(PC-12xx~14xx) Laguerre method

[robve]

Shown below is a comparison of Householder 3rd order method (4th order convergence) to the Laguerre method (3rd order convergence) for solving polynomials with complex coefficients and real coefficients. For each of the four given MachEps values, the total number of iterations to converge the roots is shown with the sum or residual errors. Note that the last root or last two roots are computed directly, hence the polynomials shown in red font require 0 iterations. Also note that the residual errors are expected to typically larger for polynomials with large coefficients, since a small change in the root z may cause a large change in p(z). Finally, note that the 3rd order polynomials are easy to solve. These are included to determine the performance and stability of the method to find a single root for a range of cases.



See also the Householder forum post, which includes details on the parameters used in the implementations.