mathnet-numerics/src/Numerics/LinearAlgebra/Factorization/LU.cs

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using System;

namespace MathNet.Numerics.LinearAlgebra.Factorization
{
    /// <summary>
    /// <para>A class which encapsulates the functionality of an LU factorization.</para>
    /// <para>For a matrix A, the LU factorization is a pair of lower triangular matrix L and
    /// upper triangular matrix U so that A = L*U.</para>
    /// <para>In the Math.Net implementation we also store a set of pivot elements for increased 
    /// numerical stability. The pivot elements encode a permutation matrix P such that P*A = L*U.</para>
    /// </summary>
    /// <remarks>
    /// The computation of the LU factorization is done at construction time.
    /// </remarks>
    /// <typeparam name="T">Supported data types are double, single, <see cref="Complex"/>, and <see cref="Complex32"/>.</typeparam>
    public abstract class LU<T> : ISolver<T>
        where T : struct, IEquatable<T>, IFormattable
    {
        static readonly T One = BuilderInstance<T>.Matrix.One;

        readonly Lazy<Matrix<T>> _lazyL;
        readonly Lazy<Matrix<T>> _lazyU;
        readonly Lazy<Permutation> _lazyP;

        protected readonly Matrix<T> Factors;
        protected readonly int[] Pivots;

        protected LU(Matrix<T> factors, int[] pivots)
        {
            Factors = factors;
            Pivots = pivots;

            _lazyL = new Lazy<Matrix<T>>(ComputeL);
            _lazyU = new Lazy<Matrix<T>>(Factors.UpperTriangle);
            _lazyP = new Lazy<Permutation>(() => Permutation.FromInversions(Pivots));
        }

        Matrix<T> ComputeL()
        {
            var result = Factors.LowerTriangle();
            for (var i = 0; i < result.RowCount; i++)
            {
                result.At(i, i, One);
            }
            return result;
        }

        /// <summary>
        /// Gets the lower triangular factor.
        /// </summary>
        public Matrix<T> L
        {
            get { return _lazyL.Value; }
        }

        /// <summary>
        /// Gets the upper triangular factor.
        /// </summary>
        public Matrix<T> U
        {
            get { return _lazyU.Value; }
        }

        /// <summary>
        /// Gets the permutation applied to LU factorization.
        /// </summary>
        public Permutation P
        {
            get { return _lazyP.Value; }
        }

        /// <summary>
        /// Gets the determinant of the matrix for which the LU factorization was computed.
        /// </summary>
        public abstract T Determinant { get; }

        /// <summary>
        /// Solves a system of linear equations, <b>AX = B</b>, with A LU factorized.
        /// </summary>
        /// <param name="input">The right hand side <see cref="Matrix{T}"/>, <b>B</b>.</param>
        /// <returns>The left hand side <see cref="Matrix{T}"/>, <b>X</b>.</returns>
        public virtual Matrix<T> Solve(Matrix<T> input)
        {
            var x = input.CreateMatrix(input.RowCount, input.ColumnCount);
            Solve(input, x);
            return x;
        }

        /// <summary>
        /// Solves a system of linear equations, <b>AX = B</b>, with A LU 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 abstract void Solve(Matrix<T> input, Matrix<T> result);

        /// <summary>
        /// Solves a system of linear equations, <b>Ax = b</b>, with A LU factorized.
        /// </summary>
        /// <param name="input">The right hand side vector, <b>b</b>.</param>
        /// <returns>The left hand side <see cref="Vector{T}"/>, <b>x</b>.</returns>
        public virtual Vector<T> Solve(Vector<T> input)
        {
            var x = input.CreateVector(input.Count);
            Solve(input, x);
            return x;
        }

        /// <summary>
        /// Solves a system of linear equations, <b>Ax = b</b>, with A LU 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 abstract void Solve(Vector<T> input, Vector<T> result);

        /// <summary>
        /// Returns the inverse of this matrix. The inverse is calculated using LU decomposition.
        /// </summary>
        /// <returns>The inverse of this matrix.</returns>
        public abstract Matrix<T> Inverse();
    }
}