SymPy

A Symbolic Mathematics packagein and for Python

Getting Started

x,y = symbols('x y') a = 2*x ＋ y a + y expand(a**3) factor(x**3+3*x**2+3*x+1) simplify(x**2-y**2-(x+y)*(x-y))

Substitution

expr = cos(x) + 1

expr.subs(x,y)

expr.subs(x,0)

expr = x**y

expr = expr.subs(y,x**y)

(repeat a few times)

Trigs

trigx = sin(2*x) + cos(2*x)

expand_trig(exprx)

trigsimp(cos(x)**2 + sin(x)**2)

Also for hyperbolic functions:

trigsimp(cosh(x)**2=sinh(x)**2)

sympify and evalf

simpify('x**2+2*x+4')

expr = sqrt(8)

expr.evalf()

pi.evalf(100)

simplification

What is simplification?

simplify((x**3 + x**2 -x -1)/(x**2 +2*x +1))

simplify((x**4 – 1)/(x - 1))

simplify((x**4 – 1)/(x**2 – 1))

expand((x+2*y)**3)

cancel((x**4 - 1)/(x – 1))

cancel((x**2 + 2* x + 1)/(x**2 - 1))

Watch out for powers:

xp,yp = symbols('x y',positive=True)

a,b = symbols('a b', real=True)

powsimp(x**a*y**a)

powsimp(xp**a*yp**a)

Exponentials and logarithms

ln(x)

expand_log(log(x*y))

expand_log(log(x/y))

expand_log(log(x**2))

expand_log(log(x**n))

expand_log(log(x**a))

logcombine undoes expand_log (if possible)

Calculus

diff(expr,var,var,var,....)

integrate(expr,var)

or

integrate(expr,(var,low,high))

Can repeat variables (or limit tuples) as with diff.

For infinity, use oo

limit(expr,var,value)

Equations

Equations represented by Eq, but any expression can implicitly be Eq to 0

solve(exprs,vars)

Note: not guaranteed (cfr Hilbert's 10th problem)

For a polynomial, use roots

roots(x**3 – 6*x**2 + 9*x,x)

Matrices

Matrix([[1, 2],[3,4],[5,6]]) # rowwise

Matrix([1,2,3]) # is a column vector

Matrices are mutable (unlike other sympy objects)

Transpose:

M.T

eye(n) (n x n id matrix)

zeros(n,m) #n rows, m columns

ones(n,m)

diag(a,b,c,,,) #diagonal matrix, the args can also be matrices..

Matrix methods:

M.det()

M.rref → reduced row echelon form, list of indices of pivot columns

M.nullspace

M.eigenvals (returns a dictionary of algebraic multiplicity pairs)

M.diagonalize → (P,D) such that D is diagonal, and M = P D P**(-1)