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Yet Another Math Parser (YAMP)

, 30 Sep 2012 CPOL
Constructing a fast math parser using Reflection to do numerics like Matlab.
YAMP
YAMPCompare
LLMathParser
MathFormula
MathParser
MathParserNet
Exceptions
MathParserTK
YAMPCompare.pidb
YAMPConsole
YAMPConsole.pidb
Exceptions
Expressions
Functions
ArgumentFunctions
LinearAlgebra
LogicFunctions
Spectroscopy
StandardFunctions
Trigonometric
Interfaces
Numerics
Decompositions
Integration
Interpolations
ODE
Optimization
Others
Solvers
Operators
AssigmentOperators
BinaryOperators
DotOperators
LogicOperators
UnaryOperators
Values
YAMP.csproj.user
YAMP.pidb
YAMP.csproj.user
YAMP.pidb
using System;
using YAMP;

namespace YAMP.Numerics
{
    /// <summary>
    /// QR Decomposition.
    /// For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n
    /// orthogonal matrix Q and an n-by-n upper triangular matrix R so that
    /// A = Q*R.
    /// The QR decompostion 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 IsFullRank()
    /// returns false.
    /// </summary>
    public class QRDecomposition : DirectSolver
    {
        #region Members

        /// <summary>
        /// Array for internal storage of decomposition.
        /// </summary>
        double[][] QR;

        /// <summary>
        /// Row and column dimensions.
        /// </summary>
        int m, n;

        /// <summary>
        /// Array for internal storage of diagonal of R.
        /// </summary>
        double[] Rdiag;

        #endregion //  Class variables

        #region Constructor

        /// <summary>
        /// QR Decomposition, computed by Householder reflections.
        /// </summary>
        /// <param name="A">Rectangular matrix</param>
        /// <returns>Structure to access R and the Householder vectors and compute Q.</returns>
        public QRDecomposition(MatrixValue A)
        {
            // Initialize.
            QR = A.GetRealArray();
            m = A.DimensionY;
            n = A.DimensionX;
            Rdiag = new double[n];

            // Main loop.
            for (int k = 0; k < n; k++)
            {
                // Compute 2-norm of k-th column without under/overflow.
                var nrm = 0.0;

                for (int i = k; i < m; i++)
                    nrm = NumericHelpers.Hypot(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 < m; i++)
                        QR[i][k] /= nrm;
                    
                    QR[k][k] += 1.0;

                    // Apply transformation to remaining columns.
                    for (int j = k + 1; j < n; j++)
                    {
                        var s = 0.0;

                        for (int i = k; i < m; i++)
                            s += QR[i][k] * QR[i][j];
                        
                        s = (-s) / QR[k][k];

                        for (int i = k; i < m; i++)
                            QR[i][j] += s * QR[i][k];
                    }
                }

                Rdiag[k] = -nrm;
            }
        }

        #endregion //  Constructor

        #region Public Properties

        /// <summary>
        /// Is the matrix full rank?
        /// </summary>
        /// <returns>true if R, and hence A, has full rank.</returns>
        virtual public bool FullRank
        {
            get
            {
                for (int j = 0; j < n; j++)
                {
                    if (Rdiag[j] == 0)
                        return false;
                }

                return true;
            }
        }

        /// <summary>
        /// Return the Householder vectors
        /// </summary>
        /// <returns>Lower trapezoidal matrix whose columns define the reflections.</returns>
        virtual public MatrixValue H
        {
            get
            {
                var X = new MatrixValue(m, n);

                for (int i = 1; i <= m; i++)
                {
                    for (int j = 1; j <= n; j++)
                    {
                        if (i >= j)
                            X[i, j].Value = QR[i - 1][j - 1];
                        else
                            X[i, j].Value = 0.0;
                    }
                }

                return X;
            }

        }

        /// <summary>
        /// Return the upper triangular factor
        /// </summary>
        /// <returns>R</returns>
        virtual public MatrixValue R
        {
            get
            {
                var X = new MatrixValue(n, n);

                for (int i = 1; i <= n; i++)
                {
                    for (int j = 1; j <= n; j++)
                    {
                        if (i < j)
                            X[i, j].Value = QR[i - 1][j - 1];
                        else if (i == j)
                            X[i, j].Value = Rdiag[i - 1];
                        else
                            X[i, j].Value = 0.0;
                    }
                }

                return X;
            }
        }

        /// <summary>
        /// Generate and return the (economy-sized) orthogonal factor
        /// </summary>
        /// <returns>Q</returns>
        virtual public MatrixValue Q
        {
            get
            {
                var X = new MatrixValue(m, n);

                for (int k = n; k > 0; k--)
                {
                    for (int i = 1; i <= m; i++)
                        Q[i, k].Value = 0.0;

                    Q[k, k].Value = 1.0;

                    for (int j = k; j <= n; j++)
                    {
                        var l = k - 1;

                        if (QR[l][l] != 0)
                        {
                            var s = 0.0;

                            for (int i = k; i <= m; i++)
                                s += QR[i - 1][l] * Q[i, j].Value;

                            s = (-s) / QR[l][l];

                            for (int i = k; i <= m; i++)
                                Q[i, j].Value += s * QR[i - 1][l];
                        }
                    }
                }

                return X;
            }
        }

        #endregion //  Public Properties

        #region Public Methods

        /// <summary>
        /// Least squares solution of A*X = B
        /// </summary>
        /// <param name="B">A Matrix with as many rows as A and any number of columns.</param>
        /// <returns>X that minimizes the two norm of Q*R*X-B.</returns>
        /// <exception cref="System.ArgumentException"> Matrix row dimensions must agree.</exception>
        /// <exception cref="System.SystemException"> Matrix is rank deficient.</exception>
        public override MatrixValue Solve(MatrixValue B)
        {
            if (B.DimensionY != m)
                throw new DimensionException(B.DimensionY, m);

            if (!this.FullRank)
                throw new MatrixFormatException("full rank");

            // Copy right hand side
            var nx = B.DimensionX;
            var X = B.GetRealArray();

            // Compute Y = transpose(Q)*B
            for (int k = 0; k < n; k++)
            {
                for (int j = 0; j < nx; j++)
                {
                    var 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 < nx; j++)
                    X[k][j] /= Rdiag[k];

                for (int i = 0; i < k; i++)
                {
                    for (int j = 0; j < nx; j++)
                        X[i][j] -= X[k][j] * QR[i][k];
                }
            }

            return new MatrixValue(X, n, nx).SubMatrix(0, n, 0, nx);
        }

        #endregion //  Public Methods
    }
}

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About the Author

Florian Rappl
Architect
Germany Germany
Florian lives in Munich, Germany. He started his programming career with Perl. After programming C/C++ for some years he discovered his favorite programming language C#. He did work at Siemens as a programmer until he decided to study Physics.

During his studies he worked as an IT consultant for various companies. After graduating with a PhD in theoretical particle Physics he is working as a senior technical consultant in the field of home automation and IoT.

Florian has been giving lectures in C#, HTML5 with CSS3 and JavaScript, software design, and other topics. He is regularly giving talks at user groups, conferences, and companies. He is actively contributing to open-source projects. Florian is the maintainer of AngleSharp, a completely managed browser engine.

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