// <copyright file="UserGramSchmidt.cs" company="Math.NET">
|
// Math.NET Numerics, part of the Math.NET Project
|
// http://numerics.mathdotnet.com
|
// http://github.com/mathnet/mathnet-numerics
|
//
|
// Copyright (c) 2009-2013 Math.NET
|
//
|
// Permission is hereby granted, free of charge, to any person
|
// obtaining a copy of this software and associated documentation
|
// files (the "Software"), to deal in the Software without
|
// restriction, including without limitation the rights to use,
|
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
// copies of the Software, and to permit persons to whom the
|
// Software is furnished to do so, subject to the following
|
// conditions:
|
//
|
// The above copyright notice and this permission notice shall be
|
// included in all copies or substantial portions of the Software.
|
//
|
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
// OTHER DEALINGS IN THE SOFTWARE.
|
// </copyright>
|
|
using System;
|
|
namespace IStation.Numerics.LinearAlgebra.Double.Factorization
|
{
|
/// <summary>
|
/// <para>A class which encapsulates the functionality of the QR decomposition Modified Gram-Schmidt Orthogonalization.</para>
|
/// <para>Any real square matrix A may be decomposed as A = QR where Q is an orthogonal mxn matrix and R is an nxn upper triangular matrix.</para>
|
/// </summary>
|
/// <remarks>
|
/// The computation of the QR decomposition is done at construction time by modified Gram-Schmidt Orthogonalization.
|
/// </remarks>
|
internal sealed class UserGramSchmidt : GramSchmidt
|
{
|
/// <summary>
|
/// Initializes a new instance of the <see cref="UserGramSchmidt"/> class. This object creates an orthogonal matrix
|
/// using the modified Gram-Schmidt method.
|
/// </summary>
|
/// <param name="matrix">The matrix to factor.</param>
|
/// <exception cref="ArgumentNullException">If <paramref name="matrix"/> is <c>null</c>.</exception>
|
/// <exception cref="ArgumentException">If <paramref name="matrix"/> row count is less then column count</exception>
|
/// <exception cref="ArgumentException">If <paramref name="matrix"/> is rank deficient</exception>
|
public static UserGramSchmidt Create(Matrix<double> matrix)
|
{
|
if (matrix.RowCount < matrix.ColumnCount)
|
{
|
throw Matrix.DimensionsDontMatch<ArgumentException>(matrix);
|
}
|
|
var q = matrix.Clone();
|
var r = Matrix<double>.Build.SameAs(matrix, matrix.ColumnCount, matrix.ColumnCount, fullyMutable: true);
|
|
for (var k = 0; k < q.ColumnCount; k++)
|
{
|
var norm = q.Column(k).L2Norm();
|
if (norm == 0.0)
|
{
|
throw new ArgumentException("Matrix must not be rank deficient.");
|
}
|
|
r.At(k, k, norm);
|
for (var i = 0; i < q.RowCount; i++)
|
{
|
q.At(i, k, q.At(i, k) / norm);
|
}
|
|
for (var j = k + 1; j < q.ColumnCount; j++)
|
{
|
var dot = q.Column(k).DotProduct(q.Column(j));
|
r.At(k, j, dot);
|
for (var i = 0; i < q.RowCount; i++)
|
{
|
var value = q.At(i, j) - (q.At(i, k) * dot);
|
q.At(i, j, value);
|
}
|
}
|
}
|
|
return new UserGramSchmidt(q, r);
|
}
|
|
UserGramSchmidt(Matrix<double> q, Matrix<double> rFull)
|
: base(q, rFull)
|
{
|
}
|
|
/// <summary>
|
/// Solves a system of linear equations, <b>AX = B</b>, with A QR factorized.
