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QuickStart Samples

Structured Linear Equations QuickStart Sample (C#)

Illustrates how to solve systems of simultaneous linear equations that have special structure in C#.

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using System;

namespace Extreme.Numerics.QuickStart.CSharp {
    // The structured matrix classes reside in the 
    // Extreme.Mathematics.LinearAlgebra namespace.
    using Extreme.Mathematics;
    using Extreme.Mathematics.LinearAlgebra;

    /// <summary>
    /// Illustrates solving symmetrical and triangular systems 
    /// of simultaneous linear equations using classes 
    /// in the Extreme.Mathematics.LinearAlgebra namespace of the Extreme 
    /// Optimization Mathematics Library for .NET.
    /// </summary>
    class StructuredLinearEquations {
        /// <summary>
        /// The main entry point for the application.
        /// </summary>
        [STAThread]
        static void Main(string[] args) {
            // To learn more about solving general systems of
            // simultaneous linear equations, see the
            // LinearEquations QuickStart Sample.
            //
            // The methods and classes available for solving
            // structured systems of equations are similar
            // to those for general equations.

            //
            // Triangular systems and matrices
            //

            Console.WriteLine("Triangular matrices:");
            // For the basics of working with triangular 
            // matrices, see the TriangularMatrices QuickStart
            // Sample.
            //
            // Let's start with a triangular matrix. Remember
            // that elements are stored in column-major order
            // by default.
            var t = Matrix.CreateUpperTriangular(
                4, 4, new double[]
                {
                    1, 0, 0, 0,
                    1, 2, 0, 0,
                    1, 4, 1, 0,
                    1, 3, 1, 2
                }, MatrixElementOrder.ColumnMajor);
            var b1 = Vector.Create(new double[] { 1, 3, 6, 3 });
            var b2 = Matrix.Create(4, 2, new double[]
                {
                    1, 3, 6, 3,
                    2, 3, 5, 8
                }, MatrixElementOrder.ColumnMajor);
            Console.WriteLine("t = {0:F4}", t);

            //
            // The Solve method
            //

            // The following solves m x = b1. The second 
            // parameter specifies whether to overwrite the
            // right-hand side with the result.
            var x1 = t.Solve(b1, false);
            Console.WriteLine("x1 = {0:F4}", x1);
            // If the overwrite parameter is omitted, the
            // right-hand-side is overwritten with the solution:
            t.Solve(b1);
            Console.WriteLine("b1 = {0:F4}", b1);
            // You can solve for multiple right hand side 
            // vectors by passing them in a DenseMatrix:
            var x2 = t.Solve(b2, false);
            Console.WriteLine("x2 = {0:F4}", x2);

            //
            // Related Methods
            //

            // You can verify whether a matrix is singular
            // using the IsSingular method:
            Console.WriteLine("IsSingular(t) = {0:F4}",
                t.IsSingular());
            // The inverse matrix is returned by the Inverse
            // method:
            Console.WriteLine("Inverse(t) = {0:F4}", t.GetInverse());
            // The determinant is also available:
            Console.WriteLine("Det(t) = {0:F4}", t.GetDeterminant());
            // The condition number is an estimate for the
            // loss of precision in solving the equations
            Console.WriteLine("Cond(t) = {0:F4}", t.EstimateConditionNumber());
            Console.WriteLine();

            //
            // Symmetric systems and matrices
            //

            Console.WriteLine("Symmetric matrices:");
            // For the basics of working with symmetric 
            // matrices, see the SymmetricMatrices QuickStart
            // Sample.
            //
            // Let's start with a symmetric matrix. Remember
            // that elements are stored in column-major order
            // by default.
            var s = Matrix.CreateSymmetric(4, new double[]
                {
                    1, 0, 0, 0,
                    1, 2, 0, 0,
                    1, 1, 2, 0,
                    1, 0, 1, 4
                }, MatrixTriangle.Upper, MatrixElementOrder.ColumnMajor);
            b1 = Vector.Create(new double[] { 1, 3, 6, 3 });
            Console.WriteLine("s = {0:F4}", s);

            //
            // The Solve method
            //

            // The following solves m x = b1. The second 
            // parameter specifies whether to overwrite the
            // right-hand side with the result.
            x1 = s.Solve(b1, false);
            Console.WriteLine("x1 = {0:F4}", x1);
            // If the overwrite parameter is omitted, the
            // right-hand-side is overwritten with the solution:
            s.Solve(b1);
            Console.WriteLine("b1 = {0:F4}", b1);
            // You can solve for multiple right hand side 
            // vectors by passing them in a DenseMatrix:
            x2 = s.Solve(b2, false);
            Console.WriteLine("x2 = {0:F4}", x2);

            //
            // Related Methods
            //

            // You can verify whether a matrix is singular
            // using the IsSingular method:
            Console.WriteLine("IsSingular(s) = {0:F4}",
                s.IsSingular());
            // The inverse matrix is returned by the Inverse
            // method:
            Console.WriteLine("Inverse(s) = {0:F4}", s.GetInverse());
            // The determinant is also available:
            Console.WriteLine("Det(s) = {0:F4}", s.GetDeterminant());
            // The condition number is an estimate for the
            // loss of precision in solving the equations
            Console.WriteLine("Cond(s) = {0:F4}", s.EstimateConditionNumber());
            Console.WriteLine();

            //
            // The CholeskyDecomposition class
            //

            // If the symmetric matrix is positive definite,
            // you can use the CholeskyDecomposition class
            // to optimize performance if multiple operations 
            // need to be performed. This class does the
            // bulk of the calculations only once. This
            // decomposes the matrix into G x transpose(G)
            // where G is a lower triangular matrix.
            //
            // If the matrix is indefinite, you need to use
            // the LUDecomposition class instead. See the
            // LinearEquations QuickStart Sample for details.
            Console.WriteLine("Using Cholesky Decomposition:");
            // The constructor takes an optional second argument
            // indicating whether to overwrite the original
            // matrix with its decomposition:
            var cf = s.GetCholeskyDecomposition(false);
            // The Factorize method performs the actual
            // factorization. It is called automatically
            // if needed.
            cf.Decompose();
            // All methods mentioned earlier are still available:
            x2 = cf.Solve(b2, false);
            Console.WriteLine("x2 = {0:F4}", x2);
            Console.WriteLine("IsSingular(m) = {0:F4}",
                cf.IsSingular());
            Console.WriteLine("Inverse(m) = {0:F4}", cf.GetInverse());
            Console.WriteLine("Det(m) = {0:F4}", cf.GetDeterminant());
            Console.WriteLine("Cond(m) = {0:F4}", cf.EstimateConditionNumber());
            // In addition, you have access to the
            // triangular matrix, G, of the composition.
            Console.WriteLine("  G = {0:F4}", cf.LowerTriangularFactor);

            // Note that if the matrix is indefinite,
            // the factorization will fail and throw a
            // MatrixNotPositiveDefiniteException.
            s[0, 0] = -99;
            cf = s.GetCholeskyDecomposition();
            try {
                cf.Decompose();
            } catch (MatrixNotPositiveDefiniteException e) {
                Console.WriteLine(e.Message);
            }
            // To avoid this, you can use the TryDecompose method,
            // which returns true if the decomposition succeeds,
            // and false otherwise:
            if (cf.TryDecompose())
                Console.WriteLine("Decomposition succeeded!");
            else
                Console.WriteLine("Decomposition failed!");

            Console.Write("Press Enter key to exit...");
            Console.ReadLine();
        }
    }
}