mathnet-numerics/src/Numerics/LinearAlgebra/Complex/DiagonalMatrix.cs

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using MathNet.Numerics.Distributions;
using MathNet.Numerics.LinearAlgebra.Storage;
using MathNet.Numerics.Properties;
using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;

namespace MathNet.Numerics.LinearAlgebra.Complex
{

#if NOSYSNUMERICS
    using Numerics;
#else
    using System.Numerics;
#endif

    /// <summary>
    /// A matrix type for diagonal matrices.
    /// </summary>
    /// <remarks>
    /// Diagonal matrices can be non-square matrices but the diagonal always starts
    /// at element 0,0. A diagonal matrix will throw an exception if non diagonal
    /// entries are set. The exception to this is when the off diagonal elements are
    /// 0.0 or NaN; these settings will cause no change to the diagonal matrix.
    /// </remarks>
    [Serializable]
    [DebuggerDisplay("DiagonalMatrix {RowCount}x{ColumnCount}-Complex")]
    public class DiagonalMatrix : Matrix
    {
        readonly DiagonalMatrixStorage<Complex> _storage;

        /// <summary>
        /// Gets the matrix's data.
        /// </summary>
        /// <value>The matrix's data.</value>
        readonly Complex[] _data;

        /// <summary>
        /// Create a new diagonal matrix straight from an initialized matrix storage instance.
        /// The storage is used directly without copying.
        /// Intended for advanced scenarios where you're working directly with
        /// storage for performance or interop reasons.
        /// </summary>
        public DiagonalMatrix(DiagonalMatrixStorage<Complex> storage)
            : base(storage)
        {
            _storage = storage;
            _data = _storage.Data;
        }

        /// <summary>
        /// Create a new square diagonal matrix with the given number of rows and columns.
        /// All cells of the matrix will be initialized to zero.
        /// Zero-length matrices are not supported.
        /// </summary>
        /// <exception cref="ArgumentException">If the order is less than one.</exception>
        public DiagonalMatrix(int order)
            : this(new DiagonalMatrixStorage<Complex>(order, order))
         {
         }

        /// <summary>
        /// Create a new diagonal matrix with the given number of rows and columns.
        /// All cells of the matrix will be initialized to zero.
        /// Zero-length matrices are not supported.
        /// </summary>
        /// <exception cref="ArgumentException">If the row or column count is less than one.</exception>
        public DiagonalMatrix(int rows, int columns)
            : this(new DiagonalMatrixStorage<Complex>(rows, columns))
        {
        }

        /// <summary>
        /// Create a new diagonal matrix with the given number of rows and columns.
        /// All diagonal cells of the matrix will be initialized to the provided value, all non-diagonal ones to zero.
        /// Zero-length matrices are not supported.
        /// </summary>
        /// <exception cref="ArgumentException">If the row or column count is less than one.</exception>
        public DiagonalMatrix(int rows, int columns, Complex diagonalValue)
            : this(rows, columns)
        {
            for (var i = 0; i < _data.Length; i++)
            {
                _data[i] = diagonalValue;
            }
        }

        /// <summary>
        /// Create a new diagonal matrix with the given number of rows and columns directly binding to a raw array.
        /// The array is assumed to contain the diagonal elements only and is used directly without copying.
        /// Very efficient, but changes to the array and the matrix will affect each other.
        /// </summary>
        public DiagonalMatrix(int rows, int columns, Complex[] diagonalStorage)
            : this(new DiagonalMatrixStorage<Complex>(rows, columns, diagonalStorage))
        {
        }

        /// <summary>
        /// Create a new diagonal matrix as a copy of the given other matrix.
        /// This new matrix will be independent from the other matrix.
        /// The matrix to copy from must be diagonal as well.
        /// A new memory block will be allocated for storing the matrix.
        /// </summary>
        public static DiagonalMatrix OfMatrix(Matrix<Complex> matrix)
        {
            return new DiagonalMatrix(DiagonalMatrixStorage<Complex>.OfMatrix(matrix.Storage));
        }

        /// <summary>
        /// Create a new diagonal matrix as a copy of the given two-dimensional array.
        /// This new matrix will be independent from the provided array.
        /// The array to copy from must be diagonal as well.
        /// A new memory block will be allocated for storing the matrix.
        /// </summary>
        public static DiagonalMatrix OfArray(Complex[,] array)
        {
            return new DiagonalMatrix(DiagonalMatrixStorage<Complex>.OfArray(array));
        }

