# C# Least Squares Example

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```using System;
using System.Globalization;

using CenterSpace.NMath.Core;

namespace CenterSpace.NMath.Core.Examples.CSharp
{
/// <summary>
/// A .NET example in C# showing how to use the least squares classes to solve
/// linear least squares problems.
/// </summary>
class LeastSquaresExample
{

static void Main( string[] args )
{

// This example uses strings representing numbers in the US locale
// so change the current culture info.  For example, "0.446"

// Calculate the slope and intercept of the linear least squares fit
// through the five points:
// (20, .446) (30, .601), (40, .786), (50, .928), (60, .950)
var A = new DoubleMatrix( "5x1[20.0  30.0  40.0  50.0  60.0]" );
var y = new DoubleVector( "[0.446 0.601 0.786 0.928 0.950]" );

// Back to your original culture.

// We want our straight line to be of the form y = mx + b, where b is
// not necessarily equal to zero. Thus we will set the third
// constructor argument to true so that we calculate the intercept
// parameter.
var lsq = new DoubleLeastSquares( A, y, true );

Console.WriteLine();

Console.WriteLine( "Y-intercept = {0}", lsq.X[0] );
Console.WriteLine( "Slope = {0}", lsq.X[1] );

// We can look at the residuals which are the difference between the
// actual value of y at a point x, and the corresponding point y on
// line for the same x.
Console.WriteLine( "Residuals = {0}", lsq.Residuals.ToString( "F3" ) );

// Finally, we can look at the residual sum of squares, which is the
// sum of the squares of the elements in the residual vector.
Console.WriteLine( "Residual Sum of Squares (RSS) = {0}", lsq.ResidualSumOfSquares.ToString( "F3" ) );

// The least squares class can also be used to solve "rank-deficient" least
// square problems:
A = new DoubleMatrix( "6x4 [0 9 -6 3  -3 0 -3 0  1 3 -1 1  1 3 -1 1  -2 0 -2 0  3 6 -1 2]" );
y = new DoubleVector( "[-3 5 -2 2 1 -2]" );

// For this problem we will specify a tolerance for computing the effective rank
// of the matrix A, and we will not have the class add an intercept parameter
// for us.
lsq = new DoubleLeastSquares( A, y, 1e-10 );
Console.WriteLine( "Least squares solution = {0}", lsq.X.ToString( "F3" ) );
Console.WriteLine( "Rank computed using a tolerance of {0}, = {1}",
lsq.Tolerance, lsq.Rank );

// You can even use the least squares class to solve under-determined systems
// (the case where A has more columns than rows).
A = new DoubleMatrix( "6x4 [-3 -1 6 -5  5 4 -6 8  7 5 0 -4  -7 4 0 3  -7 7 -8 2  3 4 2 -4]" );
y = new DoubleVector( "[-3 1 8 -2]" );
lsq = new DoubleLeastSquares( A.Transpose(), y, 1e-8 );
Console.WriteLine( "Solution to under-determined system = {0}", lsq.X.ToString( "F3" ) );
Console.WriteLine( "Rank computed using a tolerance of {0}, = {1}",
lsq.Tolerance, lsq.Rank );

Console.WriteLine();
Console.WriteLine( "Press Enter Key" );