This calculator finds the root of the equation approximately by means of the Newton's method. To get started, you need to input your task's data.
The Newton's method is numerical, as mentioned above, it finds the root of the equation approximately. However, you can predefine the accuracy of root finding before starting the calculations.
Consider the equation
The formula for finding the root of the equation is as follows:
and - are the approximate values of the root of the equation on the -th and ()-th iterations correspondingly, - the value of the function at the point , - the value of the derivative of the function at the point .
As one can see, in order to get started one need to set the point - the initial approximation to the root of the equation . The choice of the point determines whether the algorithm will converge to the solution or not. The convergence of the method is quadratic, but it deteriorates dramatically if we are looking for a multiple root of the equation, i.e. if and simultaneously , where - is the multiple root of the equation .
Calculations according to the formula above can be continued infinitely, so, in practice, a certain criteria is needed that will determine whether we need to continue calculations or not. As a rule, the criteria for stopping the calculations is used, based on the increment or on the difference between the function and zero at some point .
The criteria for stopping the calculations based on increment is given by the following formula:
i.e. the absolute value of the difference between two consecutive approximations to the root of the equation ( and ) must be less than some predefined value .
The criteria for stopping the calculations based on the difference between the function and zero is determined by the following formula:
absolute value of the difference between the function at some point and zero is less then .
At the same time, if the sequence doesn't converge to the root, than the stop criteria will not work and the root search process will continue indefinitely. To prevent this situation, in practice, the calculations are stopped after a predefined number of iterations.
The figure below shows the geometric interpretation of the process of finding the root of the equation by the Newton's method.
At the point we construct the tangent to the function . The tangent equation to the graph of the function at this point has the form:
Find the intersection point of the obtained tangent with the abscissa axis, i.e. consider a point with coordinates . Substituting the coordinates of the point specified, in the tangent equation, we get:
From this equation, find :
Continuing this process, we obtain the formula of the Newton's method given above. Due to the fact that the tangent is actually constructed at each iteration, the Newton's method is also sometimes called the tangent method.