        /// <summary>
        /// Create a new diagonal matrix and initialize each diagonal value from the provided indexed enumerable.
        /// Keys must be provided at most once, zero is assumed if a key is omitted.
        /// This new matrix will be independent from the enumerable.
        /// A new memory block will be allocated for storing the matrix.
        /// </summary>
        public static DiagonalMatrix OfIndexedDiagonal(int rows, int columns, IEnumerable<Tuple<int, Complex>> diagonal)
        {
            return new DiagonalMatrix(DiagonalMatrixStorage<Complex>.OfIndexedEnumerable(rows, columns, diagonal));
        }

        /// <summary>
        /// Create a new diagonal matrix and initialize each diagonal value from the provided enumerable.
        /// This new matrix will be independent from the enumerable.
        /// A new memory block will be allocated for storing the matrix.
        /// </summary>
        public static DiagonalMatrix OfDiagonal(int rows, int columns, IEnumerable<Complex> diagonal)
        {
            return new DiagonalMatrix(DiagonalMatrixStorage<Complex>.OfEnumerable(rows, columns, diagonal));
        }

        /// <summary>
        /// Create a new diagonal matrix and initialize each diagonal value using the provided init function.
        /// </summary>
        public static DiagonalMatrix Create(int rows, int columns, Func<int, Complex> init)
        {
            return new DiagonalMatrix(DiagonalMatrixStorage<Complex>.OfInit(rows, columns, init));
        }

        /// <summary>
        /// Create a new diagonal matrix with diagonal values sampled from the provided random distribution.
        /// </summary>
        public static DiagonalMatrix CreateRandom(int rows, int columns, IContinuousDistribution distribution)
        {
            return new DiagonalMatrix(DiagonalMatrixStorage<Complex>.OfInit(rows, columns,
                i => new Complex(distribution.Sample(), distribution.Sample())));
        }

        /// <summary>
        /// Creates a <c>DiagonalMatrix</c> for the given number of rows and columns.
        /// </summary>
        /// <param name="numberOfRows">The number of rows.</param>
        /// <param name="numberOfColumns">The number of columns.</param>
        /// <param name="fullyMutable">True if all fields must be mutable (e.g. not a diagonal matrix).</param>
        /// <returns>
        /// A <c>DiagonalMatrix</c> with the given dimensions.
        /// </returns>
        public override Matrix<Complex> CreateMatrix(int numberOfRows, int numberOfColumns, bool fullyMutable = false)
        {
            return fullyMutable
                ? (Matrix<Complex>) new SparseMatrix(numberOfRows, numberOfColumns)
                : new DiagonalMatrix(numberOfRows, numberOfColumns);
        }

        /// <summary>
        /// Creates a <see cref="Vector{T}"/> with a the given dimension.
        /// </summary>
        /// <param name="size">The size of the vector.</param>
        /// <param name="fullyMutable">True if all fields must be mutable.</param>
        /// <returns>
        /// A <see cref="Vector{T}"/> with the given dimension.
        /// </returns>
        public override Vector<Complex> CreateVector(int size, bool fullyMutable = false)
        {
            return new SparseVector(size);
        }

        #region Elementary operations

        /// <summary>
        /// Adds another matrix to this matrix.
        /// </summary>
        /// <param name="other">The matrix to add to this matrix.</param>
        /// <returns>The result of the addition.</returns>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null"/>.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If the two matrices don't have the same dimensions.</exception>
        public override Matrix<Complex> Add(Matrix<Complex> other)
        {
            if (other == null)
            {
                throw new ArgumentNullException("other");
            }

            if (other.RowCount != RowCount || other.ColumnCount != ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentOutOfRangeException>(this, other, "other");
            }

            Matrix<Complex> result;
            if (other is DiagonalMatrix)
            {
                result = new DiagonalMatrix(RowCount, ColumnCount);
            }
            else
            {
                result = new DenseMatrix(RowCount, ColumnCount);
            }

