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SymPy: Symbolic Mathematics

Hendrik Speleers

Lab Calc

2022-2023SymPy: Symbolic Mathematics

Overview

-A symbolic calculator -Algebraic manipulation

Substitution, simplification, factorization, ...

-Calculus

Series, limits, differentiation and integration

-Linear algebra

Matrix manipulation and decompositions

-Solvers

Linear and nonlinear (system of) equations

Lab Calc

2022-2023SymPy: Symbolic Mathematics

SymPy -Symbolic Python -Python extension for symbolic mathematics Performing algebraic manipulations on symbolic expressions

Evaluating expressions with arbitrary precision

Similar to: Mathematica, Maple

Keeping code as simple as possible and easily extensible -Import convention import sympy as sym

Lab Calc

2022-2023SymPy: Symbolic Mathematics

A symbolic calculator

-A simple demo

In [1]: x, y = sym.symbols('x y')

In [2]: expr = x + 3*y

In [3]: expr

Out[3]: x + 3*y

In [4]: expr + 1

Out[4]: x + 3*y + 1

In [5]: (expr + x)**2

Out[5]: (2*x + 3*y)**2

Lab Calc

2022-2023SymPy: Symbolic Mathematics

A symbolic calculator

-Symbolic variables need to be explicitly defined as symbols -Specific properties can be assigned to symbolsIn [1]: x = sym.symbols('x')

In [2]: x, y = sym.symbols('x y')

In [3]: x, y = sym.symbols('x,y')

In [4]: x, y, z = sym.symbols('x:z')

In [5]: k = sym.symbols('k', integer=True)

In [6]: x, y, z = sym.symbols('x,y,z', real=True)

Lab Calc

2022-2023SymPy: Symbolic Mathematics

A symbolic calculator

-Data types

Symbolic: sym.Symbol

Numeric: sym.Integer, sym.Rational, sym.Float

-Be careful when using integer fractions

In [1]: x = sym.Symbol('x')

In [2]: x + 1/2 # Symbol + Float

Out[2]: x + 0.5

In [3]: x + sym.Rational(1, 2) # Symbol + Rational

Out[3]: x + 1/2

Lab Calc

2022-2023SymPy: Symbolic Mathematics

A symbolic calculator

-Symbolic constants: sym.pi, sym.e, sym.oo -Symbolic functions: Number-theory functions: sym.factorial(n), sym.binomial(n, k) Power functions: sym.exp(x), sym.log(x), sym.sqrt(x), sym.root(x, n) Trigonometric functions: sym.cos(x), sym.sin(x), sym.tan(x)

Many more...

In [1]: f = 4*sym.pi**2

In [2]: sym.sqrt(f)

Out[2]: 2*pi

Lab Calc

2022-2023SymPy: Symbolic Mathematics

Basic manipulation

-Substitution

Single variable: expr.subs(old, new)

Multiple variables: expr.subs(iterable)

In [1]: expr = 1 + x*y

In [2]: expr.subs(x, sym.pi)

Out[2]: pi*y + 1

In [3]: expr.subs({x:sym.pi, y:2})

Out[3]: 1 + 2*pi

In [4]: expr.subs([(x, y+2), (y, 2)])

Out[4]: 9

Lab Calc

2022-2023SymPy: Symbolic Mathematics

Basic manipulation

-Numerical evaluation

Arbitrary precision: expr.evalf(n=15, subs=None)

In [1]: expr = sym.sqrt(8)

In [2]: expr, expr.evalf()

Out[2]: (2*sqrt(2), 2.82842712474619)

In [3]: sym.pi.evalf(100)

Out[3]: 3.1415926535897932384626433832795028841...quotesdbs_dbs3.pdfusesText_6