Name: Gaurav Dhingra

University: Indian Institute of Technology Roorkee

Email: gauravdhingra.gxyd [at] gmail.com

GitHub user name: gxyd

IRC Handle: gxyd [at] freenode.net

Blog: https://gxyd.github.io

Time Zone: IST (UTC +5:30)

Hello, I am Gaurav Dhingra a 4th year undergraduate student of Applied Mathematics at IIT Roorkee, India. I work on Ubuntu 14.04.5 LTS with vim as my primary text editor. I do my work with vim since it allows me work quickly with the code when added with vim-extensions using Vundle. I have been programming for 3 years now, I am proficient in Python, C++. I started using Python in parallel with contributing to SymPy. Python's list comprehension is a cool feature missing in other programming languages. I contributed to Pi-Producing Program which uses SymPy to generated "PI" in different ways. I like Python since as a math major Python allows me to convert mathematical ideas into code without much programming langauge barriers. Editing wikipedia articles is something that i also like so I have often edited the SymPy wikipedia article quite a few times.

By now, I have used quite a few computer algebra systems including GAP, Axiom, Maple and even after being a commiter of SymPy for than a year, I can say that the printing system of SymPy is something that is missing in other computer algebra system.

>>> print(latex(Integral(x**2/(1 + x**4), (x, 0, oo))))
\int_{0}^{\infty} \frac{x^2}{1 + x^4} dx

• (Merged) M.row_del(index) and M.col_del(index) now raise IndexError for out of bound Index #9468
• (Unmerged) M.row_insert() and M.col_insert() now act in-place for all indices #9477
• (Unmerged) transpose(MatMul(<Symbol>, Matrix)) now works properly #9507
• (Merged) y*(x*M) != x*(y*M) for non-commutative symbols x and y #9520
• (Merged) multiplicity(n, 0) raises error for any `n` #9528
• (Merged) Mod(zoo, 0) now returns nan and fixed one typo #9546
• (Merged) splitting simplify.py #9553
• (Merged) solveset(Abs(x) + 1, x) now returns EmptySet() typo fixed #9570
• (Merged) redudant declaration removed fixes issue #9573 #9574
• (Unmerged) Union of set containing UniversalSet with non-intersecting Interval #9578
• (Closed) Complement of UniversalSet with Union of Sets works #9579
• (Merged) Limit(x, x, a).free_symbols now returns set([a]) #9581
• (Merged) invert_real for sin and cos now operates correctly #9591
• (Unmerged) invert_complex added for trigonometric func. #9604
• (Merged) Interval - FiniteSet(x) is now returned unevaluated #9682
• (Closed) Intersection(Interval, FiniteSet(m, n), FiniteSet(n)) now works correct #9683
• (Merged) solveset_real(Eq(x, 1), x) fixed #9688
• (Merged) combsimp for work on factorials #9711
• (Merged) Range Function for rational functions #9719
• (Merged) imageset for Piecewise under Interval added #9763
• (Closed) solveset_real(abs(f1/f2), x) for f1,f2 polynomials in x #9772
• (Merged) not_empty_in added to codomain.py #9779
• (Merged) soleveset title_doc changed #9781
• (Closed) ImageSet(Lambda(n, nan), any_set) now returns EmptySet #9734
• (Merged) reference to sphinx docs for normalize_theta_set #9735
• (Merged) solveset(1/exp(x), x) returns S.EmptySet #9736
• (Closed) singularities((x**2 - 1)/(x**3 - 1), x) returns (1,) #9741
• (Closed) solveset_complex(sinh(x), x) returns ImageSet(Lambda(n, n*I*pi), Integers) #9750
• (Closed) [WIP] matrix.rank() with unkown .is_zero raise NotImplementedError #9793
• (Merged) solveset_real(x**3+1, x) returns FiniteSet(-1) #9804
• (Merged) Complements of FiniteSet with Symbols returned unevaluated #9814
• (Merged) test for solveset(Piecewise(((x - 2)**2, x >= 0), (0, True)), x, S.Reals) #9851
• (Unmerged) [WIP] Implementation of arbitrarily indexed Unions and Intersecions #9853
• (Merged) solveset(Abs(sin(x)) + 1, x, S.Reals) returns EmptySet #9857
• (Merged) ConditionSet API changed #9864
• (Merged) symbols -> Dummy_symbols in ComplexPlane #9867
• (Merged) ComplexPlane->ComplexRegion name changed #9873
• (Closed) Parenthesize printing of Unions having Complement #9878
• (Closed) [WIP] fuzzy_set implementation #9897
• (Merged) is_convergent method implementation for Sum #9906
• (Closed) real_bound added to Function class #9911
• (Merged) solveset(2*x + 1/(x - 10)**2, x, S.Reals) works #9930
• (Merged) Docstring Python escaping backslash in latex strings added #9938
• (Merged) is_increasing(), is_decreasing() signature changed #9951
• (Merged) Union(Interval(-oo, oo), FiniteSet(1)) returns Interval(-oo, oo) #9958
• (Closed) solve_univariate_inequality(x**2 >= 0, x) returns (-oo, oo) #9960
• (Merged) is_convergent() for Product #9982
• (Merged) Probability for empty set returns S.Zero #10004
• (Merged) oo**e for non-real complex e handling #10029
• (Merged) docs changes in rv.py #10039
• (Merged) Implementation of AccumBounds (different from Limit) in SymPy #10051
• (Merged) Intersection of sets containing common symbols handled #10079
• (Merged) X.pdf(x) for invalid x in DieDistribution raises ValueError #10083
• (Merged) imageset of first interval of singularities included #10115
• (Merged) pretty printing of Cycle works #10183
• (Merged) docs and code fixes for Combinatorics module #10229
• (Merged) .is_group for PermutationGroup now returns True #10230
• (Merged) latex printing of Cycle, Permutation #10261
• (Closed) [WIP] FiniteGroup in SymPy #10263
• (Unmerged) [WIP] FreeGroup implementation in SymPy #10350
• (Unmerged) [WIP] SymmetricGroup is now a class not function #10363
• (Merged) Simplification of an expression with exponent containing assumption symbol #10378
• (Merged) normalizing an open Interval now works #10421
• (Merged) ComplexRegion satisfies the Key Invariant property #10470
• (Closed) zeta(x) for (Re(x) - 1) nonzero is Finite #10479
• (Merged) Equality and printing of S.Complexes #10505
• (Merged) printing of Permutations for max element taken care #10558
• (Merged) Fix typo in "printing" tutorial #10917
• (Merged) Fix repr printing of Permutation and tests for str, repr printing #12031
• (Merged) Fix Subs._eval_derivative function for higher order derivative #12066

