Symbolic computing inside Python (inside R)

Will we explore basic symbolic computing inside Python. In symbolic computing, the software is able to compute \(2x = x+x\) or \(\int_0^x x^2 dx = \tfrac{x^3}{3}\) without having a numerical value for \(x\). Popular software includes, Maple, Mathematica, and Sage; Python’s sympy is a basic in-house substitute, which we will explore.

import sympy as sym

x = sym.pi   # the digit pi
print(x)
## pi
print("#############")
## #############
print(x.evalf())
## 3.14159265358979
print("#############")
## #############
print(x.evalf(100))
## 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
print("#############")
## #############
x = sym.Symbol('x')   # you got to tell Python that x is symbol
print(sym.simplify((x + x)))
## 2*x
y = sym.Symbol('y')
y=sym.integrate(3 * x ** 2, x)  # integrate 3x^2
print(y)
## x**3
print("#############")
## #############

We can also do linear algebra

a = sym.Symbol('a')
b = sym.Symbol('b')
c = sym.Symbol('c')
d = sym.Symbol('d')
M =  sym.Symbol('M')
M=sym.Matrix([[a, b], [c, d]])
print(M)
## Matrix([[a, b], [c, d]])
print("#############")
## #############
print(M.det())
## a*d - b*c
print("#############")
## #############
print(M.T)
## Matrix([[a, c], [b, d]])
print("#############")
## #############
print(M.eigenvects())
## [(a/2 + d/2 - sqrt(a**2 - 2*a*d + 4*b*c + d**2)/2, 1, [Matrix([
## [-d/c + (a/2 + d/2 - sqrt(a**2 - 2*a*d + 4*b*c + d**2)/2)/c],
## [                                                         1]])]), (a/2 + d/2 + sqrt(a**2 - 2*a*d + 4*b*c + d**2)/2, 1, [Matrix([
## [-d/c + (a/2 + d/2 + sqrt(a**2 - 2*a*d + 4*b*c + d**2)/2)/c],
## [                                                         1]])])]
print("#############")
## #############
Msub = M.subs(a,1).subs(b,1).subs(c,1).subs(d,0)
values = Msub.eigenvects()
print("#############")
## #############
print(values)
## [(1/2 - sqrt(5)/2, 1, [Matrix([
## [1/2 - sqrt(5)/2],
## [              1]])]), (1/2 + sqrt(5)/2, 1, [Matrix([
## [1/2 + sqrt(5)/2],
## [              1]])])]
print("#############")

## #############

A Markov chain

Consider the three state Markov chain on \(\{1,2,3\}\) where \[p_{11} = p = 1-p_{12}\] \[p_{22} = q=1- p_{23}\] and \[p_{31}=1.\]

We will consider the case where \(q \in (0,1)\) is fixed, but will think of \(p \to 1\).

Endnotes