and a sufficiently good starting value , an approximate calculation of the zero of the terminated Taylor expansion is obtained
the following procedure in each step. The exact derivation of the method is described in Halley's method in the section on the multi-dimensional case.
algorithm
Choose a start value , a , and sit
If or stop
Solve:, (Newton step)
Solve:, (quadratic correction)
Set ,
properties
In contrast to Newton's method, one obviously needs the 2nd derivative of the function. Increasing the order of convergence is therefore only worthwhile if the calculation of the 2nd derivative is easy in comparison with the calculation of the function value and the first derivative. Other methods are obtained by approximating the zero of the Taylor expansion. An example of this would be the Halley process .
example
As a simple one-dimensional example, the calculation of the zero point with the starting value 0 should be taken. The first derivative is the second derivative
Step 1
, ,
step 2
, ,
After the 2nd step you get the function value and can cancel.
literature
Hubert Schwetlick: Numerical solution of nonlinear equations. Deutscher Verlag der Wissenschaften, Berlin 1979, 346 pp.