My proposal is to improve the Symbolic integrator of SymPy.

SymPy's current integrator module does a pretty good job in computing whatever is thrown at it. The main algorithm used in SymPy for symbolic integration is the Risch algorithm, though there are others as well like Risch-Norman algorithm, table look up. The aim of my project is to complete the transcendental function integration of Risch algorithm that includes completing Chetna's work done in GSoC 2013 on transcendental functions integration [1], tying the loose end of previous year's GSoC and adding capability to integrate purely algebraic function.

• Completing the previous remaining work.

Complete the pull request #2380.

• Don't hardcode the value of a in the functions in cds.py. And the input should be a, not sqrt(a), (so, e.g., the input should be -1, not sqrt(-1)).
• is_deriv returning logs is wrong.
• Make sure all tests pass (check Travis).
• I ran the coverage report on the branch of Chetna and it is missing a large number of tests. For `cds.py` it is 70% and similarly for `prde.py`. So I will add necessary tests.

Kalevi had created a pull request #11761 [3] on recursive call to the param_rischDE in the function 'prde_no_cancel_b_small'.

There are a plenty of #TODO's and NotImplementedError in the integration module:

Parametric logarithmic derivative [4]: section: 7.3, the complete solution is provided by structure theorem that is not implemented. Though a heuristic method 'parametric_log_deriv_heu' is implemented which still is not complete and also requires does need writing of test cases, so if this heuristic fails, the structure theorem approach will need to be used. There doesn't seem to be any pseudo-code available for this.

Passing information for why integral was non-elementary: see line 685 in `risch.py`. For this we could use a class-level variable named `message` in class associated with class `Integral` which will be set to appropriate str value. So instead of only raising `NonElementaryIntegralException` first message is set a value. Since each place in the code that raises `NonElementaryIntegralException` has a specific reason in the algorithm why the integral must be non-elementary if the code reaches that point.

