Le mercredi 14 août 2013 10:30:43 UTC+2, Jose Guzman a écrit :
>
> Very impressive Emmanuel!
>
(Possibly) impressive but **FALSE* !*
Coming back to the subject, I wanted to see what maxima would do with the
same problem. I got the same final answer but not with the same
intermediate results !
Let's start with sage:
sage: var("x,y,l")
(x, y, l)
sage: f = x + 2*y
sage: g = 2*x^(1/4)*y^(1/2)
sage: L = f - l * (16 - g)
sage: var("x14,y12")
(x14, y12)
sage: Lt=L.subs(x=x14^4,y=y12^2)
sage: Lt
x14^4 + 2*((x14^4)^(1/4)*sqrt(y12^2) - 8)*l + 2*y12^2
sage: Lt.simplify_full()
x14^4 + 2*l*x14*y12 + 2*y12^2 - 16*l
One notes that simplify_full() replaces ((x14^4)^(1/4)*sqrt(y12^2) with
x14*y12.
Let's compare with maxima :
(%i1) f:x+2*y;
(%o1) 2*y+x
(%i2) g:2*x^(1/4)*y^(1/2);
(%o2) 2*x^(1/4)*sqrt(y)
(%i3) L:f-l*(16-g);
(%o3) 2*y-l*(16-2*x^(1/4)*sqrt(y))+x
(%i4) Lt:ratsimp(factor(radcan(expand(subst([x=x14^4,y=y12^2],L)))));
(%o4) 2*l*abs(x14)*abs(y12)+2*y12^2+x14^4-16*l
One notes that radcan() replaces the same subexpression with
abs(y12)*abs(x14), which is quite different, analysis-wise...
Maxima is unable to solve the gradient of this expression :
(%i5) solve([diff(Lt,x14)=0,diff(Lt,y12)=0,diff(Lt,l)=0],[x14,y12,l]);
(%o5) []
(%i6) load(to_poly_solve);
(%o6) "/usr/share/maxima/5.30.0/share/to_poly_solve/to_poly_solve.mac"
(%i7) %solve([diff(Lt,x14)=0,diff(Lt,y12)=0,diff(Lt,l)=0],[x14,y12,l]);
(%o7) %solve([2*l*abs(x14)*y12/abs(y12)+4*y12 = 0,
2*l*x14*abs(y12)/abs(x14)+4*x14^3 = 0,
2*abs(x14)*abs(y12)-16 = 0],[l,x14,y12])
On the other hand, sage *does* solve it (albeit quite slowly...). We get
four symmetric solutions :
sage: SI=solve([diff(Lto,x14)==0,diff(Lto,y12)==0,diff(Lto,l)==0],
[x14,y12,l],solution_dict=True)
sage: SI
[{l: -4, y12: -4, x14: -2}, {l: -4, y12: 4, x14: -2}, {l: -4, y12: -4, x14:
2}, {l: -4, y12: 4, x14: 2}]
All those solutions check for the transformed problem :
sage: SIcheck=map(lambda s:[diff(Lt,x14).subs(s).expand(),\
diff(Lt,y12).subs(s).expand(),\
diff(Lt,l).subs(s).expand()], SI)
....: ....: sage:
sage: SIcheck
[[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]
Getting back to the original variable gives us a quadruple root :
sage: S=map(lambda s:{x:(s.get(x14))^4,\
y:(s.get(y12))^2,\
l:s.get(l)},SI)
....: ....: sage:
sage: Scheck=map(lambda s:bool(diff(L,x14).subs(s)==0) &\
bool(diff(L,y12).subs(s).expand()==0) &\
bool(diff(L,l).subs(s)==0), S)
....: ....: sage:
sage: S
[{x: 16, l: -4, y: 16}, {x: 16, l: -4, y: 16}, {x: 16, l: -4, y: 16}, {x:
16, l: -4, y: 16}]
sage: Scheck
[True, True, True, True]
Some lessons :
- While derived from maxima's radcan(), simplify_full() does different
things ! In this case, getting rid of the abs() calls was the Wrong
Thing (TM) do to.
- Always, always, always check the answers from solve() (both maxima's
and sage's) if this is even remotely possible.
- (Todo) check sage's documentation to learn how to obtain valid
substitutions of absolute values to radicals-of-powers and
power-of-radicals expressions...
- Don't post at midnight after a day of hospital work...
HTH,
Emmanuel Charpentier
[ Snip... ]
--
You received this message because you are subscribed to the Google Groups
"sage-support" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-support+unsubscr...@googlegroups.com.
To post to this group, send email to sage-support@googlegroups.com.
Visit this group at http://groups.google.com/group/sage-support.
For more options, visit https://groups.google.com/groups/opt_out.
