# Structured Linear Equations QuickStart Sample (C#)

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

View this sample in: Visual Basic F# IronPython

``````using System;
// The structured matrix classes reside in the
// Extreme.Mathematics.LinearAlgebra namespace.
using Extreme.Mathematics;
using Extreme.Mathematics.LinearAlgebra;

namespace Extreme.Numerics.QuickStart.CSharp {
/// <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 {
static void Main(string[] args) {
// The license is verified at runtime. We're using
// https://www.extremeoptimization.com/trial-key

// 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.
//
// 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.
//
// 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 Factor 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());
// 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...");