Module: sage.calculus.test_sympy
A Sample Session using Sympy
In this first part, we do all of the examples in the Sympy tutorial (http://code.google.com/p/sympy/wiki/Tutorial), but using Sage instead of Sympy.
sage: a = Rational((1,2)) sage: a 1/2 sage: a*2 1 sage: Rational(2)^50 / Rational(10)^50 1/88817841970012523233890533447265625 sage: 1.0/2 0.500000000000000 sage: 1/2 1/2 sage: pi^2 pi^2 sage: float(pi) 3.1415926535897931 sage: RealField(200)(pi) 3.1415926535897932384626433832795028841971693993751058209749 sage: float(pi + exp(1)) 5.85987448204883... sage: oo != 2 True
sage: var('x y') (x, y) sage: x + y + x - y 2*x sage: (x+y)^2 (y + x)^2 sage: ((x+y)^2).expand() y^2 + 2*x*y + x^2 sage: ((x+y)^2).subs(x=1) (y + 1)^2 sage: ((x+y)^2).subs(x=y) 4*y^2
sage: limit(sin(x)/x, x=0) 1 sage: limit(x, x=oo) +Infinity sage: limit((5^x + 3^x)^(1/x), x=oo) 5
sage: diff(sin(x), x) cos(x) sage: diff(sin(2*x), x) 2*cos(2*x) sage: diff(tan(x), x) sec(x)^2 sage: limit((tan(x+y) - tan(x))/y, y=0) 1/cos(x)^2 sage: diff(sin(2*x), x, 1) 2*cos(2*x) sage: diff(sin(2*x), x, 2) -4*sin(2*x) sage: diff(sin(2*x), x, 3) -8*cos(2*x)
sage: cos(x).taylor(x,0,10) 1 - x^2/2 + x^4/24 - x^6/720 + x^8/40320 - x^10/3628800 sage: (1/cos(x)).taylor(x,0,10) 1 + x^2/2 + 5*x^4/24 + 61*x^6/720 + 277*x^8/8064 + 50521*x^10/3628800
sage: matrix([[1,0], [0,1]]) [1 0] [0 1] sage: var('x y') (x, y) sage: A = matrix([[1,x], [y,1]]) sage: A [1 x] [y 1] sage: A^2 [x*y + 1 2*x] [ 2*y x*y + 1] sage: R.<x,y> = QQ[] sage: A = matrix([[1,x], [y,1]]) sage: print A^10 [x^5*y^5 + 45*x^4*y^4 + 210*x^3*y^3 + 210*x^2*y^2 + 45*x*y + 1 10*x^5*y^4 + 120*x^4*y^3 + 252*x^3*y^2 + 120*x^2*y + 10*x] [ 10*x^4*y^5 + 120*x^3*y^4 + 252*x^2*y^3 + 120*x*y^2 + 10*y x^5*y^5 + 45*x^4*y^4 + 210*x^3*y^3 + 210*x^2*y^2 + 45*x*y + 1] sage: var('x y') (x, y)
And here are some actual tests of sympy:
sage: from sympy import Symbol, cos, sympify, pprint sage: from sympy.abc import x
sage: e = sympify(1)/cos(x)**3; e cos(x)**(-3) sage: f = e.series(x, 0, 10); f 1 + (3/2)*x**2 + (11/8)*x**4 + (241/240)*x**6 + (8651/13440)*x**8 + O(x**10)
And the pretty-printer:
sage: pprint(e) 1 ------- 3 cos (x) sage: pprint(f) 2 4 6 8 3*x 11*x 241*x 8651*x 1 + ---- + ----- + ------ + ------- + O(x**10) 2 8 240 13440
And the functionality to convert from sympy format to Sage format:
sage: e._sage_() 1/cos(x)^3 sage: print e._sage_() 1 ------- 3 cos (x) sage: e._sage_().taylor(x._sage_(), 0, 8) 1 + 3*x^2/2 + 11*x^4/8 + 241*x^6/240 + 8651*x^8/13440 sage: f._sage_() 8651*x^8/13440 + 241*x^6/240 + 11*x^4/8 + 3*x^2/2 + 1
Mixing SymPy with Sage:
sage: import sympy sage: sympy.sympify(var("y"))+sympy.Symbol("x") x + y sage: o = var("omega") sage: s = sympy.Symbol("x") sage: t1 = s + o sage: t2 = o + s sage: print type(t1) <class 'sage.calculus.calculus.SymbolicArithmetic'> sage: print type(t2) <class 'sage.calculus.calculus.SymbolicArithmetic'> sage: print t1, t2 x + omega x + omega sage: e=sympy.sin(var("y"))+sage.all.cos(Symbol("x")) sage: print type(e) <class 'sage.calculus.calculus.SymbolicArithmetic'> sage: print e sin(y) + cos(x) sage: e=e._sage_() sage: print type(e) <class 'sage.calculus.calculus.SymbolicArithmetic'> sage: print e sin(y) + cos(x) sage: e = sage.all.cos(var("y")**3)**4+var("x")**2 sage: e = e._sympy_() sage: print e x**2 + cos(y**3)**4