            Add(other, result);
            return result;
        }

        /// <summary>
        /// Adds another matrix to this matrix.
        /// </summary>
        /// <param name="other">The matrix to add to this matrix.</param>
        /// <param name="result">The matrix to store the result of the addition.</param>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null"/>.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If the two matrices don't have the same dimensions.</exception>
        protected override void DoAdd(Matrix<Complex> other, Matrix<Complex> result)
        {
            var diagOther = other as DiagonalMatrix;
            var diagResult = result as DiagonalMatrix;

            if (diagOther == null || diagResult == null)
            {
                base.DoAdd(other, result);
            }
            else
            {
                Control.LinearAlgebraProvider.AddArrays(_data, diagOther._data, diagResult._data);
            }
        }

        /// <summary>
        /// Subtracts another matrix from this matrix.
        /// </summary>
        /// <param name="other">The matrix to subtract.</param>
        /// <returns>The result of the subtraction.</returns>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null"/>.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If the two matrices don't have the same dimensions.</exception>
        public override Matrix<Complex> Subtract(Matrix<Complex> other)
        {
            if (other == null)
            {
                throw new ArgumentNullException("other");
            }

            if (other.RowCount != RowCount || other.ColumnCount != ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentOutOfRangeException>(this, other, "other");
            }

            Matrix<Complex> result;
            if (other is DiagonalMatrix)
            {
                result = new DenseMatrix(RowCount, ColumnCount);
            }
            else
            {
                result = new DiagonalMatrix(RowCount, ColumnCount);
            }

            Subtract(other, result);
            return result;
        }

        /// <summary>
        /// Subtracts another matrix from this matrix.
        /// </summary>
        /// <param name="other">The matrix to subtract.</param>
        /// <param name="result">The matrix to store the result of the subtraction.</param>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null"/>.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If the two matrices don't have the same dimensions.</exception>
        protected override void DoSubtract(Matrix<Complex> other, Matrix<Complex> result)
        {
            var diagOther = other as DiagonalMatrix;
            var diagResult = result as DiagonalMatrix;

            if (diagOther == null || diagResult == null)
            {
                base.DoSubtract(other, result);
            }
            else
            {
                Control.LinearAlgebraProvider.SubtractArrays(_data, diagOther._data, diagResult._data);
            }
        }

        /// <summary>
        /// Copies the values of the given array to the diagonal.
        /// </summary>
        /// <param name="source">The array to copy the values from. The length of the vector should be
        /// Min(Rows, Columns).</param>
        /// <exception cref="ArgumentNullException">If <paramref name="source"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If the length of <paramref name="source"/> does not
        /// equal Min(Rows, Columns).</exception>
        /// <remarks>For non-square matrices, the elements of <paramref name="source"/> are copied to
        /// this[i,i].</remarks>
        public override void SetDiagonal(Complex[] source)
        {
            if (source == null)
            {
                throw new ArgumentNullException("source");
            }

            if (source.Length != _data.Length)
            {
                throw new ArgumentException(Resources.ArgumentArraysSameLength, "source");
            }

            Array.Copy(source, _data, source.Length);
        }

        /// <summary>
        /// Copies the values of the given <see cref="Vector{T}"/> to the diagonal.
        /// </summary>
        /// <param name="source">The vector to copy the values from. The length of the vector should be
        /// Min(Rows, Columns).</param>
        /// <exception cref="ArgumentNullException">If <paramref name="source"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If the length of <paramref name="source"/> does not
        /// equal Min(Rows, Columns).</exception>
        /// <remarks>For non-square matrices, the elements of <paramref name="source"/> are copied to
        /// this[i,i].</remarks>
        public override void SetDiagonal(Vector<Complex> source)
        {
            var denseSource = source as DenseVector;
            if (denseSource == null)
            {
                base.SetDiagonal(source);
                return;
            }

            if (_data.Length != denseSource.Values.Length)
            {
                throw new ArgumentException(Resources.ArgumentVectorsSameLength, "source");
            }

            Array.Copy(denseSource.Values, _data, denseSource.Values.Length);
        }

        /// <summary>
        /// Multiplies each element of the matrix by a scalar and places results into the result matrix.
        /// </summary>
        /// <param name="scalar">The scalar to multiply the matrix with.</param>
        /// <param name="result">The matrix to store the result of the multiplication.</param>
        /// <exception cref="ArgumentNullException">If the result matrix is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If the result matrix's dimensions are not the same as this matrix.</exception>
        protected override void DoMultiply(Complex scalar, Matrix<Complex> result)
        {
            if (scalar == 0.0)
            {
                result.Clear();
                return;
            }

            if (scalar == 1.0)
            {
                CopyTo(result);
                return;
            }

            var diagResult = result as DiagonalMatrix;
            if (diagResult == null)
            {
                base.DoMultiply(scalar, result);
            }
            else
            {
                if (!ReferenceEquals(this, result))
                {
                    CopyTo(diagResult);
                }

