The Risch Algorithm for Symbolic Integration in SymPy
Aaron Meurer
July 14, 2011
1 / 9 Over the summer of 2010, I worked for the Python Software Organization with the SymPy project under the Google Summer of Code program to implement the transcendental Risch Algorithm in SymPy.
2 / 9 End result: risch integrate() function
Orders of magnitude faster than old integrate() function
Able to solve much more
And it can prove that integrals are nonelementary!
3 / 9 In[1]:f1=(1+exp(x**2))*(1-x-4*x**2*exp(x**2)*log(x)- ...:x*exp(x**2)+4*x**3*exp(x**2)+exp(x**2))/(x*(x-log(x))**2)
In[2]:f1 Out[2]: 1 ⎛ ⎛2⎞⎞⎛ ⎛2⎞ ⎛2⎞ ⎛2⎞ ⎛2⎞⎞ ⎜ ⎝x⎠⎟⎜ 2 ⎝x⎠ ⎝x⎠ 3 ⎝x⎠ ⎝x⎠⎟ ⎝1+e ⎠⋅⎝1-x-4⋅x⋅e⋅log(x)-x⋅e +4⋅x⋅e +e ⎠ ────────────────────────────────────────────────────────────────────── 2 x⋅(x-log(x))
In[3]:factor(cancel(risch_integrate(f1,x))) Out[3]: 1 2 ⎛ ⎛2⎞⎞ ⎜ ⎝x⎠⎟ ⎝1+e ⎠ ──────────── x-log(x)
In[4]:%timeitrisch_integrate(f1,x) 1loops,bestof3:390msperloop 1
1
4 / 9 In[5]:risch_integrate(exp(-x**2),x) Out[5]: ⌠ ⎮ 2 ⎮ -x ⎮e dx ⌡
In[7]:risch_integrate(1/log(x),x) Out[7]: 1 ⌠ ⎮ 1 ⎮──────dx ⎮log(x) ⌡
In[10]:risch_integrate(exp(x)*log(x),x) Out[10]: 1 ⌠ ⎮ x ⎮e⋅log(x)dx ⌡
In[11]:risch_integrate(exp(exp(x)),x) Out[11]: 1 ⌠ ⎮ ⎛x⎞ ⎮ ⎝e⎠ ⎮e dx ⌡
1 5 / 9 Let’s look at how long it takes to compute R x10ex dx and R x20ex dx.
In[14]:%timeitintegrate(x**10*exp(x),x) 1loops,bestof3:2.75sperloop
In[15]:%timeitintegrate(x**20*exp(x),x) 1loops,bestof3:31.5sperloop 1
In[16]:%timeitrisch_integrate(x**10*exp(x),x) 10loops,bestof3:75.9msperloop 1
In[17]:%timeitrisch_integrate(x**20*exp(x),x) 10loops,bestof3:98msperloop 1
1 (integrate() is the old function and risch integrate() is the new function)
6 / 9 Time of integrate() and risch integrate() to evaluate R x nex dx
10
8
6 Time (s)
4
2
0 0 100 200 300 400 500 n
The new implementation is both orders of magnitude faster, and asymptotically faster! 7 / 9 def no_cancel_b_small(b, c, n, DE): """ Poly Risch Differential Equation - No cancelation: deg(b) small enough.
Given a derivation D on k[t], n either an integer or +oo, and b, c in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or PolyRischDENoCancel2(b, c, D, n) /* Poly Risch d.e. – no cancellation */ deg(D) >= 2, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], /* Given a derivation D on k[t], n either an integer or + , and b, c k[t] with or a solution q in k[t] of this equation with deg(q) <= n, or the deg(b) < δ(t) 1 and either D = d/dt or δ(t) 2, return either∞ “no solution”,∈ in tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any which case the− equation Dq + bq = c has no solution≥ of degree at most n in k[t], or solution q in k[t] of degree at most n of Dq + bq == c, y == q - h a solution q k[t] of this equation with deg(q) n, or the tuple (h, b , c ) such that 0 0 is a solution in k of Dy + b0*y == c0. h k[t], b , c∈ k[t], and for any solution q k≤[t] of degree at most n of 0 0 """ Dq∈+ bq = c, y∈= q h is a solution in k of Dy∈ + b y = c . */ − 0 0 q= Poly(0, DE.t) q 0 ← while c = 0 do while notc.is_zero: if n =6 0 then m 0 else m deg(c) = δ(t) + 1 ← ← ifn ==0: if n < 0 or m < 0 or m > n then return “no solution” m=0 if m > 0 then p (lc(c)/(mλ(t)))tm ← else: else /* m = 0 */ m=c.degree(DE.t)-DE.d.degree(DE.t)+1 if deg(b) = deg(c) then return “no solution” if deg(b)6 = 0 then return (q, c, b) if not0<=m<= n: # n < 0 or m < 0 or m > n p lc(c)/lc(b) q q←+ p raise NonElementaryIntegralException n ← m 1 c ←c −Dp bp ifm>0: return← q − − p= Poly(c.as_poly(DE.t).LC()/(m*DE.d.as_poly(DE.t).LC())*DE.t**m, DE.t, expand=False) else: ifb.degree(DE.t) !=c.degree(DE.t): raise NonElementaryIntegralException ifb.degree(DE.t) ==0: return (q, b.as_poly(DE.T[DE.level-1]), c.as_poly(DE.T[DE.level-1])) p= Poly(c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC(), DE.t, expand=False)
q=q+p n=m-1 c=c- derivation(p, DE)-b*p
returnq
8 / 9 You can download SymPy at www.sympy.org
Or pip install sympy (We just released 0.7.0)
My development branch is at www.github.com/asmeurer/sympy/tree/integration3 (it isn’t merged into the main repo yet)
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