# GSoC 2017 Application Ranjith Kumar: Implementing Solvers for SymEngine

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# Personal Background

### Details

University : Indian Institute of Technology, Kharagpur

GitHub : ranjithkumar007

Time-zone : IST (UTC +5:30)

I am a second year undergraduate student pursuing Computer Science and Engineering at Indian Institute of Technology, Kharagpur.

I have been programming for about 2 years now. I started with C and C++. I've used C++ mainly for competitive programming contests where i got comfortable with STL. I learnt Python last year. I like the ease with which one can code in python. I also learnt web-scraping as a part of mini-project under Harsh Gupta. I developed an Image Processing based project that analyses the impact of colors in movie posters and also mentored junior students in an IEEE certified Image Processing workshop. Apart from C++/C and Python, I did projects in Java and JavaScript. I am quite comfortable using git and github for version control.

I am also a robotics enthusiast and have been part of Kharagpur Robosoccer Students' Group (KRSSG) team as Artificial Intelligence team member. KRSSG made India's first ever participation in RoboCup's Humanoid Simulation League last year in Germany. I am working on Humanoid simulation since March 2016 and have implemented startegies for goalkeeper and positioning using vornoi and delaunay triangulation.

My educational background is sufficient for the project. I have taken the following mathematical courses: Discrete Mathematics, Calculus, Probability and Statistics; programming courses: Algorithms-I, Programming and Data Structures.

Abinash Meher introduced me to SymEngine last December and I have started contributing ever since. I learnt many new topics like lambda functions,CRTP and got familiar with CMake,makefile while contributing.

### Platform details

OS : Ubuntu 14.04

Editor : SublimeText 3

### Contributions to SymEngine

• (Open) Hensel lift and zassenhaus in Integer Domain #1227; This is a rebase over #1066. Completed Hensel-lift in integer-domain. This is important for implementing solvers as factor() method needs hensel_lift in integer domain.

• (Merged) increase coverage and minor bug fixes #1217; Several bugs were fixed along the due course of adding thorough test cases. CodeCoverage increased by more than 9.3% with this PR.

• (Merged) Fix for #1154 #1165; It fixed a bug in pow. Earlier, pow(base, expo) returned zero if base is zero but it should depend on the sign of expo. Apparently, this fixed issue #1154.

• (Merged) use ComplexInf instead of throwing DivisionByZero error #1170; Instead of throwing error if denominator is zero, It is advisable to return ComplexInf or Nan depending on the numerator.

• (Merged) edit a misleading variable name #1163; It was a minor fix for a misleading variable name. This is my first merged PR.

• (Merged) Improve Log #1172; This PR improved Log function by implementing Log for negative arguments. To facilitate implementation of Inf values as suggested by isuruf, Evaluate Class was added for Infinity.

• (Merged) add asech to evaluate class #1189; Implemented asech function in evaluate class.

• (Merged) add erf and erfc to evaluate class #1192; Implemented erf and erfc function in evaluate class.

• (Merged) move Log to functions.h #1195; As Log is a OneArgFunction, It should better be placed inside functions.cpp rather than pow.cpp

• (Merged) add Symbolic constants #1196; Added two more symbolic constants namely Catalan's constant and GoldenRatio.

• (Merged) add evaluate class for nan #1202; Implemented evaluate class for nan.

• (Merged) change SymEngineException to NotImplemented wherever applicable #1201; While implementing symbolic constants, I found many occurences where NotImplementedError could have been used instead of SymEngineException. Fixed with this PR.

• (Merged) implement digamma,trigamma using polygamma #1203; Implemented digamma and trigamma functions using already implemented polygamma.

• (Merged) bug fix in upper and lower gamma #1204; fixed a non-trivial bug in upper and lower gamma. Along with this, I added base cases for both these using erf and erfc functions.

• (Merged) add xor,xnor and tests #1206; Implemented missing xor and xnor in the logic module.

• (Open) more error functions #1208; Implemented inverses of error function,complementary error function and Two-argument error function. Also, Added Derivatives of all these three functions.

• (Open) auto eval if base is E and expo is log() #1207; Evaluating expressions of type e^(log) automatically and of type e^(log +-* log...) only when expand() is invoked.

• (Merged) remove redundant check #1216;

• (Merged) remove outdated comments #1193;

• (Merged) Make formatting test fail more clear #1240; Added a travis test dedicated only for checking formatting.

