# Least Squares in C# QuickStart Sample

Illustrates how to solve least squares problems using classes in the Extreme.Mathematics.LinearAlgebra namespace in C#.

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``````using System;
// The DenseMatrix and LeastSquaresSolver classes reside in the
// Extreme.Mathematics.LinearAlgebra namespace.
using Extreme.Mathematics;
using Extreme.Mathematics.LinearAlgebra;

namespace Extreme.Numerics.QuickStart.CSharp
{
/// <summary>
/// Illustrates solving least squares problems using the
/// LeastSquaresSolver class in the Extreme.Mathematics.LinearAlgebra
/// namespace of Extreme Numerics.NET.
/// </summary>
class LeastSquares
{
static void Main(string[] args)
{
// The license is verified at runtime. We're using
// https://numerics.net/trial-key
// A least squares problem consists in finding
// the solution to an overdetermined system of
// simultaneous linear equations so that the
// sum of the squares of the error is minimal.
//
// A common application is fitting data to a
// curve. See the CurveFitting sample application
// for a complete example.

// Let's start with a general matrix. This will be
// the matrix a in the left hand side ax=b:
var a = Matrix.Create(6, 4, new double[]
{
1, 1, 1, 1, 1, 1,
1, 2, 3, 4, 5, 6,
1, 4, 9, 16, 25, 36,
1, 2, 1, 2, 1, 2
}, MatrixElementOrder.ColumnMajor);
// Here is the right hand side:
var b = Vector.Create(new double[] { 1, 3, 6, 11, 15, 21 });
var b2 = Matrix.Create(6, 2, new double[]
{
1, 3, 6, 11, 15, 21,
1, 2, 3, 4, 5, 7
}, MatrixElementOrder.ColumnMajor);
Console.WriteLine("a = {0:F4}", a);
Console.WriteLine("b = {0:F4}", b);

//
// The LeastSquaresSolver class
//

var x = a.LeastSquaresSolve(b);
var qr = a.GetQRDecomposition();
qr.LeastSquaresSolve(b);
// The following creates an instance of the
// LeastSquaresSolver class for our problem:
var solver = new LeastSquaresSolver<double>(a, b);
// We can specify the solution method: normal
// equations or QR decomposition. In most cases,
// a QR decomposition is the most desirable:
solver.SolutionMethod = LeastSquaresSolutionMethod.QRDecomposition;
// The Solve method calculates the solution:
x = solver.Solve();
Console.WriteLine("x = {0:F4}", x);
// The Solution property also returns the solution:
Console.WriteLine("x = {0:F4}", solver.Solution);
// More detailed information is available from
// The values of the right hand side predicted
// by the solution:
Console.WriteLine("Predictions = {0:F4}", solver.GetPredictions());
// The residuals (errors) of the solution:
Console.WriteLine("Residuals = {0:F4}", solver.GetResiduals());
// The total sum of squares of the residues:
Console.WriteLine("Residual square error = {0}",
solver.GetResidualSumOfSquares());

//
// Direct normal equations
//

// Alternatively, you can create a least squares
// solution by providing the normal equations
// directly. This may be useful when it is easy
// to calculate the normal equations directly.
//
// Here, we'll just calculate the normal equation:
var aTa = SymmetricMatrix<double>.FromOuterProduct(a);
var aTb = b * a; // a.Transpose() * b;
// We find the solution by solving the normal equations
// directly:
x = aTa.Solve(aTb);
Console.WriteLine("x = {0:F4}", x);
// However, properties of the least squares solution, such as
// error estimates and residuals are not available.

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