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Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
Two-dimensional generalization of the Muller
root-finding algorithm and its applications
Plamen Fiziev, Denitsa Staicova
Sofia University

Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
Why another root-finding algorithm
Well-known algorithms and their two-dimensional generalisations:
Methods: Convergence Complex Generalisation
Bisection Linear (~1) No* Yes (The bisection method
in higher dimensions, Wood)
Newton Quadratic (~2) Yes Newton-Raphson's
method
Secant Superlinear (~1.6) Yes Broyden's method
Brent
The problem:
Secant, bisection and
inverse quadratic
interpolation
No*
Unknown
There are cases where those algorithms are not efficient enough, for
example in systems featuring the Heun functions!

Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
The Heun functions
Particular local solutions of a second-order linear ordinary differential
equation of the Heun type with 4 regular singularities.(see P.Fiziev. Class.
Quant. Grav.27:135001, 2010 /arXiv:0908.4234v4 [gr-qc]/ and references therein.)
Those singularities may coalesce to form an irregular singularity and thus
the ODEs become the confluent Heun equations with their respective
solutions (confluent Heun functions, biconfluent Heun functions, double
confluent Heun functions, triconfluent Heun functions).
Since the Heun functions generalize the hypergeometric function (also Lame,
Mathieu and spheroidal wave functions), they would appear in many nonlinear
problems from quantum mechanics to astrophysics. Recent paper on
the application of the Heun functions in physics (arXiv:1101.0471v1 [math-ph]).
The only software package that can work numerically with the Heun functions
is MAPLE. Not without problems! Numerical errors, bad derivatives and
slow evaluations in the complex plane.
We need a root-finding algorithm that avoids derivatives and that has good
convergence!

The one-dimensional Muller algorithm
The one dimensional Muller algorithm: for every 3 points
and the values of the function at them:
we find as:
Advantages:
Superlinear convergence (~1.84)
Works in the complex plane
Works well with special functions like the Heun function („The Spectrum of
Electromagnetic Jets from Kerr Black Holes and Naked Singularities in the Teukolsky
Perturbation Theory““ D.S., P.Fiziev, Astrophys Space Sci (2011) 332: 385-401,
arXiv:1002.0480v3 [astro-ph.HE])
Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011

Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
Two-dimensional generalization of the Muller algorithm
The idea:
Ideally, to solve the system [F 1 (x,y)=0, F 2 (x,y)=0], one expresses the relation
y(x) from one of the equations, it is substituted in the other equation to find x
and then returning back to y(x), one finds y.
In most cases, however, finding directly y(x) is impossible.
In the two-dimensional Muller algorithm, we find the linear relation y(x)
approximately and then we follow the above-described procedure.
To find the approximate relation y(x) at each iteration, we form the plane
passing trough 3 points of one of the functions:
C 1 x i +C 2 y i +C 3 = F 2 (x i ,y i ) ,i=n-2,n-2,n
and then we intersect it with the plane z=0 to find the line y(x).
We substitute y(x) in F 1 (x,y) and use the one-dimensional Muller algorithm to
find x n+1 .
We then substitute x n+1 in F 2 (x,y) to find y n+1 .

Two-dimensional Muller
algorithm
Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
M1
M2

Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
Numerical
experiments:
For complex (x,y):
t N

For complex (x,y):
t N

In the general case:
t B

Physical application: The problem of Quasi-Normal Modes
(QNM) of a non-rotating black hole
The Regge-Wheeler equations:
:
(θ = π-10 -7 , |r|=20, ε- parameter)
The results:
l=1.99(9)
(17 digits)

Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
Summary
The one-dimensional Muller method was generalized to two-dimensions
using linear relation y(x).
The two-dimensional Muller method confirmed its usability in both the test
systems and in real physical problem featuring the Heun functions.
It proved to work comparably to the well-known Newton method and Broyden
method.
In certain systems, the two-dimensional Muller method works better than the
other two algorithms.
For more details: arXiv:1005.5375v1 [cs.NA]

Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011
Acknowledgements
This article was supported by the Foundation "Theoretical and
Computational Physics and Astrophysics", by the Bulgarian National Scientific
Fund under contracts DO-1-872, DO-1-895, DO-02-136, and Sofia University
Scientific Fund, contract 185/26.04.2010.
Thank you for your attention!