• (Merged) fix bug in eval mpc #1210; Earlier, acsc and asec evaluated inverses of acos and asin. But It should be reverse. This trivial bug existed because of lack of test cases. So I started working on #1217.

### Contributions to SymPy

• (Merged) fix bug in extract_multiplicatively() #12270; Inherently, Fixed issue #12254.

• (Merged) add xnor and tests #12119; Added Xnor in logic module of SymPy. I encountered it, While I was going through the docs of SymPy to implement xor,xnor for SymEngine.

# Project Overview

Solving equations and system of equations is a crucial feature of Computer Algebra Systems and Polynomial factorization is one of the fundamental tools to implement solvers. Currently, SymEngine has implementation of Univariate Polyomials, MultiVariate Polynomials, Sets and Finite Fields. Most of the necessary tools for Solvers have been developed last year by Srajan Garg and Nishant Nikhil.

SymPy has a robust solvers module. I would like to port basic solvers like univariate solvers, trigonometric solvers and System of linear Equations to SymEngine. SymEngine's Set module is not matured enough, So I will also improve Sets to facilitate Solvers. In SymPy, Currently Major work is going around replacing solve with solveset. So, rather that portingsolve, I would port solveset to SymEngine. After implementing basic solvers module, I would also implement some applications of solvers like eigenvectors and eigenvalues of a matrix.

# Details of Project

### Wrappers for Factorisation

As Isuru pointed out in the mailing list discussion, rather than implementing SymEngine's own implementation of factorisation during GSoC period, I would add Flint wrappers for factorising the polynomials.

void fmpz_poly_factor(fmpz_poly_factor_t fac, const fmpz_poly_t G);


and fmpz_poly_factor_t is a typedef as defined below :

typedef struct {
fmpz c;
fmpz_poly_struct *p;
slong *exp;
slong num;
slong alloc;
} fmpz_poly_factor_struct;


This struct would be converted into std::pair<integer_class, std::set<std::pair<Poly, unsigned>,Poly::FactorLess>> to return factors.

SymEngine already has wrappers for fmpz_poly_struct.

Earlier Plan for factorisation (Just for future reference) -

Before Proceeding to Solvers, I would like to complete Polynomial factorisation. Currently, SymEngine has Cantor Zassenhaus method for factoring polynomials in finite fields and also hensel lifting to convert roots back to integer domain. Remaining Aspect here is implementing a factor() method that actually returns the factors in integer domain. For this, I need to implement the get square free part() method which in turn needs computation of gcd() in integer domain which is still unimplemented in SymEngine.

### Implementing GCD :

The GCD of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This algorithm is similar to trivial gcd that one learns in schools :

gcd(f, g):
if deg(f) < deg(g)
swap f, g

if g = 0
return f
else
return gcd(g, f mod g)


However this method turns out to be very slow(O(N^2)). This paper describes a faster and heuristics based algorithm.

The algorithm computes the polynomial GCD by evaluating polynomials f and g at certain points and computing (fast) integer GCD of those evaluations. The polynomial GCD is recovered from the integer image by interpolation. The final step is to verify if the result is the correct GCD. If the verfication turns out to be false, then algorithm tries to run again with a different set of points. Algorithm runs for at most 6(GCDHUE_LIMIT) times.

LCM can be found directly if we can compute GCD as

lcm(f,g) = (f*g)/gcd(f,g)


function prototypes would be similar to other functions in Univariate Polynomials.

Function Prototype :

template <typename Poly> RCP<const Poly> gcd_poly(const Poly &a, const Poly &b)


Example :


In[] : gcd_poly(x**2-3*x+2,x-2)
Out[] : x - 2

In[] : gcd_poly(x**2-3*x+2,x+2)
Out[] : x**3−x**2−4*x+4


Wrappers for FLINT and Piranha are already implemented in SymEngine.

### Completing Factorisation

Once GCD is complete, zz_get_square_free_part() is implemented and then setup for zz_factor() is ready and can be easily implemented along the lines of implementation in SymPy.

Function Prototype

std::set<std::pair<integer_class,RCP<const UIntPoly>>, FactorKeyLess>> zz_factor() const;


Examples :

In[] : zz_factor(x**3-1)
Out[] : [{1,x-1},{1,x**2+x+1}]

In[] : zz_factor(x**2-1)
Out[] : [{2,x-1}]


As it is clear from the function prototype, We need to implement a custom functor FactorKeyLess for defining the order of storing in set.

