Systems of Linear Equations

Abstract

One of the commonest problems of numerical computation is to solve a system of simultaneous linear equations

$$\left. {\begin{array}{*{20}{c}} {{a_{11}}{x_1}}& + &{{a_{12}}{x_2}}& + & \ldots & + &{{a_{1n}}{x_n}}& = &{{b_1}} \\ {{a_{21}}{x_1}}& + &{{a_{22}}{x_2}}& + & \ldots & + &{{a_{2n}}{x_n}}& = &{{b_2}} \\ &&&&{}&&&{}& \\ &&&&{}&&&{}& \\ {{a_{n1}}{x_1}}& + &{{a_{n2}}{x_2}}& + & \ldots & + &{{a_{nn}}{x_n}}& = &{{b_n}} \end{array}} \right\}$$

(7.1)

The coefficients a _ij for 1 ≼ i, j ≼ n and the right hand sides b _i for 1 ≼ i ≼ n are given; the problem is to find numerical values for the unknowns x₁, …, x _n which satisfy the n equations.

Our little systems have their day.

Lord Tennyson, In Memoriam