Advanced Engineering Mathematics, 7th Edition

8.16 Method of Least Squares

INTRODUCTION

When performing experiments we often tabulate data in the form of ordered pairs (x₁, y₁), (x₂, y₂), . . . , (x_n, y_n), with each x_i distinct. Given the data, it is then often desirable to be able to extrapolate or predict y from x by finding a mathematical model, that is, a function that approximates or “fits” the data. In other words, we want a function f(x) such that

f(x₁) ≈ y₁,     f(x₂) ≈ y₂,     . . .,     f(x_n) ≈ y_n.

But naturally we do not want just any function but a function that fits the data as closely as possible.

In the discussion that follows we shall confine our attention to the problem of finding a linear polynomial f(x) = ax + b or straight line that “best fits” the data (x₁, y₁), (