### Improve Set module :

Mathematically solving an equation means returning the set where the equation is true.Sets are the most generic and can consistently represent all type of solutions. Also, We can use ImageSet i proposed to implement, to represent even generic solutions like {n*pi,for all n in Interval(-oo,+oo)} for equation sin(x) = 0. SymPy is also rewriting its solve() to a new solveset() which would eventually be renamed to solve() depricating older solve(). Status of Sets Module in SymEngine : Presently we have the following sub-modules implemented in Sets.

• FiniteSet
• EmptySet
• UniversalSet
• Interval
• Union
• Intersection

We need a more robust implementation of Sets to facilitate solvers module. For this i am planning to implement the following in sets :

• Complement
• ConditionSet
• ImageSet

#### Complement :

It is the relative complement of a set with another set. All elements that are present in A and not in B come under Complement(A,B). If Only One argument is passed, A is assumed to be Universal Set. Complement would be implemented as a dervied class of Set.

Prototype :

class Complement : public Set {};   #Class Prototype
RCP<const Set> set_complement(const RCP<const Set> &a,const RCP<const Set> &b)  # Function Prototype



Example :

In[] : complement(FiniteSet(0, 1, 2, 3, 4, 5), FiniteSet(1))
Out[] : {0,2,3,4,5}


#### ConditionSet:

It contains a Set of elements which satisfies a given condition. Mathematically, Its set-builder form is {x | condition(x) is True for x in S}. Similar to Complement, ConditionSet would be derived class of Set

Prototype :

class ConditionSet : public Set {};   #Class Prototype


Example :

setC = ConditionSet(x, x**2 > 4, Interval(0,100))
setC->contains(5) #True
setC->contains(2) #False


#### ImageSet :

It is an Image of a set under a mathematical function. If this function can't compute the image, it returns an unevaluated ImageSet object. Its set-builder form is { f(x) | x \in this }.

Prototype :

class ImageSet : public Set {};   #Class Prototype


Example :

In[] : imageset(x, x**2, Interval(0, 4))
Out[] : [0, 16]


### Univariate Solvers -

#### Polynomial solvers

First, I would like to implement Polynomial solvers.

Definition -

• Cyclotomic Polynomial: the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients, which is a divisor of x**n-1 and is not a divisor of x**k-1 for any k < n. Its roots are all nth primitive roots of unity e**(2*i*pi*k/n), where k runs over the positive integers not greater than n and coprime to n. Help from here

Examples :

  x**6+x**5+x**4+x**3+x**2+x+1
x**4+1


This module is divided into three sub modules.

• Polynomials of lower degree(degree<=4) :

• Linear (degree = 1)
• root is trivial that is x = -b/a
• finding roots is straight forward here as well using school-grade formula.
• Cubic (degree = 3)
• general formula exists for cubic polynomials. Reference from wikipedia.
• Quartic (degree = 4)
• In general ,we can find roots of quartic polynomials using Descartes-Euler algorithm as explained here.
• cyclotomic polynomials - For solving cyclotomic polynomials, we need to know its order n. this can be found using inverse totient function after which roots can be calculated easily as e**(k*2*pi*I/n) , for 1<=k<=n and n, k being co-prime.

• integrating zz_factor() to solve polynomials that don't fall in the above categories.

• remaining set of polynomials are factorised using zz_factor() method that internally converts coefficients to galois fields, finds roots using Cantor-Zassenhaus algorithm and converts them back to integer domain using hensel-lifting. After factorising, We have a set of factors whose roots can be found using above methods and final set of roots are returned.

All these Polynomial root-finding algorithms would be wrapped under a single function solve_poly() which is directly callable but would be recommended to call through solve() which internally calls solve_poly(). Essential helper functions like _invert_real() would be implemented along the way.

Prototype :

template <typename Poly> RCP<const Set> solve_poly(const Poly& p,RCP<const Symbol> &x,
RCP<const Set> domain = Interval(-Inf,+Inf))


Examples :

In[] : solve(x**2-2*x+1,x,FiniteSet(2,3,4))
Out[] : {}

In[] : solve_poly(x**2-2*x+1,x,FiniteSet(2,3,4))
Out[] : {}

In[] : solve(x**2-2*x+1,x,FiniteSet(1,2,3,4))
Out[] : {1}

In[] : solve(x**2-3*x+2,x,Interval(0,100))
Out[] : {1,2}



#### Trigonometric Solvers

For implementing trigonometric solvers, We need helpers for simplifying trigonometric expressions. Currently, SymEngine has a naive trig_simplify() which i am planning to expand on lines with the fu model in SymPy.

