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

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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!

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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!

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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

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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 .

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Two-dimensional Muller

algorithm

Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011

M1

M2

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Fourth IRC-CoSiM Workshop, Tryavna, 28.04-1.05.2011

Numerical

experiments:

For complex (x,y):

t N

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For complex (x,y):

t N

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In the general case:

t B

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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)

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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]

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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!