Community Bonding Period

Organization : SymPy
Solvers: Completing Solveset

23 April 2016 - 22 May 2016

by Shekhar Prasad Rajak — Posted on May 22, 2016

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Week 1 ( 23 April 2016 - 30 April 2016)

Excited about my GSoC Blog

I created my blog on around Nov-Dec 2015, mostly for GSoC ( confident :p ). But the template was still on github with random contents. Blog repo

So it was the time to replace these contents and edit the template and host on my personal site

On 23 April 2016 00:30 (IST) GSoC projects are announced, next day in the evening I edited the blog template and made it for GSoC. I added two posts ( contribution before GSoC and project) and hosted late night.

In this week I had Lab exams and Minor tests also. I had some chat/Email with the mentors about the project as well.

Week 2, 3 ( 1 May 2016 - 14 May 2016)

It was End Examination time ( 5 - 12 May). In this period I tried to be active in gitter chat and Mailing list. I worked on my previously opened PRs and new issues.

Some of them are : pull/10733, pull/10550. I related some new issues with old PRs and tried to fix them.

I tried to add new feature in decompogen method in this PR pull/11085

We had first hangout on 2 May 2016 , in which Kshitij and me choose our work for next week. I choose to work on Non Linear System of Equations and Multivariate Equation solver for Solveset.

Minutes of meeting is in Wiki page GSoC-2016-Solvers-Progress#meeting-01

Week 4, 5 ( 15 May 2016 - 23 May 2016)

After the End semester examination, we had 2nd meeting on 15 May 2016. We ( Harsh, Amit, kshitij and me) discussed about what should be done before coming on meeting and documenting ideas/mplementation on blog or somewhere.

I started working on nlinsolve to solve Non Linear System of Equations.

Auditing SymPy’s non linear system of equation solver

A non-linear system of equations is a system in which at least one of the variables has an exponent other than 1 and/or there is a product of variables in one of the equations.

How old Solver handle this :

  • SymPy uses _solve_system function to solve all kind of systems( linear, non linear, Multivariate Equations).

  • _solve_system, try to converts all the equations into Poly and checks for linear system. If all the Poly equations are linear then solve this using minsolve_linear_system or solve_linear_system.

  • If all the equations are not linear then _solve_poly_system try to solve them. If system is not zero dimensional then returns NotImplementedError. _solve_poly_system uses Groebner basis of the system, when any one equation in the basis contains only symbol, then this reduced system is solvable.

  • When solve_poly_system returns solutions for some variables, then one by one solve the equation using solve for all the remain unsolved variables.

Problems in old Solver method :

  • solve_poly_system can return solution for zero-dimensional only. So when polynomial Equation system is positive dimensional system (infinite solution) then NotImplementedError is returned by _solve_poly_system.

  • Elimination, substitution and solve_poly_system is in one function _solve_system. So it is messy.

  • When it solves for unsolved symbol, after substituting known values in the equation using solve then we don’t get solution in general form. eg. if equation is sin(x) + y = 0 and known solution is y = -1 then solve returns pi/2.

  • When it solves for unsolved symbol, after substituting known values in the equation using solve then we don’t get solution in general form. eg. if equation is sin(x) + y = 0 and known solution is y = -1 then solve returns pi/2.

My idea and implementation :

nlinsolve

I opened a PR pull/11111 with initial code.

Idea :
  • The main idea was to use solve_poly_system method ( sympy/solvers/polysys.py ) which works pretty well in old solvers, to solve zero-dimensional system.

  • solve_poly_system is used to solve equations which are polynomial.

  • If all the equations are polynomail and have finite solution (zero-dimensional) then solve_poly_system solve them.

  • When system has all the equatons polynomial but not zero-dimensional then it has infinite number of solutions, To handle these system I thought about method used in this PDF page: 9 , examples : 2.2

I implemented nlinsolve_matrix using these steps for 2 equations and 2 variables. The code can be found in this gist

This method give equation on y ( if 2 variables are x and y. Matrix created using coefficients of x power). But Matrix returns 0 for positive dimensional system ( I only checked for 2 equations 2 variables poisitive dimentional system). So we can’t get an equation of y. After seeing this issue I didn’t generalize this method for more equations and variables.</p>

  • I read more about positive dimensional systems from Kshitij’s proposal and internet. I found another way of solving them using RUR (Rational Univariate Representation), some good examples/PDFs are :

    • http://www.maplesoft.com/support/help/Maple/view.aspx?path=Groebner/RationalUnivariateRepresentation
    • http://www.m-hikari.com/imf-2010/5-8-2010/ayadIMF5-8-2010.pdf
    • href=”http://research.cs.tamu.edu/keyser/Papers/AMSDimacs2005.pdf
    • http://homepages.math.uic.edu/~jan/mcs563/lec12.pdf

These links contains examples but not the clearly the methods for positive dimensional system.

  • I am still finding a good way to get the RUR.

  • If we have system in which some of the equations are polynomial and some are not then substitution method is best (I didn’t find other symbolic way).

  • First solve the polynomial equations if present for all the combinations of symbols according to the number of polynomial equations. Store the solutions with symbols in dict. Each solution may contains symbols that can be solved using substitution method.

  • Now solve the non polynomial equations using knows solutions( one by one substitution of known solution ) and solve. After solving substitute in the previous solution if that symbol was in there.

  • If number of equations and number of variables are equal then we get solution for each variable independently otherwise solutions may depend on each other.

  • Work is still in progress for RUR and substitution method.

Problems I am facing :

1. How can we get RUR for the system of equation to get solution for +ve Dimensional.

What I think : After reading the articles and discussion with @jksoum , I found that Groebner basis and concepts of Gauss-Jordan method for linear systems is helpful to eliminate the variabels.
Now I am trying Groebner Basis of the system( which is reduced form and can be solved using proper substitution method). The main thing is elimination systematically and ordering of monomials.
After solving the groebner basis expressions we may get the answer.

2. Solveset may return Imageset/FiniteSet/Intersection/complement during substitution method.

What I think : Sometimes solveset returns real solution with intersection with R (-oo to oo) or in ImageSet. If solution is in ImageSet then need to check the expression ( if there is unsolved symbol in substitution steps).

If we get ConditionSet then NotImplementedError.

If we are solving a + 1/d for d then solveset will return FiniteSet(-1/a) complement FiniteSet(0). so need to consider first args and store the complement in dict.

3. There may be Complex solution but no real solution so need to consider both complex and real solutions (inside substitution step) ?

What I think : For complex/real solution we will get next solution( after substituting that solution). Need to check complex solution if no real solution from solveset (inside substitution method)

I continued the work and next week I tried to fix these problems and almost succeed.My further work on nlinsolve is here

Solveset for non symbol

I opened a PR pull/11128 to add this feature.

Idea :

  • First detect the symbol ( for which we are solving equation ) is not a symbol and replace it with some temporary symbol in both symbol and f ( expression )

  • We need to use Symbol only to replace, because all the other methods/functions are made for symbols .I realized this after failing many times :) .

  • After this there may be non invertable terms like Derivative, Integrals ( function is not needed here). If want to solve for f(x) then f(x) is replaced from symbol and f as well.

I defined a method check_noninverts(f, symbol) for this.

  • Now solveset treat them as Symbol and solves. At the end we need to replace back these temporary variables. For this I defined swap_back(result, swapped_symbol, non_inverts).This works for ConditionSet also. Most of the time in Trigonometric Equations doesn’t contains Non invertable terms so ImageSet was not needed.

  • All checks have passed after little modifications.



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