                Control.LinearAlgebraProvider.ScaleArray(scalar, _data, diagResult._data);
            }
        }

        /// <summary>
        /// Multiplies this matrix with another matrix and places the results into the result matrix.
        /// </summary>
        /// <param name="other">The matrix to multiply with.</param>
        /// <param name="result">The result of the multiplication.</param>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null" />.</exception>
        /// <exception cref="ArgumentNullException">If the result matrix is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If <strong>this.Columns != other.Rows</strong>.</exception>
        /// <exception cref="ArgumentException">If the result matrix's dimensions are not the this.Rows x other.Columns.</exception>
        public override void Multiply(Matrix<Complex> other, Matrix<Complex> result)
        {
            if (other == null)
            {
                throw new ArgumentNullException("other");
            }

            if (result == null)
            {
                throw new ArgumentNullException("result");
            }

            if (ColumnCount != other.RowCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, other);
            }

            if (result.RowCount != RowCount || result.ColumnCount != other.ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, result);
            }

            var m = other as DiagonalMatrix;
            var r = result as DiagonalMatrix;

            if (m == null || r == null)
            {
                base.Multiply(other, result);
            }
            else
            {
                var thisDataCopy = new Complex[r._data.Length];
                var otherDataCopy = new Complex[r._data.Length];
                Array.Copy(_data, thisDataCopy, (r._data.Length > _data.Length) ? _data.Length : r._data.Length);
                Array.Copy(m._data, otherDataCopy, (r._data.Length > m._data.Length) ? m._data.Length : r._data.Length);

                Control.LinearAlgebraProvider.PointWiseMultiplyArrays(thisDataCopy, otherDataCopy, r._data);
            }
        }

        /// <summary>
        /// Multiplies this matrix with another matrix and returns the result.
        /// </summary>
        /// <param name="other">The matrix to multiply with.</param>
        /// <exception cref="ArgumentException">If <strong>this.Columns != other.Rows</strong>.</exception>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null" />.</exception>
        /// <returns>The result of multiplication.</returns>
        public override Matrix<Complex> Multiply(Matrix<Complex> other)
        {
            if (other == null)
            {
                throw new ArgumentNullException("other");
            }

            if (ColumnCount != other.RowCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, other);
            }

            var result = other.CreateMatrix(RowCount, other.ColumnCount);
            Multiply(other, result);
            return result;
        }

        /// <summary>
        /// Multiplies this matrix with a vector and places the results into the result matrix.
        /// </summary>
        /// <param name="rightSide">The vector to multiply with.</param>
        /// <param name="result">The result of the multiplication.</param>
        /// <exception cref="ArgumentNullException">If <paramref name="rightSide"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentNullException">If <paramref name="result"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If <strong>result.Count != this.RowCount</strong>.</exception>
        /// <exception cref="ArgumentException">If <strong>this.ColumnCount != <paramref name="rightSide"/>.Count</strong>.</exception>
        public override void Multiply(Vector<Complex> rightSide, Vector<Complex> result)
        {
            if (rightSide == null)
            {
                throw new ArgumentNullException("rightSide");
            }

            if (ColumnCount != rightSide.Count)
            {
                throw DimensionsDontMatch<ArgumentException>(this, rightSide, "rightSide");
            }

            if (result == null)
            {
                throw new ArgumentNullException("result");
            }

            if (RowCount != result.Count)
            {
                throw DimensionsDontMatch<ArgumentException>(this, result, "result");
            }

            if (ReferenceEquals(rightSide, result))
            {
                var tmp = result.CreateVector(result.Count);
                Multiply(rightSide, tmp);
                tmp.CopyTo(result);
            }
            else
            {
                // Clear the result vector
                result.Clear();

                // Multiply the elements in the vector with the corresponding diagonal element in this.
                for (var r = 0; r < _data.Length; r++)
                {
                    result[r] = _data[r] * rightSide[r];
                }
            }
        }

