Computer Algebra Systems

• What are they?
  
• Area of study == Symbolic computation
• Symbolic computation vs Numerical computation
• No longer subset of scientific computing
• Cryptography and Artificial Intelligence

Why do you need them fast?

• Heavy scientific studies require such
• Celestial mechanics is one such area (Piranha)
• Symbolic algorithms are naturally more heavy
• SymEngine was born with this goal
• A cousin to the present SymPy

SymPy

• SymPy is a pure Python library for symbolic mathematics
• Code - Simple, comprehensible and easily extensible
• SymPy is standalone

>>> from sympy.integrals import laplace_transform
>>> from sympy.abc import t, s, a
>>> laplace_transform(t**a, t, s)
(s**(-a)*gamma(a + 1)/s, 0, -re(a) < 1)

This simple :o

Examples

>>> from sympy import *
>>> init_printing()
>>> x, t, z, nu = symbols('x t z nu')
>>> diff(sin(x)*exp(x), x)
 x           x       
ℯ ⋅sin(x) + ℯ ⋅cos(x)
>>> integrate(sin(x**2), (x, -oo, oo))
  ___   ___
╲╱ 2 ⋅╲╱ π 
───────────
     2     
>>> limit(sin(x)/x, x, 0)
1

Examples

>>> solve(x**2 - 2, x)
⎡   ___    ___⎤
⎣-╲╱ 2 , ╲╱ 2 ⎦
>>> y = Function('y')
>>> dsolve(Eq(y(t).diff(t, t) - y(t), exp(t)), y(t))
           -t   ⎛     t⎞  t
y(t) = C₂⋅ℯ   + ⎜C₁ + ─⎟⋅ℯ
                ⎝     2⎠
>>> Matrix([[1, 2], [2, 2]]).eigenvals()
⎧      ____         ____       ⎫
⎪3   ╲╱ 17        ╲╱ 17    3   ⎪
⎨─ + ──────: 1, - ────── + ─: 1⎬
⎪2     2            2      2   ⎪
⎩                              ⎭
>>> besselj(nu, z).rewrite(jn)
  ___   ___
╲╱ 2 ⋅╲╱ z ⋅jn(ν - 1/2, z)
──────────────────────────
            ___
          ╲╱ π

SymPy

Core capabilities|Polynomials |Calculus

         Solving equations |Combinatorics |Discrete math

          Matrices |Geometric Algebra| Geometry |Plotting |Physics 

Statistics| Cryptography| Parsing |Printing

Why SymEngine was born?

Python too slow!

but...

Lot of algorithms and modules implemented

Pythonic interface is awesome (\m/)

Need to speed up

hence,

Write the core in C++...
...and use Python wrappers to SymPy

SymEngine

• SymEngine is a fast symbolic manipulation library, C++
• SymEngine was previously called CSymPy
• Standalone CAS

RCP<const Basic> x = symbol("x");
RCP<const Basic> a = x;
RCP<const Basic> c = integer(2);

a = add(a, mul(c, pow(x, integer(3))));

Examples

>>> from symengine import *
>>> x = Symbol("x")
>>> y = Symbol("y")
>>> z = Symbol("z")
>>> ((2*x+y)**2).expand() 
4*x**2 + 4*x*y + y**2
>>> ((2*x**2+3*y)**2).expand() 
4*x**4 + 12*x**2*y + 9*y**2
>>> ((1/(y*z) - y*z)*y*z).expand() 
1 - y**2*z**2

Arithmetic

Examples

>>> from symengine.lib.symengine_wrapper import *
>>> binomial(5, 2) 
10
>>> divides(5, 2)
False
>>> prime_factors(100)
[2, 2, 5, 5]
>>> prime_factor_multiplicities(90) 
{Integer(2): 1, Integer(3): 2, Integer(5): 1}
>>> Sieve.generate_primes(50) 
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
>>> totient(-15) 
8
>>> legendre(14, 15) 
-1
>>> kronecker(9, 2) 
1

Number theory

Examples

>>> from symengine import * 
>>> A = DenseMatrix(2, 2, [1, 2, 3, 4])
>>> B = DenseMatrix(2, 2, [1, 0, 0, 1])
>>> A.det()
-2
>>> A.inv()
[-2, 1]
[3/2, -1/2]

>>> A + B
[2, 2]
[3, 5]

>>> A * B 
[1, 2]
[3, 4]

Matrix

SymEngine

Plans

* Write the core in C++
* SymEngine is hot swappable SymPy core

* Make Sage use SymEngine as it's symbolic engine

Eventually...

* Full support C, Julia, Ruby and many more wrappers

* Make algorithms as efficient as possible

* Core to multiple CAS

SymPy and SymEngine

In [1]: from sympy import *

In [2]: import symengine

In [3]: var("x y")
Out[3]: (x, y)

In [4]: a = sin(x) + y**2

In [5]: type(a)
Out[5]: sympy.core.add.Add

In [6]: z = symengine.Symbol("z")

In [7]: type(a + z)
Out[7]: sympy.core.add.Add

In [8]: type(z + a)
Out[8]: symengine.lib.symengine_wrapper.Add

Python wrapper to C++ code

The Problem!

C++ function/file

Interface

Use it from Python

Solutions

• Using Python C API

• Pyrex

• Boost.Python

• SWIG

• SIP

• Interrogate

• PyCXX

• Ctypes

• Py++

• f2py

• PyD

• Robin

Cython -

How SymEngine does it?

Cython

• Cython is a blend of Python and C.
• Improving Python performace
  
• Wrapping or exposing libraries
  

Improving Python performance

Python

C/C++

def fib(n):
    a,b = 1,1
    for i in range(n):
        a, b = a+b, a
    return a 

int fib(int n)
{
    int tmp, i, a, b;
    a = b = 1;
    for(i=0; i<n; i++) {
        tmp = a; a += b; b = tmp;
    }
    return a;
} 

def fib(int n):
    cdef int i, a, b
    a,b = 1,1
    for i in range(n):
        a, b = a+b, a
    return a 

Cython

Underlying magic

Cython source file
fib.pyx

C extension file

fib.c

Python Extension Module

fib.so

Wrapping code

cdef extern from "string.h":
    # Describe the interface for the functions used.
    int strlen(char *c)

def get_len(char *message):
    # strlen can now be used from Cython code (but not Python)…
    return strlen(message) 

Cython code

Call from Python

>>> import len_extern
>>> len_extern.strlen
Traceback (most recent call last):
AttributeError: 'module' object has no attribute 'strlen'
>>> len_extern.get_len("woohoo!")