For the algebraic function integration, a description is given on Trager's page 13 to obtain canonical forms for the ones which can't be evaluated as well as for those which are not integrable(non-elementary).

• See prde.py line number 456, implement Parametric Risch DE.

(a). theta_i = log(z), where z \in K(x, theta1, theta2, ..., theta_(i-1)), i.e. theta_i is a written as a logarithm of field extension of theta1, theta2, ..., theta(i-1).

(b). theta_i = z = exp(y) for y \in k_i i.e. theta_i is written as an exponential of field extension of theta1, theta2, ..., theta(i-1).

(c). theta_i = v = tan(u) for u \in k_i i.e. theta_i is written as a tangent of field extension of theta1, theta2, ..., theta(i-1).

(d). theta_i = u = atan(v) for v \in k_i i.e. theta_i is written as a tangent of field extension of theta1, theta2, ..., theta(i-1).

• Implement the integration of algebraic extensions, currently only transcendental extensions are implemented. Read conversations after message [6]

Computing Integral basis: Algorithm for computing integral basis for our function field, though computing it is an ongoing research and computing it is no harder than computing Pusieux expansion.

A class named `IntegralBasis(f, x, y)` Returns the integral basis of the algebraic function field of.

The `.integral_basis` method of `IntegralBasis`

• Parameters*: `f`: sympy.Expr

• Returns*: list, Expr i.e A list of rational functions representing an integral basis.

This will help in: This will determine the nature of poles of an algebraic function.

How are we going to do that? Present as chapter 2 in [9]. Anything necessary to complete this? This algorithm constructs the algebraic part of answer by a generalization of Hermite's reduction. So going through the Hermite reduction present in Bronstein [10] will help.

Algorithm to find absolutely irreducible factors of multivariate polynomials

Algorithm finding the genus of a curve

What will happen in case the problem is not integrable? Even when the problem is not integrable it will be reduced to a simpler version of the original problem.

A good resource for integration of algebraic functions is [8].

The Axiom computer algebra system has a powerful integration module and algebraic function integration is also there in it, so it would not be very hard to verify the correctness of the code with some fairly sophisticated examples. Also since it is open source, its documentation can also be easily accessed.

Considering my last year's experience with GSoC, we didn't followed much according to the time (for good reasons). But still having a time-line helps in acting as a self check.

• Will go through and complete the pull request #2380 on Coupled Differential System.
• Complete the Parametric risch differential equation PR #11761.

• Support for trigonometric extensions.

• Purely algebraic function integration.

Last summer also I worked on SymPy for Google Summer of Code, which was my first experience with open source, now its been exactly 2 years since my first commit got merged in SymPy. My project was to extend implementations in Computational Group Theory, and it was successful. I decided to continue as a developer of SymPy. I blog at https://gxyd.github.io, which was also my blog for last summer Google Summer of Code project.

I will devote 30+ hours/week (updated weekly hour work, see faq). My summer break starts on 1st May. I can use this time to get a head start into the tasks at hand too. Also, I can efficiently use the time before the summers to gather all the necessary information and details on my project, so I can begin work without any delays, so I will try to work from there onwards too.

We have a gitter channel on [7] which we plan to continue using, though time for communication between me and Aaron (probably he will be mentoring) since of time difference, but with a secondary mentor like Kalevi(his knowledge is always helpful), I think we can get things done in a better way. Since of my last GSoC project I have some algorithmic exposure to abstract algebra.

I have gone through the Lazard-Rioboo-Trager algorithm (since because complete Risch algorithm is a generalization of this) and chapter 5 of Manuel Bronstein [10].

[1] #2380, #2184, #2077, #2034

[2] Risch algorithm for symbolic integration discussion thread

[3] https://github.com/sympy/sympy/pull/11761

[4] https://github.com/sympy/sympy/blob/master/sympy/integrals/prde.py#L566

[5] ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation Pages 207-211

[6] https://gitter.im/jksuom/sympy?at=58d035cffe6a638b1ae6f2b4

[7] https://gitter.im/jksuom/sympy

[8] James Harold Davenport, On the Integration of Algebraic Functions

[9] Integration of Algebraic Functions by Barry Marshall Trager

[10] Manuel Bronstein Symbolic Integration 1 Transcendental Functions