        /// <summary>
        /// Left multiply a matrix with a vector ( = vector * matrix ) and place the result in the result vector.
        /// </summary>
        /// <param name="leftSide">The vector to multiply with.</param>
        /// <param name="result">The result of the multiplication.</param>
        /// <exception cref="ArgumentNullException">If <paramref name="leftSide"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentNullException">If the result matrix is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If <strong>result.Count != this.ColumnCount</strong>.</exception>
        /// <exception cref="ArgumentException">If <strong>this.RowCount != <paramref name="leftSide"/>.Count</strong>.</exception>
        public override void LeftMultiply(Vector<Complex> leftSide, Vector<Complex> result)
        {
            if (leftSide == null)
            {
                throw new ArgumentNullException("leftSide");
            }

            if (RowCount != leftSide.Count)
            {
                throw DimensionsDontMatch<ArgumentException>(this, leftSide, "leftSide");
            }

            if (result == null)
            {
                throw new ArgumentNullException("result");
            }

            if (ColumnCount != result.Count)
            {
                throw DimensionsDontMatch<ArgumentException>(this, result, "result");
            }

            if (ReferenceEquals(leftSide, result))
            {
                var tmp = result.CreateVector(result.Count);
                LeftMultiply(leftSide, tmp);
                tmp.CopyTo(result);
            }
            else
            {
                // Clear the result vector
                result.Clear();

                // Multiply the elements in the vector with the corresponding diagonal element in this.
                for (var r = 0; r < _data.Length; r++)
                {
                    result[r] = _data[r] * leftSide[r];
                }
            }
        }

        /// <summary>
        /// Computes the determinant of this matrix.
        /// </summary>
        /// <returns>The determinant of this matrix.</returns>
        public override Complex Determinant()
        {
            if (RowCount != ColumnCount)
            {
                throw new ArgumentException(Resources.ArgumentMatrixSquare);
            }

            return _data.Aggregate(Complex.One, (current, t) => current * t);
        }

        /// <summary>
        /// Returns the elements of the diagonal in a <see cref="DenseVector"/>.
        /// </summary>
        /// <returns>The elements of the diagonal.</returns>
        /// <remarks>For non-square matrices, the method returns Min(Rows, Columns) elements where
        /// i == j (i is the row index, and j is the column index).</remarks>
        public override Vector<Complex> Diagonal()
        {
            // TODO: Should we return reference to array? In current implementation we return copy of array, so changes in DenseVector will
            // not influence onto diagonal elements
            return new DenseVector((Complex[])_data.Clone());
        }

        /// <summary>
        /// Multiplies this matrix with transpose of another matrix and places the results into the result matrix.
        /// </summary>
        /// <param name="other">The matrix to multiply with.</param>
        /// <param name="result">The result of the multiplication.</param>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null" />.</exception>
        /// <exception cref="ArgumentNullException">If the result matrix is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If <strong>this.Columns != other.Rows</strong>.</exception>
        /// <exception cref="ArgumentException">If the result matrix's dimensions are not the this.Rows x other.Columns.</exception>
        public override void TransposeAndMultiply(Matrix<Complex> other, Matrix<Complex> result)
        {
            var otherDiagonal = other as DiagonalMatrix;
            var resultDiagonal = result as DiagonalMatrix;

            if (otherDiagonal == null || resultDiagonal == null)
            {
                base.TransposeAndMultiply(other, result);
                return;
            }

            Multiply(otherDiagonal.Transpose(), result);
        }

        /// <summary>
        /// Multiplies this matrix with transpose of another matrix and returns the result.
        /// </summary>
        /// <param name="other">The matrix to multiply with.</param>
        /// <exception cref="ArgumentException">If <strong>this.Columns != other.Rows</strong>.</exception>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null" />.</exception>
        /// <returns>The result of multiplication.</returns>
        public override Matrix<Complex> TransposeAndMultiply(Matrix<Complex> other)
        {
            var otherDiagonal = other as DiagonalMatrix;
            if (otherDiagonal == null)
            {
                return base.TransposeAndMultiply(other);
            }

            if (ColumnCount != otherDiagonal.ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, otherDiagonal);
            }

            var result = other.CreateMatrix(RowCount, other.RowCount);
            TransposeAndMultiply(other, result);
            return result;
        }

