Definition:Method of Least Squares (Approximation Theory)

This page is about Method of Least Squares in the context of Approximation Theory. For other uses, see Method of Least Squares.

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.

Historical Note

The method of least squares was invented by Carl Friedrich Gauss.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): least squares: 1.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): method of least squares
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): least squares: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): method of least squares