int j = 0; j < count; j++)
X[k, j] /= Rdiag[k];
for (int i = 0; i < k; i++)
for (int j = 0; j < count; j++)
X[i, j] -= X[k, j] * qr[i, k];
}
double[,] r = new double[n, count];
for (int i = 0; i < n; i++)
for (int j = 0; j < count; j++)
r[i, j] = X[i, j];
return r;
}
/// <summary>Least squares solution of <c>X * A = B</c></summary>
/// <param name="value">Right-hand-side matrix with as many columns as <c>A</c> and any number of rows.</param>
/// <returns>A matrix that minimized the two norm of <c>X * Q * R - B</c>.</returns>
/// <exception cref="T:System.ArgumentException">Matrix column dimensions must be the same.</exception>
/// <exception cref="T:System.InvalidOperationException">Matrix is rank deficient.</exception>
public double[,] SolveTranspose(double[,] value)
{
if (value == null)
throw new ArgumentNullException("value", "Matrix cannot be null.");
if (value.GetLength(1) != qr.GetLength(0))
throw new ArgumentException("Matrix row dimensions must agree.");
if (!this.FullRank)
throw new InvalidOperationException("Matrix is rank deficient.");
// Copy right hand side
int count = value.GetLength(0);
double[,] X = value.Transpose();
int m = qr.GetLength(0);
int n = qr.GetLength(1);
// Compute Y = transpose(Q)*B
for (int k = 0; k < n; k++)
{
for (int j = 0; j < count; j++)
{
double s = 0.0;
for (int i = k; i < m; i++)
s += qr[i, k] * X[i, j];
s = -s / qr[k, k];
for (int i = k; i < m; i++)
X[i, j] += s * qr[i, k];
}
}
// Solve R*X = Y;
for (int k = n - 1; k >= 0; k--)
{
for (int j = 0; j < count; j++)
X[k, j] /= Rdiag[k];
for (int i = 0; i < k; i++)
for (int j = 0; j < count; j++)
X[i, j] -= X[k, j] * qr[i, k];
}
double[,] r = new double[count, n];
for (int i = 0; i < n; i++)
for (int j = 0; j < count; j++)
r[j, i] = X[i, j];
return r;
}
/// <summary>Least squares solution of <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>A matrix that minimized the two norm of <c>Q * R * X - B</c>.</returns>
/// <exception cref="T:System.ArgumentException">Matrix row dimensions must be the same.</exception>
/// <exception cref="T:System.InvalidOperationException">Matrix is rank deficient.</exception>
public double[] Solve(double[] value)
{
if (value == null)
throw new ArgumentNullException("value");
if (value.GetLength(0) != qr.GetLength(0))
throw new ArgumentException("Matrix row dimensions must agree.");
if (!this.FullRank)
throw new InvalidOperationException("Matrix is rank deficient.");
// Copy right hand side
double[] X = (double[])value.Clone();
int m = qr.GetLength(0);
int n = qr.GetLength(1);
// Compute Y = transpose(Q)*B
for (int k = 0; k < n; k++)
{
double s = 0.0;
for (int i = k; i < m; i++)
s += qr[i, k] * X[i];
s = -s / qr[k, k];
for (int i = k; i < m; i++)
X[i] += s * qr[i, k];
}
// Solve R*X = Y;
for (int k = n - 1; k >= 0; k--)
{
X[k] /= Rdiag[k];
for (int i = 0; i < k; i++)
X[i] -= X[k] * qr[i, k];
}
return X;
}
/// <summary>Shows if the matrix <c>A</c> is of full rank.</summary>
/// <value>The value is <see langword="true"/> if <c>R</c>, and hence <c>A</c>, has full rank.</value>
public bool FullRank
{
get
{
int columns = qr.GetLength(1);
for (int i = 0; i < columns; i++)
{
if (this.Rdiag[i] == 0)
{
return false;
}
}
return true;
}
}
/// <summary>Returns the upper triangular factor <c>R</c>.</summary>
public double[,] UpperTriangularFactor
{
get
{
int n = this.qr.GetLength(1);
double[,] x = new double[n, n];
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
if (i < j)
{
x[i, j] = qr[i, j];
}
else if (i == j)
{
x[i, j] = Rdiag[i];
}
else
{
x[i, j] = 0.0;
}
}
}
return x;
}
}
/// <summary>Returns the orthogonal factor <c>Q</c>.</summary>
public double[,] OrthogonalFactor
{
get
{
int rows = qr.GetLength(0);
int cols = qr.GetLength(1);
double[,] x = new double[rows, cols];
for (int k = cols - 1; k >= 0; k--)
{
for (int i = 0; i < rows; i++)
{
x[i, k] = 0.0;
}
x[k, k] = 1.0;
for (int j = k; j < cols; j++)
{
if (qr[k, k] != 0)
{
double s = 0.0;
for (int i = k; i < rows; i++)
{
s += qr[i, k] * x[i, j];
}
s = -s / qr[k, k];
for (int i = k; i < rows; i++)
{
x[i, j] += s * qr[i, k];
}
}
}
}
return x;
}
}
/// <summary>Returns the diagonal of <c>R</c>.</summary>
public double[] Diagonal
{
get { return Rdiag; }
}
/// <summary>Least squares solution of <c>A * X = I</c></summary>
public double[,] Inverse()
{
if (!this.FullRank)
throw new InvalidOperationException("Matrix is rank deficient.");
// Copy right hand side
int m = qr.GetLength(0);
int n = qr.GetLength(1);
double[,] X = new double[m, m];
// Compute Y = transpose(Q)
for (int k = n - 1; k >= 0; k--)
{
for (int i = 0; i < m; i++)
X[k, i] = 0.0;
X[k, k] = 1.0;
for (int j = k; j < n; j++)
{
if (qr[k, k] != 0)
{
double s = 0.0;
for (int i = k; i < m; i++)
s += qr[i, k] * X[j, i];
s = -s / qr[k, k];
for (int i = k; i < m; i++)
X[j, i] += s * qr[i, k];
}
}
}
// Solve R*X = Y;
for (int k = n - 1; k >= 0; k--)
{
for (int j = 0; j < m; j++)
X[k, j] /= Rdiag[k];
for (int i = 0; i < k; i++)
for (int j = 0; j < m; j++)
X[i, j] -= X[k, j] * qr[i, k];
}
return X;
}
}
}
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