        #endregion

        /// <summary>
        /// Returns the transpose of this matrix.
        /// </summary>
        /// <returns>The transpose of this matrix.</returns>
        public override Matrix<Complex> Transpose()
        {
            var ret = new DiagonalMatrix(ColumnCount, RowCount);
            Array.Copy(_data, ret._data, _data.Length);
            return ret;
        }

        /// <summary>Calculates the L1 norm.</summary>
        /// <returns>The L1 norm of the matrix.</returns>
        public override Complex L1Norm()
        {
            return _data.Aggregate(double.NegativeInfinity, (current, t) => Math.Max(current, t.Magnitude));
        }

        /// <summary>Calculates the L2 norm.</summary>
        /// <returns>The L2 norm of the matrix.</returns>
        public override Complex L2Norm()
        {
            return _data.Aggregate(double.NegativeInfinity, (current, t) => Math.Max(current, t.Magnitude));
        }

        /// <summary>Calculates the Frobenius norm of this matrix.</summary>
        /// <returns>The Frobenius norm of this matrix.</returns>
        public override Complex FrobeniusNorm()
        {
            var norm = _data.Sum(t => t.Magnitude * t.Magnitude);
            return Math.Sqrt(norm);
        }

        /// <summary>Calculates the infinity norm of this matrix.</summary>
        /// <returns>The infinity norm of this matrix.</returns>
        public override Complex InfinityNorm()
        {
            return L1Norm();
        }

        /// <summary>Calculates the condition number of this matrix.</summary>
        /// <returns>The condition number of the matrix.</returns>
        public override Complex ConditionNumber()
        {
            var maxSv = double.NegativeInfinity;
            var minSv = double.PositiveInfinity;
            foreach (var t in _data)
            {
                maxSv = Math.Max(maxSv, t.Magnitude);
                minSv = Math.Min(minSv, t.Magnitude);
            }

            return maxSv / minSv;
        }

        /// <summary>Computes the inverse of this matrix.</summary>
        /// <exception cref="ArgumentException">If <see cref="DiagonalMatrix"/> is not a square matrix.</exception>
        /// <exception cref="ArgumentException">If <see cref="DiagonalMatrix"/> is singular.</exception>
        /// <returns>The inverse of this matrix.</returns>
        public override Matrix<Complex> Inverse()
        {
            if (RowCount != ColumnCount)
            {
                    throw new ArgumentException(Resources.ArgumentMatrixSquare);
            }

            var inverse = (DiagonalMatrix)Clone();
            for (var i = 0; i < _data.Length; i++)
            {
                if (_data[i] != 0.0)
                {
                    inverse._data[i] = 1.0 / _data[i];
                }
                else
                {
                    throw new ArgumentException(Resources.ArgumentMatrixNotSingular);
                }
            }

            return inverse;
        }

        /// <summary>
        /// Returns a new matrix containing the lower triangle of this matrix.
        /// </summary>
        /// <returns>The lower triangle of this matrix.</returns>
        public override Matrix<Complex> LowerTriangle()
        {
            return Clone();
        }

        /// <summary>
        /// Puts the lower triangle of this matrix into the result matrix.
        /// </summary>
        /// <param name="result">Where to store the lower triangle.</param>
        /// <exception cref="ArgumentNullException">If <paramref name="result"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If the result matrix's dimensions are not the same as this matrix.</exception>
        public override void LowerTriangle(Matrix<Complex> result)
        {
            if (result == null)
            {
                throw new ArgumentNullException("result");
            }

            if (result.RowCount != RowCount || result.ColumnCount != ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, result, "result");
            }

            if (ReferenceEquals(this, result))
            {
                return;
            }

            result.Clear();
            for (var i = 0; i < _data.Length; i++)
            {
                result.At(i, i, _data[i]);
            }
        }

        /// <summary>
        /// Returns a new matrix containing the lower triangle of this matrix. The new matrix
        /// does not contain the diagonal elements of this matrix.
        /// </summary>
        /// <returns>The lower triangle of this matrix.</returns>
        public override Matrix<Complex> StrictlyLowerTriangle()
        {
            return new DiagonalMatrix(RowCount, ColumnCount);
        }

