# Solving Least Squares Problems

**Numerics.NET**
supports a simple mechanism for solving linear least squares problems.

## The LeastSquaresSolver Class

The LeastSquaresSolver<T> class implements the calculation of least squares solutions. The constructor takes two arguments: the matrix of observations, and the vector of outcomes:

Least squares problems can be solved using a variety of techniques. The method to use is specified by the SolutionMethod property. This property is of type LeastSquaresSolutionMethod and can take on the following values:

Value | Description |
---|---|

NormalEquations | The solution is found by solving the normal equations. This method is often the fastest, but is also less stable. |

QRDecomposition | The solution is found by first calculating the QR decomposition of the observation matrix. This is the default. |

SingularValueDecomposition | The solution is found by first calculating the singular value decomposition of the observation matrix. This is the slowest but most robust method. |

NonNegative | The solution is restricted to values that are positive or zero. |

The Solve method performs the actual calculation. This method returns a Vector<T> containing the least squares solution. This vector is also available through the Solution property.

Various other methods let you retrieve more information about the least squares solution. The GetPredictions method returns the vector of the outcomes predicted by the solution. The GetResiduals method returns the residuals: the difference between the predicted and actual outcome.

By default, the least squares solver uses an algorithm based on the QR decomposition of the matrix of observations. Depending on the numerical properties of the observation matrix, it may be acceptable to calculate the solution using the normal equations. This is faster, especially for problems with many more observations than variables, but the numerical accuracy tends to be questionable for larger numbers of variables.

Methods using a complete orthogonal decomposition and the singular value decomposition will be supported in the future.

## Directly Solving the Normal Equations

In some instances, it is convenient and cheap to accumulate the normal equations directly. If the condition of the observation matrix is good enough, a least squares solution may be obtained by solving the accumulated equations. The following example illustrates this procedure:

```
SymmetricMatrix aTa = SymmetricMatrix.FromOuterProduct(a);
Vector aTb = b * a;
Vector x = aTa.Solve(aTb);
Console.WriteLine("x = {0}", x.ToString("F4"));
```

## Non-Negative Least Squares

It is sometimes necessary to restrict the solution to values that are positive or zero. To compute such a non-negative least squares solution, set the SolutionMethod property to NonNegative.

## References

G. H. Golub, C. F. Van Loan, (3rd Ed), Johns Hopkins University Press, 1996.