elliptic

Tuesday, March 20, 2012

inverselliptic2

Hi all, glad to present the first try to the inverse elliptic integral of the second kind. Please try it yourself and give us a feedback!

% modulus and phase in degrees
[phi,alpha] = meshgrid(0:5:90, 0:2:90);
% values of integrals
[F,E] = elliptic12(pi/180*phi, sin(pi/180*alpha).^2);
% values of inverse
invE = inverselliptic2(E, sin(pi/180*alpha).^2);

Refs:

http://code.google.com/p/elliptic/source/browse/trunk/inverselliptic2.m
J. P. Boyd, "Numerical, Perturbative and Chebyshev Inversion of the Incomplete Elliptic Integral of the Second Kind", Applied Mathematics and Computation (January 2012)

Friday, March 5, 2010

DLT

It seems that Descending Landen Transformation converges quite well for complex arguments.
Just need to check for cuts on a complex plane and singularities.
YES!!!

Friday, February 26, 2010

About Bayes and data mining

Wiki says

In probability theory, Bayes' theorem, often called Bayes' law or Bayes' rule, and named after Rev. Thomas Bayes (pronounced /bejz/), shows how one conditional probability (such as the probability of a hypothesis given observed evidence) depends on its inverse (in this case, the probability of that evidence given the hypothesis).

The key idea is that the probability of event A (e.g., having breast cancer) given event B (having a positive mammogram) depends not only on the relationship between A and B (i.e., the accuracy of mammograms) but on the absolute probability of A independent of B (i.e., the incidence of breast cancer in general). For example, even if mammograms are 95% accurate, a positive mammogram is still most likely to be a false positive, given the relative rarity of breast cancer (about 1 in a 1000).

This result violates most people's intuition.

Curios and mysterious math even for today's times :)

Wednesday, February 17, 2010

The On-Line Encyclopedia of Integer Sequences

In my research I found extremely useful the On-Line Encyclopedia of Integer Sequences

http://www.research.att.com/~njas/sequences/

Here are some of my observations

when you lost any hope to find one mathematician that understand something about your particular function you may find him here. Just find a sequence (like A120362) and then look for the author!
the main of this encyclopedia is that you'll get immediate and exact formula for the expansion, so you just insert it into you favorite math program and thats it!
you get refs and discussion about this particular function its representations, converges problem and complex variants

So, let me take A120362: Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind. We get immediately the expansion

$E(\varphi,m)=\int_0^\varphi \sqrt{1-m\,sin^2 \theta}\,d\theta$

where

$E(\varphi,m)=\sum_{n=1,3,5,..}\Big[\sum_{s=0,1,...,(n-1)/2} a\big((n+1)/2,s\big)m^s\Big]\frac{\varphi^n}{n!}$

and

$a(n)=1,0,-1,0,4,-3,0,-16,60,-45,0,...$

for

$n=1,...,40$

So, hope the project will exist and advance!

Wednesday, February 10, 2010

Weierstraß elliptic functions

Hi the lonely reader :)

Trying to conclude the elliptic package I have to pass one very complicated stair. This the Weierstraß elliptic function ℘. The theory you can find here

   http://en.wikipedia.org/wiki/Weierstrass_elliptic_function

Here are just my plans to afford the task. Since the Weierstrass's elliptic function ℘ posses one very nice representation in terms of theta functions.
   $\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)} + e_2(\tau).$
where
      $e_1(\tau) = {\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{01}^4(0;\tau)),$
   $e_2(\tau) = -{\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau)),$
   $e_3(\tau) = {\pi^2 \over {3}}(\vartheta_{10}^4(0;\tau) - \vartheta_{01}^4(0;\tau)).$

are the roots of the equation X³ − g₂X − g₃ where g₂ and g₃ are invariants and depend only on τ, being modular forms.

The another approach (always citing the wiki page) is to calculate the Weierstrass elliptic function in terms of the Jacobi's elliptic functions. The basic relations are
         $\wp(z) = e_{3} + \frac{e_{1} - e_{3}}{\mathrm{sn}^{2}\,w} = e_{2} + \left( e_{1} - e_{3} \right) \frac{\mathrm{dn}^{2}\,w}{\mathrm{sn}^{2}\,w} = e_{1} + \left( e_{1} - e_{3} \right) \frac{\mathrm{cn}^{2}\,w}{\mathrm{sn}^{2}\,w}$
where e_1-3 are the three roots described above and where the modulus k of the Jacobi functions equals

$k \equiv \sqrt{\frac{e_{2} - e_{3}}{e_{1} - e_{3}}}$

and their argument w equals
$w \equiv z \sqrt{e_{1} - e_{3}}.$

So, will try and will see.

Saturday, February 6, 2010

Just hello!

Hello to everybody! This is my first post on the elliptic blog. Some words about. Some years ago I was "crawling" all the internet for elliptic integrals and found NO good implementation of fast algorithm for elliptic functions on Matlab. Mathematica/Maple/Octave didn't not satisfy me with velocity and other reasons.

Since the problem that had been posed in my thesis heavily depends on elliptic integrals I decided to write it by myself. So did this and posted it on mathworks.com

http://www.mathworks.com/matlabcentral/fileexchange/8805-elliptic-integrals-and-functions

still some time I start to receive requests to extend the elliptic integrals on complex arguments and to make behave the numerical version for wider range of arguments (i mean periodicity properties). So, after some work I collected some scripts to the package called elliptic and released under super open license BSD on googlecode. Now you may find it here

http://elliptic.googlecode.com/

Thats all the story.

I'm searching for collaborators or tester or developers to join the project. Any requests-suggestions-comments you can make here on the blog or in the group

http://groups.google.com/group/pelliptic

Thank you.