Page - 24 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

24 1 TheFirstFewSteps from sympy import * x, y = symbols(’x y’) z = x*y print z whichcauses the symbolic resultx*y to appearon the screen. Note that nonumer- ical valuewas assigned to anyof the variables in the symbolic computation. Only the symbolswereused, aswhenyoudo symbolicmathematics byhandonapiece ofpaper. SymboliccomputationsinPythonmakeuseof theSymPypackage. Eachsymbol is representedbyastandardvariable, but thenameof the symbolmust bedeclared with Symbol(name) for a single symbol, or symbols(name1 name2 ...) for multiplesymbols.. Thefollowingscriptexample_symbolic.pyisaquickdemon- strationof someof thebasic symbolicoperations that are supported inPython. from sympy import * x = Symbol(’x’) y = Symbol(’y’) print 2*x + 3*x - y # Algebraic computation print diff(x**2, x) # Differentiates x**2 wrt. x print integrate(cos(x), x) # Integrates cos(x) wrt. x print simplify((x**2 + x**3)/x**2) # Simplifies expression print limit(sin(x)/x, x, 0) # Finds limit of sin(x)/x as x->0 print solve(5*x - 15, x) # Solves 5*x = 15 Other symbolic calculations like Taylor series expansion, linear algebra (with matrix and vector operations), and (some) differential equation solving are also possible. Symbolic computations are also readily accessible through the (partly) free on- line toolWolframAlpha4,which applies thevery advancedMathematica5 package assymbolicengine. ThedisadvantagewithWolframAlphacomparedto theSymPy package is that the results cannot automatically be imported into your code and used for further analysis. On theotherhand,WolframAlphahas the advantage that it displaysmanyadditionalmathematical results related to thegivenproblem. For example, ifwe type2x + 3x - y inWolframAlpha, it not only simplifies the ex- pression to 5x - y, but it also makes plots of the function f.x;y/ D 5x y, solves the equation 5x y D 0, and calculates the integral R R.5x Cy/dxdy. ThecommercialProversioncanalso showastep-by-stepof the analytical compu- tations in theproblem. Youare strongly encouraged to try out these commands in WolframAlpha: diff(x^2, x)ordiff(x**2, x) integrate(cos(x), x) simplify((x**2 + x**3)/x**2) 4http://www.wolframalpha.com 5http://en.wikipedia.org/wiki/Mathematica