Seite - 112 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

112 5 SomeMorePythonEssentials 5.3.2 SymPy:SomeBasicFunctionality Thefollowingscriptexample_symbolic.pygivesa quickdemonstrationof some of the basic symbolicoperations thatare supported in Python. import sympy as sym x, y = sym.symbols(’x y’) print(2*x + 3*x - y) # Algebraic computation print(sym.diff(x**2, x)) # Differentiates x**2 wrt. x print(sym.integrate(sym.cos(x), x)) # Integrates cos(x) wrt. x print(sym.simplify((x**2 + x**3)/x**2)) # Simplifies expression print(sym.limit(sym.sin(x)/x, x, 0)) # lim of sin(x)/x as x->0 print(sym.solve(5*x - 15, x)) # Solves 5*x = 15 Another useful possibility with sympy, is that sympy expressions may be converted to lambda functions, which then may be used as “normal” Python functionsfornumericalcalculations.Anexamplewill illustrate. Letususesympy toanalyticallyfindthederivativeof thefunctionf(x)=5x3+ 2x2−1,and thenmakebothfand its derivative intoPythonfunctions: import sympy as sym x = sym.symbols(’x’) f_expr = 5*x**3 + 2*x**2 - 1 # symbolic expression for f(x) dfdx_expr = sym.diff(f_expr, x) # compute f’(x) symbolically # turn symbolic expressions into functions f = sym.lambdify([x], f_expr) # f = lambda x: 5*x**3 + 2*x**2 - 1 dfdx = sym.lambdify([x], dfdx_expr) # dfdx = lambda x: 15*x**2 + 4*x print(f(1), dfdx(1)) # call and print, x = 1 Note the arguments to lambdify. The first argument[x] specifies the argument that the generatedfunctionf (and the functiondfdx) is supposed to take,while the secondargumentf_expr(anddfdx_expr)specifiestheexpressiontobeevaluated. Whenexecuted, theprogramprints6 and19,correspondingtof(1)anddfdx(1), respectively. Other symbolic calculations for, e.g., Taylor series3 expansion, linear algebra (with matrix and vector operations), and (some) differential equation solving are alsopossible. 5.3.3 SymbolicCalculationswithSomeOtherTools Symbolic computations are also readily accessible through the (partly) free online tool WolframAlpha,4 which applies the very advanced Mathematica5 package as symbolic engine. The disadvantage with WolframAlpha compared to the SymPy 3 See, e.g., https://en.wikipedia.org/wiki/Taylor_series. 4 http://www.wolframalpha.com. 5 http://en.wikipedia.org/wiki/Mathematica.