Assume that the coefficient matrix A of a system of equations, A = , is a regular n x n matrix. Then the system of equations has a unique solution . Convert the equation A = (in same way or another) into the form
G is a so-called iteration matrix. G and can be chosen in different ways so that we obtain different iteration methods. Form the following iteration formula:
Proposition 2. If the matrix equations A = ja = G + are equivalent and if ||G|| 1, then the sequence ( (k)) of vectors defined by (2) converges to the solution of the equation A = .
|||(k + 1) - || = ||G (k) + - |||| = ||||G (k) - G|| = ||G ( (k) - )||||||G|| || (k) - ||||||G|| 2 || (k - 1) - ||||:||||G||k + 1 || (0) - || - > 0, k - > .|
The condition ||G|| 1 is sufficient for the convergence, but not necessary, i.e., the iteration sequence may converge even if ||G|| 1.
It can be shown that a necessary and sufficient condition for convergence is
where (G) is the spectral radius of the matrix G,
The error in the approximation (k) can be estimated by the formula
||| (k) - || = || (k) - (k + 1) + (k + 1) - |||||| (k + 1) - (k)|| + || (k + 1) - ||||||G|| || (k) - (k - 1)|| + ||G|| || (k) - ||.|