Updates:
Cc: Vinzent.Steinberg
Labels: -NeedsReview
Comment #158 on issue 1694 by smichr: solve has many issues with fractions
http://code.google.com/p/sympy/issues/detail?id=1694
1694 is doing well.
It is weak in advanced Lambert pattern recognition and handles trig
function via Euler substitutions. It should be as robust in solvable
multi-system equations as it is in single equations now. If you have some
favorite functions to test, that would be useful for feedback.
I think it was Vinzent that posted the following equations. I have held
those as a sort of torture test for the solver and am happy to say that
there are only two failures in solving for the roots of the function and
its first three derivatives. Here is the test and the output is appended.
def dotest_german():
from sympy import limit, oo, diff
x = Symbol('x')
a = Symbol('a')
# partially taken from German "Abitur", Saxonia, advanced math course
# see http://www.sn.schule.de/~matheabi/
f1 = x**3 - 9*x
f2 = 4*x * (1 - a*sqrt(x))
f3 = x**x
f4 = 1/a * log(1/a**3 - x**2)
f5 = exp(a*x) * (x - 1)**2
f6 = 10*exp(-a*x) * sin(a*x)
f7 = 1 - log(4*x**2 + a)
f8 = (x**3 + a*x) / (x**2 - a)
f9 = (S.Half*x - a) * exp(1/a*x)
f10 = 2*x * log(x) - a*x
f11 = 1/a * (x - 2*a) * sqrt(x)
f12 = -exp(2*a*x) + 4*exp(a*x) + 5
f13 = (2*x - 1) * exp(a*x)
f14 = x**2 * exp(1 - a*x)
f15 = a*x**2 * (1 - log(x**2/a))
f16 = exp(x) / (8*(a + x)**2)
f17 = exp(x/2) + a/x
f18 = 1 / log(a*x)
f19 = (a**2 + 1) * (sin(a*x) + cos(a*x))
f20 = x/a * sqrt(a**2 - x)
f21 = (x - 1)**2 * (x + 4) / (x + a) / (x + 2)
for ii,f in enumerate(eval('[' + ', '.join('f%i' % i for i in range(1,
22)) + ']')):
print 'graph discussion for f%i = %s' % (ii+1,f)
if f==f11:
pass
if f == f2 or f==f11 or f==f20:
limits = False
else:
limits = True
df=diff(f,x)
d2f=diff(df,x)
d3f=diff(d2f,x)
print 'f=',f
print 'df=',df
print 'd2f=',d2f
print 'd3f=',d3f
#graph_discussion(f, limits=limits)
#continue
if 1:
print 'limits'
if f not in [f6]:
print '->oo',limit(f, x, oo)
else: #f=f6
print 'ValueError: Cannot determine the sign of sin(a*x)'
if f not in [f6]:
print '-->-oo',limit(f, x, -oo)
try:
#graph_discussion(f, limits=limits)
try:
print 'soln',solve(f,x)
except:
print 'FAIL0',f
stop()
try:
if f==f21:
print solve(df,x,simplified=0)
else:
print solve(df,x)
except:
print 'FAIL1',df
stop()
try:
if f==f21:
print solve(d2f,x, simplified=False)
else:
print solve(d2f,x)
except:
print 'FAIL2',d2f
stop()
if f in [f3]: #f3 can't be solved by mathematicac, either; f21
is a 4th order
continue
try:
if f==f21:
print solve(d3f,x,simplified=False)
else:
print solve(d3f,x)
except:
print 'FAIL3',d3f
stop()
except:# Exception as e:
print 'other FAILURE!'#print type(e), e
print
====== OUTPUT ===========
graph discussion for f3 = x**x
f= x**x
df= x**x*(1 + log(x))
d2f= x**x*(1 + log(x))**2 + x**x/x
d3f= 3*x**x*(1 + log(x))/x + x**x*(1 + log(x))**3 - x**x/x**2
limits
->oo oo
-->-oo oo
soln [0]
[0, exp(-1)]
FAIL2 x**x*(1 + log(x))**2 + x**x/x
other FAILURE!
graph discussion for f17 = a/x + exp(x/2)
f= a/x + exp(x/2)
df= -a/x**2 + exp(x/2)/2
d2f= 2*a/x**3 + exp(x/2)/4
d3f= -6*a/x**4 + exp(x/2)/8
limits
->oo oo
-->-oo 0
soln FAIL0 a/x + exp(x/2)
other FAILURE!
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