Consider the following situation - If user's input equation is of form 2*sin**2(x)+2*cos**2(x)+x-2 then instead of any generic algorithm for solving this equation, We can use the simple Identity sin**2(x)+cos**2(x) = 1 and the equation now becomes deadly simple.

Fu et al's algorithm would be employed to simplify such equations.

In an Abstract way, This algorithm uses basic trigonometric transformations commonly stated in textbooks and expressed as TR1,TR2 and then simplifies the equation. Simplification of the generated expression tree is done in a bottom-up fashion.

Example..

TR7 -"Lowering the degree of cos(x)**2"

In[] : TR7(cos(x)**2)
Out[]: cos(2*x)/2 + 1/2

TR10i - "Sum of products to function of sum"

In[] : TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x)
Out[] : 2*sqrt(2)*x*sin(x + pi/6)


After we simplify the equation using trig_simplify(), the resultant equation is rewritten in terms of exp() by using conversion formulae described well here. Next step is to expand the equation possibly a rational and substitute exp(I*x) with a dummy symbol say x0 and solve the equation for the x0 and resubsititute solutions of x0 with exp(I*x). Now, We can find actual x easily by implementing an invert function that calculates its inverse .

Similar to solve_poly, solve_trig would be wrapped inside solve().

Prototype :

template <typename Poly> RCP<const Set> solve_trig(const Poly& p,RCP<const Symbol> &x,
RCP<const Set> domain = Interval(-Inf,+Inf))


Examples :

In[] : solve(tan(x)-1,x)
Out[] : {2*n*pi+5*pi/4|n in Z} U {2*n*pi+pi/4|n in Z}

In[] : solve(tan(x)-1,x,Interval(-pi/2,pi/2))
Out[] : {pi/4}

In[] : solve_trig(sin(x)**2+cos(x)**2-1,x)
Out[] : R



#### From Basic to Poly :

Prototype for solve :

template <typename Poly> RCP<const Set> solve(RCP<const Basic> b,RCP<const Symbol> &x,
RCP<const Set> domain = Interval(-Inf,+Inf))


As it is clear from Prototype, solve() would take Basic as input whereas solve_poly and solve_trig take Poly as input.So while wrapping them in solve, We need to go from Basic to polynomial.This can be done using basiconversions already implemented in SymEngine.

Say, RCP<const Basic> inp = add(pow(x, i2), add(mul(x, i3),i2)) and symbol is x that becomes the generator,say gen.So now we can use RCP<const UIntPoly> convertedinp = from_basic<UIntPoly>(inp,gen) to convert to polynomial x**2+3*x+2.

### Linear System of Equations :

Solving system of linear equations is an important feature of solvers in a CAS, these arise quite frequently in "real life", problems and can be incredibly complex sometimes. SymPy has linsolve() for solving such equations. Multivariate Polynomials are currently supported for Integer coefficients only. I am planning to solve only those equations that are representable with this(Integral coefficients).

Approach :

• First the system of equations are converted into matrix form by implementing a new function lineareq_to_matrix that converts N linear equations with M variables into an augmented matrix comprising of N*M sized matrix A amd N*1 matrix b.

• Next is to Solve Ax = b using Gauss Jordan elimination. Currently, this is implemented in SymEngine but it only supports square dense matrices. I will Overload this function to return output as Set and solve for both square and non-square matrices.

• There may be zero, one, or infinite solutions. If one solution exists, it will be returned. If infinite solutions exist, it will be returned parametrically. If no solutions exist, It will return EmptySet, else it returns a FiniteSet of Solutions.

Prototype :

RCP<const Set> guass_jordan_solve(const DenseMatrix &A,const DenseMatrix &b);

DenseMatrix lineareq_to_matrix(std::set<RCP<const MIntPoly>,RCPBasicKeyLess>);

RCP<const Set> linsolve(std::set<RCP<const MIntPoly>,RCPBasicKeyLess>eqns,std::set<RCP<const Symbol>,
RCPBasicKeyLess> symbols);

RCP<const Set> linsolve(const DenseMatrix aug,std::set<RCP<const Symbol>,


Examples :

auto A = DenseMatrix([[1, 1, 1], [2, 3, 5], [4, 0, 5]])
auto b = DenseMatrix([5, 8, 2])
In[] : guass_jordan_solve(A,b)
Out[] : [3,4,-2]

