ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 09 Aug 2013 05:04:27 -0500Evaluating values of the Weierstrass $\wp$-functionhttps://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/I would like to know how can we evaluate the Weierstrass $\wp$-functions. That is, I would like to find $\wp(\theta,\omega,i\omega)$ for some $\theta,\omega\in\mathbb{R}$.
I'm only able to find a function which outputs the Laurent series of the Weierstrass $\wp$-function when an elliptic curve has been entered. Should I evaluate that laurent series?Tue, 06 Aug 2013 04:39:19 -0500https://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/Answer by Luca for <p>I would like to know how can we evaluate the Weierstrass $\wp$-functions. That is, I would like to find $\wp(\theta,\omega,i\omega)$ for some $\theta,\omega\in\mathbb{R}$.</p>
<p>I'm only able to find a function which outputs the Laurent series of the Weierstrass $\wp$-function when an elliptic curve has been entered. Should I evaluate that laurent series?</p>
https://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/?answer=15328#post-id-15328The function `weierstrass_p()` of `EllipticCurve` returns the Laurent expansion of $\wp$ at the origin, hence evaluating it only gives a reasonable approximation near it.
Nothing comes to my mind to do this kind of numerical evaluation straightforwardly in Sage. If you have access to Maple, you could give a shot at the NumGFun package, part of the AlgoLib library <http://algo.inria.fr/libraries/>. It has support for the numerical evaluation of functions satisfying linear differential equations with polynomial coefficients.
The author of NumGFun gave a talk on it at Sage Days 49 <http://www.marc.mezzarobba.net/#expose-sd49>. As you can read in the slides, nothing of it is already in Sage, but there are plans for the close future. See, for example, <http://trac.sagemath.org/ticket/14996>, which will add functionality similar to what you need.Wed, 07 Aug 2013 04:29:35 -0500https://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/?answer=15328#post-id-15328Comment by Luca for <p>The function <code>weierstrass_p()</code> of <code>EllipticCurve</code> returns the Laurent expansion of $\wp$ at the origin, hence evaluating it only gives a reasonable approximation near it.</p>
<p>Nothing comes to my mind to do this kind of numerical evaluation straightforwardly in Sage. If you have access to Maple, you could give a shot at the NumGFun package, part of the AlgoLib library <a href="http://algo.inria.fr/libraries/">http://algo.inria.fr/libraries/</a>. It has support for the numerical evaluation of functions satisfying linear differential equations with polynomial coefficients.</p>
<p>The author of NumGFun gave a talk on it at Sage Days 49 <a href="http://www.marc.mezzarobba.net/#expose-sd49">http://www.marc.mezzarobba.net/#expose-sd49</a>. As you can read in the slides, nothing of it is already in Sage, but there are plans for the close future. See, for example, <a href="http://trac.sagemath.org/ticket/14996">http://trac.sagemath.org/ticket/14996</a>, which will add functionality similar to what you need.</p>
https://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/?comment=17172#post-id-17172Nice, I didn't know that. Then you may try using the formulas given here: <http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Relation_to_Jacobi_elliptic_functions>Fri, 09 Aug 2013 05:04:27 -0500https://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/?comment=17172#post-id-17172Comment by Eviatar Bach for <p>The function <code>weierstrass_p()</code> of <code>EllipticCurve</code> returns the Laurent expansion of $\wp$ at the origin, hence evaluating it only gives a reasonable approximation near it.</p>
<p>Nothing comes to my mind to do this kind of numerical evaluation straightforwardly in Sage. If you have access to Maple, you could give a shot at the NumGFun package, part of the AlgoLib library <a href="http://algo.inria.fr/libraries/">http://algo.inria.fr/libraries/</a>. It has support for the numerical evaluation of functions satisfying linear differential equations with polynomial coefficients.</p>
<p>The author of NumGFun gave a talk on it at Sage Days 49 <a href="http://www.marc.mezzarobba.net/#expose-sd49">http://www.marc.mezzarobba.net/#expose-sd49</a>. As you can read in the slides, nothing of it is already in Sage, but there are plans for the close future. See, for example, <a href="http://trac.sagemath.org/ticket/14996">http://trac.sagemath.org/ticket/14996</a>, which will add functionality similar to what you need.</p>
https://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/?comment=17176#post-id-17176The Jacobi elliptic functions in #14996 are already in Sage, the patch just improves them.Wed, 07 Aug 2013 19:45:41 -0500https://ask.sagemath.org/question/10419/evaluating-values-of-the-weierstrass-wp-function/?comment=17176#post-id-17176