One of the commonest problems of numerical computation is to solve a system of simultaneous linear equations
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% ypa0JaamOyamaaBaaaleaacaWGUbaabeaaaaGccaGL9baaaaa!AEF1!EquationSource$$ \left. \matrix
{a_{11}}{x_1} + {a_{12}}{x_2} + ... + {a_{1n}}{x_n} = {b_1} \hfill \cr
{a_{21}}{x_1} + {a_{22}}{x_2} + ... + {a_{2n}}{x_n} = {b_2} \hfill \cr
{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}. \hfill \cr
{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}. \hfill \cr
{a_{n1}}{x_1} + {a_{n2}}{x_2} + ... + {a_{nn}}{x_n} = {b_n} \hfill \cr
\endmatrix \right\} $$
((7.1))
The coefficients aij for 1≤ i,j ≤ n and the right hand sides bi for 1≤ I ≤ n are given; the problem is to find numerical values for the unknowns x1,,xn which satisfy the n equations.
