Lipschitz's condition makes simple geometrical sense. Let's take not a function graph of y=f (x) two any points of M1 and M2 with coordinates (x1, f (x) and (x2, f (x). Let's write the equation of the straight line passing through these points:
Especially quickly process of consecutive approximations if in point the derivative of the (x) function addresses in zero meets. In this case as approaching , value (x) aspires to zero. As:
The formulated theorem makes very simple sense. Let's say that the function carries out display of a point x to a point of y= (x). Then Lipschitz's condition from constant <1 means that the display is squeezing: the distance between points of x1 and x2 is more, than distance between their images y1= (x and y2= (x.
The found point is interesting to that it is the only general point for all pieces of the constructed sequence Using a continuity of function f (x), we will prove that it is a root of the equation of f (x) =
Convergence of iterative sequence to an equation root (it can be used for approximate definition of a root with any degree of accuracy. For this purpose it is necessary to carry out enough iterations only.
The method of tangents connected with a name of I. Newton is one of the most effective numerical methods of the solution of the equations. The idea of a method is very simple. Let's take a derivative point of x0 and we will write down in it the tangent equation to a function graph of f (x):
The first problem can be solved, having broken this interval into rather large number of intervals where the equation would have exactly one root: on the ends of intervals had values of different signs. There where this condition is not satisfied, to cast away those intervals.
The most universal is the method of halving (dichotomy): he only demands a function continuity. Other methods impose stronger restrictions. In many cases this advantage of a method of a fork can be essential.
while in the Newton method the mistake decreases quicker (corresponding to =. But in a method on each iteration it is necessary to calculate both function, and a derivative, and in a method of secants – only function. Therefore at the identical volume of calculation in a method of secants it is possible to make twice more iterations and to receive higher precision. That is more acceptable at numerical calculations on the COMPUTER, than a method of tangents.
From all ways with what it is possible (to transform the equation to a look (we choose what provides the simplest creation of schedules of y1=1 (x) and y2=2(x). In particular it is possible to take 2(x) = 0 and then we will come to creation of a function graph (which points of intersection with direct y2=2(x)=0, i.e. with abscissa axis, and are required roots of the equation (.