In[] : lineareq_to_matrix({3*x+2*y+z-5,x+3*y+5*z+9,4*x+3*y-5*z-2})
Out[] : DenseMatrix([3,2,1,-5],[1,3,5,9], [4,3,-5,-2])

auto aug = DenseMatrix([[1, 1, 1, 5], [2, 3, 5, 8], [4, 0, 5, 2]])
In[] : linsolve(aug,[x,y,z])
Out[] : {(3, 4, -2)}

In[] : linsolve({x+2*y -3*z-2,6*x+3*y-9*z-6,7*x+14*y-21*z-13},[x,y,z]}
Out[] : EmptySet()



### Applications of Solvers

Solvers have wide range of Applications. Some of them are addressed here.

EigenVectors and EigenValues
EigenVectors and EigenValues are frequently computed in Matrix Algebra. Currently SymEngine has a matrix module but due to lack of solvers, Eigenvectors couldn't be implemented. This issue is also raised here by Thilina. Computing them is easy and the algorithm is the standard text-book one. Breifly ,

• Compute the determinant of A - (lamda)*I .It is a polynomial in lamda of degree atmost n.
• Find the roots of this polynomial, by solving det(A - (lamda)*I = 0. The n roots are the n eigenvalues of A. They make A - (lamda)*I singular.
• For each eigenvalue , solve (A - (lamda)*I)*x = 0 to find an eigenvector x.

### Previous Implementation in SymPy

Sympy has a robust Solvers,polynomial and sets module. Harsh Gupta started rewriting solve as solve_set in 2014 and several improvements took place since then.

Solvers module is implemented at sympy.solvers and majority of the work in my project would revolve specifically around algorithms implemented in solveset.py.

Trignometric Simplification is implemeted at sympy.simplify and fu algorithm is implemented at fu.py.

Sets module is implemented at sympy.sets.

# Timeline

I have my summer vacations starting from 1st May. I am already well-acquainted with the community.In between, I have no other commitments and will be easily able to give 40 - 50 hours a week.My college will restart in mid July but I will still be able to contribute atleast 35-40 hours since there will be no exams or tests until GSoC period ends. I will also maintain a blog so that my mentors can regulate my work, and also monitor my progress.

#### Pre-GSoC

• I already went through Major Portion of SymPy's code for solvers.I would like to Inspect remaining part of it.
• Discuss the choice and implementation of the structures and algorithms with mentors
• Atleast Write and verify Pseudo-Code for fu algorithm,gcd_hueristic and other complex algorithms so that i can go at full throttle in implementing them.

#### Week 1

• Write Flint Wrappers for factor and thorough tests.

#### Week 2 & 3

• Implement the following in set module :
• Complement
• ConditionSet
• ImageSet

#### Week 4

Implement Polynomial solvers :

• Start working on Polynomial solvers by implementing solvers for lower degree polynomials(degree <= 4).
• Implement helper functions for cyclotomic polynomials like _inverse_totient.

Before Phase 1 Evaluations,Set-Up for Solvers would be finished and initial work of Polynomial solvers would be initiated.

#### Week 5

• Implement solvers for cyclotomic polynomials.
• Integrate zz_factor() with solve_poly().
• Wrap up polynomial solvers into solve.

#### Week 6, 7 & 8

Implementing trigonometric solvers.

• Implement fu algorithm for trigonometric expression simplication.
• Add rewrite_as_exp() for all trigonometric and hyperbolic functions
• Complete trigonometric solvers and wrap it in solve().

Before Phase 2 Evaluations,polynomial solvers and trigonometric solvers would be completed.

#### Week 9 & 10

Implement System of Linear equations

• implement Guass-Jordan Elimination method for solving Matrices of the form Ax=b.
• implement lineareq_to_matrix function.
• implement linsolve() for solving system of equations.

#### Week 11

• Implement some Applications of Solvers Like Finding EigenValues and EigenVectors
• Make report of all the work done on Solvers during the GSoC period.

Buffer time.

#### Week 13

• I will try to implement my initial plan for SymEngine's own factorisation module.
• Implement gcd function in Integral domain.
• Implement lcm using gcd function

#### Post GSoC

I would like to continue contributing to SymEngine and SymPy even after my GSoC period.

• Complete SymEngine's factorisation module by implementing zz_get_square_free_part() and zz_factor().
• Isuru already started making polynomials and series classes work with other Symbolic types.I would like to take up the remaining work as given in the TODO list here.