Hi,
Yes, the general implementation will require some work. Continuous
functions are the easy ones. Piecewise functions handling would have to be
consider.
Additionally I already found some other missing cases in my first attempt,
some functions might reach a minimum at -oo or +oo. This has some small
improvements:
def _minimize(expr, x):
f = Lambda(x, expr)
f1 = f(x).diff(x)
f2 = Lambda(x,f1.diff(x))
m = None
mp = None
for p in solve(f1,x):
if Gt(f2(p),0)==True and ((m == None) or Lt(f(p),m)==True):
m=f(p)
mp = p
if f(oo) < m:
m = f(oo)
mp=oo
if f(-oo)<m:
m=f(-oo)
mp=-oo
return (m,mp)
def _maximize(expr, x):
(m,mp) = _minimize(-expr, x)
return (-m, mp)
def test_U13():
assert _minimize(x**4 - x + 1, x)[0] == -3*2**Rational(1,3)/8 + 1
It would be nice to support functions or more than one variable, for
example a special function for 2 variables could be:
def _minimize2(expr, x, y):
f = Lambda((x,y), expr)
f1x = f(x,y).diff(x)
f1y = f(x,y).diff(y)
d = solve((f1x,f1y),(x,y))
m = f(d[x],d[y])
mp=(d[x],d[y])
if f(oo,oo) < m:
m = f(oo,oo)
mp = (oo,oo)
if f(-oo,-oo) < m:
m = f(-oo,-oo)
mp = (-oo,-oo)
if f(oo,-oo) < m:
m = f(oo,-oo)
mp = (oo,-oo)
if f(-oo,oo) < m:
m = f(-oo,oo)
mp = (-oo,oo)
return (m,mp)
def _maximize2(expr, x, y):
(m,mp) = _minimize2(-expr, x, y)
return (-m,mp)
def test_U14():
f = 1/(x**2 + y**2 + 1)
assert [_minimize2(f,x,y)[0], _maximize2(f,x,y)[0]] == [0,1]
Best Regards,
Pablo
On Wednesday, October 9, 2013 9:00:39 PM UTC+2, Tim Lahey wrote:
>
> Hi,
>
> I can see the utility for something like this in general. But, I think the
> general case should have an option for function boundaries (e.g., x >=0). I
> ran into an error once in a paper where they defined the maximum in a
> manner similar to this and it turned out that the maximum was at the
> boundary (x=0 in that case).
>
> Cheers,
>
> Tim.
>
> ---
> Tim Lahey, Ph.D.
> Post-Doctoral Fellow
> Systems Design Engineering
> University of Waterloo
>
> On 2013-10-09, at 2:29 PM, Pablo Puente <[email protected]<javascript:>>
> wrote:
>
> > Hi,
> >
> > I could not find a function in Sympy that finds the minimum of a
> function symbolically.
> >
> > In case it is not implemented yet, would it make sense to add it?
> >
> > minimize() and maximize() are used in several of the Wester test cases.
> This is an example of how I implemented one of the test cases. Maybe can be
> used as a basis:
> >
> > def _minmaximize(expr, x, si):
> > f = Lambda(x, expr)
> > f1 = f(x).diff(x)
> > f2 = Lambda(x,f1.diff(x))
> > m = None
> > mp = None
> > for p in solve(f1,x):
> > if Gt(si*f2(p),0)==True and ((m == None) or
> Lt(si*(f(p)-m),0)==True):
> > m=f(p)
> > mp = p
> > return (m,mp)
> >
> > def _minimize(expr, x):
> > return _minmaximize(expr, x, +1)
> >
> > def _maximize(expr, x):
> > return _minmaximize(expr, x, -1)
> >
> > def test_U13():
> > assert _minimize(x**4 - x + 1, x)[0] == -3*2**Rational(1,3)/8 + 1
> >
> >
> >
> > --
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>
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