Definition:Method of Least Squares (Approximation Theory)

Definition
Let there be a set of points $\set {\tuple {x_k, y_k}: k \in \set {1, 2, \ldots, n} }$ plotted on a Cartesian $x y$ plane which correspond to measurements of a physical system.

Let it be required that a straight line is to be fitted to the points.

The method of least squares is a technique of producing a straight line of the form $y = m x + c$ such that:
 * the points $\set {\tuple {x_k', y_k'}: k \in \set {1, 2, \ldots, n} }$ are on the line $y = m x + c$
 * $\forall k \in \set {1, 2, \ldots, n}: y_k' = y_k$
 * $\ds \sum_n \paren {x_k' = x_k}^2$ is minimised.


 * LeastSquares.png