        /// <summary>
        /// Puts the strictly lower triangle of this matrix into the result matrix.
        /// </summary>
        /// <param name="result">Where to store the lower triangle.</param>
        /// <exception cref="ArgumentNullException">If <paramref name="result"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If the result matrix's dimensions are not the same as this matrix.</exception>
        public override void StrictlyLowerTriangle(Matrix<Complex> result)
        {
            if (result == null)
            {
                throw new ArgumentNullException("result");
            }

            if (result.RowCount != RowCount || result.ColumnCount != ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, result, "result");
            }

            result.Clear();
        }

        /// <summary>
        /// Returns a new matrix containing the upper triangle of this matrix.
        /// </summary>
        /// <returns>The upper triangle of this matrix.</returns>
        public override Matrix<Complex> UpperTriangle()
        {
            return Clone();
        }

        /// <summary>
        /// Puts the upper triangle of this matrix into the result matrix.
        /// </summary>
        /// <param name="result">Where to store the lower triangle.</param>
        /// <exception cref="ArgumentNullException">If <paramref name="result"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If the result matrix's dimensions are not the same as this matrix.</exception>
        public override void UpperTriangle(Matrix<Complex> result)
        {
            if (result == null)
            {
                throw new ArgumentNullException("result");
            }

            if (result.RowCount != RowCount || result.ColumnCount != ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, result, "result");
            }

            result.Clear();
            for (var i = 0; i < _data.Length; i++)
            {
                result.At(i, i, _data[i]);
            }
        }

        /// <summary>
        /// Returns a new matrix containing the upper triangle of this matrix. The new matrix
        /// does not contain the diagonal elements of this matrix.
        /// </summary>
        /// <returns>The upper triangle of this matrix.</returns>
        public override Matrix<Complex> StrictlyUpperTriangle()
        {
            return new DiagonalMatrix(RowCount, ColumnCount);
        }

        /// <summary>
        /// Puts the strictly upper triangle of this matrix into the result matrix.
        /// </summary>
        /// <param name="result">Where to store the lower triangle.</param>
        /// <exception cref="ArgumentNullException">If <paramref name="result"/> is <see langword="null" />.</exception>
        /// <exception cref="ArgumentException">If the result matrix's dimensions are not the same as this matrix.</exception>
        public override void StrictlyUpperTriangle(Matrix<Complex> result)
        {
            if (result == null)
            {
                throw new ArgumentNullException("result");
            }

            if (result.RowCount != RowCount || result.ColumnCount != ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentException>(this, result, "result");
            }

            result.Clear();
        }

        /// <summary>
        /// Creates a matrix that contains the values from the requested sub-matrix.
        /// </summary>
        /// <param name="rowIndex">The row to start copying from.</param>
        /// <param name="rowCount">The number of rows to copy. Must be positive.</param>
        /// <param name="columnIndex">The column to start copying from.</param>
        /// <param name="columnCount">The number of columns to copy. Must be positive.</param>
        /// <returns>The requested sub-matrix.</returns>
        /// <exception cref="ArgumentOutOfRangeException">If: <list><item><paramref name="rowIndex"/> is
        /// negative, or greater than or equal to the number of rows.</item>
        /// <item><paramref name="columnIndex"/> is negative, or greater than or equal to the number
        /// of columns.</item>
        /// <item><c>(columnIndex + columnLength) &gt;= Columns</c></item>
        /// <item><c>(rowIndex + rowLength) &gt;= Rows</c></item></list></exception>
        /// <exception cref="ArgumentOutOfRangeException">If <paramref name="rowCount"/> or <paramref name="columnCount"/>
        /// is not positive.</exception>
        public override Matrix<Complex> SubMatrix(int rowIndex, int rowCount, int columnIndex, int columnCount)
        {
            var target = rowIndex == columnIndex
                ? (Matrix<Complex>)new DiagonalMatrix(rowCount, columnCount)
                : new SparseMatrix(rowCount, columnCount);

            Storage.CopySubMatrixTo(target.Storage, rowIndex, 0, rowCount, columnIndex, 0, columnCount, skipClearing: true);
            return target;
        }

