An iterative process
Let's copy for convenience the equation (1) in the form of: (3) That it is possible to receive by exchanging: . Let – zero approach, i.e. the initial approached value of a root of the equation (3). Then as the following, the 1-st approach we’ll accept
the following, the 2-nd approach will be
etc., as the
(4)
Here there is a main question: whether
comes nearer to the true solution of the equation (3) at unlimited increasing
if
at
all
values
calculated from solving process (4):
1)
,
2)
,
If in some points the derivative on the module is less than 1, and in other points it is greater than 1, to tell anything certain about convergence of iterative process impossible. It can both to converge, and to diverge. If an iterative process diverges, the reason of it often is the unsuccessful choice of zero approach. So, on Fig.1 it is shown, that the choice of zero approach essentially influences convergence of an iterative process. It directly is connected with, whether there is a zero approach in the field of where the conditions of convergence of iterative process are satisfied.
Fig.1. Dependence of iterative process convergence on a choice of zero approach
The process (4) is considered completed, if – the given accuracy of the solution. |

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>> Algebraic and Transcendental Equations
>> Method of iterations