|
/// </summary>
|
/// <param name="input">The right hand side <see cref="Matrix{T}"/>, <b>B</b>.</param>
|
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>X</b>.</param>
|
public override void Solve(Matrix<double> input, Matrix<double> result)
|
{
|
// The solution X should have the same number of columns as B
|
if (input.ColumnCount != result.ColumnCount)
|
{
|
throw new ArgumentException("Matrix column dimensions must agree.");
|
}
|
|
// The dimension compatibility conditions for X = A\B require the two matrices A and B to have the same number of rows
|
if (Q.RowCount != input.RowCount)
|
{
|
throw new ArgumentException("Matrix row dimensions must agree.");
|
}
|
|
// The solution X row dimension is equal to the column dimension of A
|
if (Q.ColumnCount != result.RowCount)
|
{
|
throw new ArgumentException("Matrix column dimensions must agree.");
|
}
|
|
var inputCopy = input.Clone();
|
|
// Compute Y = transpose(Q)*B
|
var column = new double[Q.RowCount];
|
for (var j = 0; j < input.ColumnCount; j++)
|
{
|
for (var k = 0; k < Q.RowCount; k++)
|
{
|
column[k] = inputCopy.At(k, j);
|
}
|
|
for (var i = 0; i < Q.ColumnCount; i++)
|
{
|
double s = 0;
|
for (var k = 0; k < Q.RowCount; k++)
|
{
|
s += Q.At(k, i) * column[k];
|
}
|
|
inputCopy.At(i, j, s);
|
}
|
}
|
|
// Solve R*X = Y;
|
for (var k = Q.ColumnCount - 1; k >= 0; k--)
|
{
|
for (var j = 0; j < input.ColumnCount; j++)
|
{
|
inputCopy.At(k, j, inputCopy.At(k, j) / FullR.At(k, k));
|
}
|
|
for (var i = 0; i < k; i++)
|
{
|
for (var j = 0; j < input.ColumnCount; j++)
|
{
|
inputCopy.At(i, j, inputCopy.At(i, j) - (inputCopy.At(k, j) * FullR.At(i, k)));
|
}
|
}
|
}
|
|
for (var i = 0; i < FullR.ColumnCount; i++)
|
{
|
for (var j = 0; j < input.ColumnCount; j++)
|
{
|
result.At(i, j, inputCopy.At(i, j));
|
}
|
}
|
}
|
|
/// <summary>
|
/// Solves a system of linear equations, <b>Ax = b</b>, with A QR factorized.
|
/// </summary>
|
/// <param name="input">The right hand side vector, <b>b</b>.</param>
|
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>x</b>.</param>
|
public override void Solve(Vector<double> input, Vector<double> result)
|
{
|
// Ax=b where A is an m x n matrix
|
// Check that b is a column vector with m entries
|
if (Q.RowCount != input.Count)
|
{
|
throw new ArgumentException("All vectors must have the same dimensionality.");
|
}
|
|
// Check that x is a column vector with n entries
|
if (Q.ColumnCount != result.Count)
|
{
|
throw Matrix.DimensionsDontMatch<ArgumentException>(Q, result);
|
}
|
|
var inputCopy = input.Clone();
|
|
// Compute Y = transpose(Q)*B
|
var column = new double[Q.RowCount];
|
for (var k = 0; k < Q.RowCount; k++)
|
{
|
column[k] = inputCopy[k];
|
}
|
|
for (var i = 0; i < Q.ColumnCount; i++)
|
{
|
double s = 0;
|
for (var k = 0; k < Q.RowCount; k++)
|
{
|
s += Q.At(k, i) * column[k];
|
}
|
|
inputCopy[i] = s;
|
}
|
|
// Solve R*X = Y;
|
for (var k = Q.ColumnCount - 1; k >= 0; k--)
|
{
|
inputCopy[k] /= FullR.At(k, k);
|
for (var i = 0; i < k; i++)
|
{
|
inputCopy[i] -= inputCopy[k] * FullR.At(i, k);
|
}
|
}
|
|
for (var i = 0; i < FullR.ColumnCount; i++)
|
{
|
result[i] = inputCopy[i];
|
}
|
}
|
}
|
}
|
