shubhamtibra/Holonomic_Discussion

This gitter chat room is used to discuss about the GSoC'16 project "Implementation of Holonomic Function".

Kalevi Suominen

@jksuom

I think it diverges always when

p > q+1

, even if some of the

a_i

are negative (non-integers, of course).

p = q+1

, the radius of convergence is 1.

This should follow from the simple ratio test.

Subham Tibra

@shubhamtibra

Yes you are right. The series only terminates if there is one or more nonpositive integer in a- parameters.

Subham Tibra

@shubhamtibra

I actually didn't consider complex roots in the final solution yet. I will add them too.

Kalevi Suominen

@jksuom

There should be no essential difference in the treatment. (Probably no difference at all.)

Subham Tibra

@shubhamtibra

Hi Kalevi. I probably won't be able to contribute for the next couple of days. Will be travelling back to college.

Kalevi Suominen

@jksuom

Hi. There is no need to be coding all the time, though you have been doing that. Besides, you have already completed about all that I would have expected.

Subham Tibra

@shubhamtibra

Thanks, very glad to know that. Good Night. :)

Ondřej Čertík

@certik

@shubhamtibra I went over all issues, and reopened those that are not fixed yet.

Thanks for fixing all the other issues.

How does the conversion to hypergeometric function work?

I mean, to MeijerG function.

Will it always work?

I.e. if the holonomic function is a linear combination of MeijerG functions, will this always work?

Subham Tibra

@shubhamtibra

NOT ALWAYS. The current algorithm rather converts first to a linear combination of hypergeometric functions and then converts each hyper in the combination to meijerg. So if a function have G-function representation but not hypergeometric, it won't work.

There doesn't seem to exist an algorithm converting Holonomic functions directly to G-functions.

Kalevi Suominen

@jksuom

This method will only produce G-functions whose Slater expansion consists of a single hypergeometric series (m = 1). There does not seem to exist a general (symbolic) method for introducing other G-functions.

Subham Tibra

@shubhamtibra

Hi Kalevi! I couldn't write a blog post last week. Would it be Okay if I write about last week in this weeks blog post?
Also I am thinking about writing a code that itself finds the domain for the polynomial coefficients in the differential equation in a new PR. Is there anything else you have in mind we should do first?

Kalevi Suominen

@jksuom

I think it is quite ok. I know what you have accomplished, and if the others will learn that a little later, it should not matter too much. Automatically finding the coefficient domain would surely be appreciated by the users. That would probably solve the issue sympy/sympy#11323 raised by Ondřej.

Subham Tibra

@shubhamtibra

Yes, the issue is my motivation for implementing it.

Subham Tibra

@shubhamtibra

We can also add support for singular initial conditions in operations like add, multiply, integrate etc. .

Kalevi Suominen

@jksuom

It seems that initial conditions at singular points are particularly useful when holonomic functions shall be connected with G-functions and other know special functions. On the other hand, initial conditions at regular points are suitable for numerical work.

Subham Tibra

@shubhamtibra

Good Evening Kalevi! I wanted to discuss a question regarding conditions at singular point in integration.
For instance we want to integrate

cos(x)^{2}/x

with singular_ics = [(-1, [1, 0, -1])], i.e. the series is of the form

x^{-1}(1 + 0 - x^{2} ....)

1/x - x + ...

. If we integrate the initial terms of the series to compute singular_ics of the result, the term 1/x will become log(x). In cases where there is no term of x with power -1 we can compute the singular_ics directly by integration of initial terms of the series. What should be the singular_ics for this case?

Kalevi Suominen

@jksuom

General holonomic functions also have these logarithmic terms. They appear at regular singular points when some roots of the indicial equation differ by an integer (or there is a multiple root). Such expressions may become quite complicated in general if several roots are involved. In case of two roots the situation might be controllable. See e.g. the series expansions of the Bessel functions of the second kind. But it may be too ambitious to include them in this gsoc project. I am not sure of how the initial conditions should be defined. Probably something like this: log(x) multiplied by an initial condition of the more general type (i.e. x**s multiplied by a constant and optionally some low order terms), in addition of the usual initial conditions at a regular singular point.

Kalevi Suominen

@jksuom

Look at http://dlmf.nist.gov/10.8, Power Series, to see how the logarithm enters.

Subham Tibra

@shubhamtibra

This looks complicated. Should we implement it later?

Kalevi Suominen

@jksuom

I think it would be too complicated for this project.

So we should probably stop with NotImplementedError when something like this will appear.

Subham Tibra

@shubhamtibra

Hi Kalevi! I can't figure out any algorithm to compute singular_ics after multiplication of two holonomic functions. Do you have any thoughts on that?

Kalevi Suominen

@jksuom

It would seem that if you have two (partial) representations of the type

x^{s_1} f_1(x)

and

x^{s_2} f_2 (x)

the product would contain

x^{s_1+s_2}f_1(x)f_2(x)

with fairly obvious initial conditions. But maybe this is not what you wanted to ask?

Subham Tibra

@shubhamtibra

I actually was asking this.

Subham Tibra

@shubhamtibra

I got it now. :)

Kalevi Suominen

@jksuom

Hi. If the tests passed in Travis even with the typos, that would probably mean that those lines were not tested. Or could there be another explanation?

Subham Tibra

@shubhamtibra

Those lines should have been executed on travis because they did when I ran the tests locally. The lines were returning the final answer after multiplication. I am also stumped.

Kalevi Suominen

@jksuom

In any case, sol must have been available in a dictionary, if not in the local dictionary, then in the dictionary of one of the calling routines. Its value may have been correct by chance.

Subham Tibra

@shubhamtibra

I think you are right.

Subham Tibra

@shubhamtibra

Hi Kalevi. I've added things I could think of in sympy/sympy#11422. You can also give your thoughts now.

Kalevi Suominen

@jksuom

Hi. I just saw it and wrote some comments.

Kalevi Suominen

@jksuom

I wonder if some additional initial conditions should be added to the result of expr_to_holonomic.

Subham Tibra

@shubhamtibra

Do you mean the method should return this new initial condition for more types of functions? It currently computes this initial condition only for the algebraic functions.

Kalevi Suominen

@jksuom

It seems that the condition may not suffice to define the function. For example, all constant multiples of x have value 0 at the origin. The coefficient of the first nonzero power is needed.

Ondřej Čertík

@certik

@shubhamtibra what's the problem regarding sympy/sympy#11323 ?

Did I do something wrong when using symbolic variables as coefficients?

Subham Tibra

@shubhamtibra

No. Of course not. What you used should be the ideal way of doing it while using symbolic variables. At present there are additional keyword arguments like specifying the domain, and the number of initial conditions to fix the problem. :)

Kalevi Suominen

@jksuom

I think it is otherwise ok now, but we have to use singular initial conditions since the equation is singular at 0.

Subham Tibra

@shubhamtibra

Yes. We should have an algorithm for computing singular_ics for more functions.

Subham Tibra

@shubhamtibra

I changed the title of the PR to a more descriptive one.

Subham Tibra

@shubhamtibra

Hi Kalevi! I was thinking next to add functionality in expr_to_holonomic so it can calculate singular_ics for more types of functions. Do you have any other thing in mind to do first?

Kalevi Suominen

@jksuom

No. I think that could be the next task.

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