        /// <summary>
        /// Creates a new  <see cref="SparseMatrix"/> and inserts the given column at the given index.
        /// </summary>
        /// <param name="columnIndex">The index of where to insert the column.</param>
        /// <param name="column">The column to insert.</param>
        /// <returns>A new <see cref="SparseMatrix"/> with the inserted column.</returns>
        /// <exception cref="ArgumentNullException">If <paramref name="column "/> is <see langword="null" />. </exception>
        /// <exception cref="ArgumentOutOfRangeException">If <paramref name="columnIndex"/> is &lt; zero or &gt; the number of columns.</exception>
        /// <exception cref="ArgumentException">If the size of <paramref name="column"/> != the number of rows.</exception>
        public override Matrix<Complex> InsertColumn(int columnIndex, Vector<Complex> column)
        {
            if (column == null)
            {
                throw new ArgumentNullException("column");
            }

            if (columnIndex < 0 || columnIndex > ColumnCount)
            {
                throw new ArgumentOutOfRangeException("columnIndex");
            }

            if (column.Count != RowCount)
            {
                throw new ArgumentException(Resources.ArgumentMatrixSameRowDimension, "column");
            }

            var result = new SparseMatrix(RowCount, ColumnCount + 1);

            for (var i = 0; i < columnIndex; i++)
            {
                result.SetColumn(i, Column(i));
            }

            result.SetColumn(columnIndex, column);

            for (var i = columnIndex + 1; i < ColumnCount + 1; i++)
            {
                result.SetColumn(i, Column(i - 1));
            }

            return result;
        }

        /// <summary>
        /// Creates a new  <see cref="SparseMatrix"/> and inserts the given row at the given index.
        /// </summary>
        /// <param name="rowIndex">The index of where to insert the row.</param>
        /// <param name="row">The row to insert.</param>
        /// <returns>A new  <see cref="SparseMatrix"/> with the inserted column.</returns>
        /// <exception cref="ArgumentNullException">If <paramref name="row"/> is <see langword="null" />. </exception>
        /// <exception cref="ArgumentOutOfRangeException">If <paramref name="rowIndex"/> is &lt; zero or &gt; the number of rows.</exception>
        /// <exception cref="ArgumentException">If the size of <paramref name="row"/> != the number of columns.</exception>
        public override Matrix<Complex> InsertRow(int rowIndex, Vector<Complex> row)
        {
            if (row == null)
            {
                throw new ArgumentNullException("row");
            }

            if (rowIndex < 0 || rowIndex > RowCount)
            {
                throw new ArgumentOutOfRangeException("rowIndex");
            }

            if (row.Count != ColumnCount)
            {
                throw new ArgumentException(Resources.ArgumentMatrixSameRowDimension, "row");
            }

            var result = new SparseMatrix(RowCount + 1, ColumnCount);

            for (var i = 0; i < rowIndex; i++)
            {
                result.At(i, i, At(i, i));
            }

            result.SetRow(rowIndex, row);

            for (var i = rowIndex + 1; i < result.RowCount; i++)
            {
                result.At(i, i - 1, At(i - 1, i - 1));
            }

            return result;
        }

        /// <summary>
        /// Permute the columns of a matrix according to a permutation.
        /// </summary>
        /// <param name="p">The column permutation to apply to this matrix.</param>
        /// <exception cref="InvalidOperationException">Always thrown</exception>
        /// <remarks>Permutation in diagonal matrix are senseless, because of matrix nature</remarks>
        public override void PermuteColumns(Permutation p)
        {
            throw new InvalidOperationException("Permutations in diagonal matrix are not allowed");
        }

        /// <summary>
        /// Permute the rows of a matrix according to a permutation.
        /// </summary>
        /// <param name="p">The row permutation to apply to this matrix.</param>
        /// <exception cref="InvalidOperationException">Always thrown</exception>
        /// <remarks>Permutation in diagonal matrix are senseless, because of matrix nature</remarks>
        public override void PermuteRows(Permutation p)
        {
            throw new InvalidOperationException("Permutations in diagonal matrix are not allowed");
        }

        /// <summary>
        /// Gets a value indicating whether this matrix is symmetric.
        /// </summary>
        public override bool IsSymmetric
        {
            get
            {
                return true;
            }
        }

        #region Static constructors for special matrices.

        /// <summary>
        /// Initializes a square <see cref="DiagonalMatrix"/> with all zero's except for ones on the diagonal.
        /// </summary>
        /// <param name="order">the size of the square matrix.</param>
        /// <returns>A diagonal identity matrix.</returns>
        /// <exception cref="ArgumentException">
        /// If <paramref name="order"/> is less than one.
        /// </exception>
        public static DiagonalMatrix Identity(int order)
        {
            var m = new DiagonalMatrix(order);
            for (var i = 0; i < order; i++)
            {
                m._data[i] = 1.0;
            }

            return m;
        }

        #endregion
    }
}