EXPLANATION FILE OF PROGRAM ZNEWTON =================================== Newton's Method in the Complex Domain ------------------------------------- When the generallocation of a root is known reasonably well, it is possible to apply the rapidly convergent Newton technique discussed in another chapter to very accurately determine its value. Recall the form of the Newton iteration equation in real space: x = x - f(xi)/f'(xi) i+1 i The analogous form in the complex plane is z = z - f(zi)/f'(zi) (7.4.1) i+1 i If we represent f(z) with the form µ(x,y) + i v(x,y), then the ratio in the above equation becomes f µ + iv µ µx + v µy v µx - µ µy --- = ------ = ------------- + i ------------ (7.4.2) f' µx + µy µx² + µy² µx² + µy² where µx = dµ/dx µy = dµ/dy (partial derivatives) Equation (7.4.1) can then be broken down into x and y components as follows: µ µx + v µy x = x - ----------- i+1 i µx² + µy² (7.4.3) v µx - µ µy y = y - ------------ i+1 i µx² + µy² If µ(x,y), v(x,y), dµ/ dx, and dv/dy are available, then Newton's method can be simply applied using equation (7.4.3). The program Znewton performs this operation. The functions and variables em- ployed are: Variables --------- X0, Y0 Initial guess E Convergence criterion Return on (xi+1 - xi)²+(yi+1 - yi)² < E² N Maximum number of iterations X, Y Calculated root K Actual number of iterations performed Functions --------- U = µ(x,y) V = v(x,y) U1 = dµ/dx U2 = dµ/dy Two examples of the use of this program are shown. ln the first example, the initial guess was close to one of the roots and the convergence was very rapid (quadratic after the second iteration). ln the second example, the initial guess was not close to the root, and the error roughly halved on each itera- tion up to fourth, after which the algorithm quickly closed in on the root. Undoubtedly, there are other functions for which the initial convergence rate is even slower, [ustifying the use of an algorithm such as ZClRCLE. Newton's method in the complex plane suffers from the same instabilities or weaknesses as the corresponding algorithm in real space. High curvature can very dramatically slow convergence, and zero derivatives can be disastrous, especially when round-off error is factored in. ln any case, it is best to start the iteration near the root, and to avoid regions where df/dz = O. From [BIBLI 01]. ---------------------------------------- End of file Znewton.txt