| # Sensitivity Analysis Least-Squares > Ordinary LS | Variants | Sensitivity | Applications Relative error analysis Condition number We consider the OLS problem: with the data matrix (known); is the measurement (known); is the vector to be estimated (unknown). We assume that is full column rank. Then, the solution to the OLS problem is unique, and can be written as a linear function of the measurement vector : with the pseudo-inverse of . Again, since is full column rank, We are interested in analyzing the impact of perturbations in the vector , on the resulting solution . We begin by analyzing the absolute errors in the estimate, and then turn to the analysis of relative errors. ## Bound on the absolute error We first try to find a bound on the error in the solution given a bound on the error in the input. Let us assume a simple model of potential perturbations: we assume that belongs to a unit ball: , where is given. We will assume for simplicity; the analysis is easily extended to any . Let us define to be the estimate corresponding to the measurement , that is: Since , we have We obtain that where is the matrix norm, which is defined for a matrix as In other words, is simply the maximum output norm that we can attain with a unit-norm input. (The matrix norm can be computed from an SVD of the matrix .) The interpretation is that is the largest absolute error that can result from an absolute error in the input that bounded by . ## Bound on the relative error analysis We now turn to the problem of analyzing the impact of relative errors in the measurement, on the relative error in the solution. To simplify, let us assume that the matrix is square () and full rank (invertible). The development before simplifies to so that Likewise, the equation allows to bound with respect to : Combining the two inequalities we obtain: ## Condition number As seen above, when is square, invertible, the quantity measures the sensitivity to relative errors in the system . The quantity above is called the condition number of . The condition number is always greater than . The closer to , the more confident we can be about the solution when is subject to small relative errors. We can generalize this notion to non-necessarily square matrices that have full rank. Many practical algorithms to solve linear equations (or OLS) use the condition number (or an estimate) to evaluate the quality of the solution.