Le lundi 29 novembre 2021 à 16:33:41 UTC+1, cyrille piatecki a écrit :
Thanks Emmanuel for your precious answer. But It generates some few new > questions : > - is there a place in the documentation where I can find the information > on `solve()` and mainly its options ? > The documentation <https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/relation.html>, of course… - if I understand clearly z_{6497} is an integer but how to fix it to zero > --- when the number change at each iteration > That’s why I used a methods sequence to designate it, rather than using its name… - sympy seems to be the good approach > Beware : see below… but it is not self evident that to call y one must typpeset sol2[0][x] > It is, because algorithm="sympy" will cause the results to be expressed as dictionaries and D[x] is the canonical way to get the value of the entry of dictionary D indexed by x. Basic Python… - the giac way is certainly the better but it keeps no track of the > variable's order. > Again, ask for a solution dictionary. As for algorithm="giac", I have seen it go pear-shaped a couple times… Now for the various expression of solutions : consider : print(table([[u,solve(FOC,[x,y,l], solution_dict=True, algorithm=u)] for u in ["maxima", "fricas", "sympy", "giac"]])) maxima [{l: 1/p_y, x: (p_x/p_y)^(1/a)*e^(2*I*pi*z3541/a), y: -(p_x*(p_x/p_y)^(1/a)*e^(2*I*pi*z3541/a) - R)/p_y}] fricas [{l: 1/p_y, x: (p_x/p_y)^(1/a)*e^(2*I*pi*z3892/a), y: -(p_x*(p_x/p_y)^(1/a)*e^(2*I*pi*z3892/a) - R)/p_y}] sympy [{x: (p_x/p_y)^(1/a), l: 1/p_y, y: -(p_x*(p_x/p_y)^(1/a) - R)/p_y}] giac [{x: (p_x/p_y)^(1/a), y: -(p_x*(p_x/p_y)^(1/a) - R)/p_y, l: 1/p_y}] Both maxima and fricas try to explicitly express the set of solutions of the equation z^a==p_x/p_y, which is a set of a complexes if a is a positive integer. (I leave to you (as en exercise ;-) to determine what it means (if any…) if a is rational, algebraic or transcendental, real or complex…). OTOH, both sympy and giac use the notation (p_x/p_y)^(1/a) to *implicitly* denote *the very same set of solutions to the very same equation. One could say that tey are glossing over whatever maxima and fricas insist on. Choose your poison… HTH, > I have tried my solution assuming l>0 on the 3 conditions but it changes > nothing. > > > > ----- Mail d’origine ----- > De: Emmanuel Charpentier <emanuel.c...@gmail.com> > À: sage-support <sage-s...@googlegroups.com> > Envoyé: Mon, 29 Nov 2021 11:03:37 +0100 (CET) > Objet: [sage-support] Re: Constrained optimization with strange result. > > Variables of the form z_xxxx are *integer* variables created by Maxima, > which attempts to give you *also* the complex roots, if any, thus > ignoring the assumptions on x, y and l. Note that : > > sage: solve(FOC[0], x) > --------------------------------------------------------------------------- > > [ Snip… ] > > TypeError: Computation failed since Maxima requested additional constraints; > using the 'assume' command before evaluation *may* help (example of legal > syntax is 'assume(l>0)', see `assume?` for more details) > Is l positive, negative or zero? > sage: with assuming(l>0): print(solve(FOC[0], x)) > [ > x == (l*p_x)^(1/a) > ] > sage: with assuming(l<0): print(solve(FOC[0], x)) > [ > x^a == l*p_x > ] > sage: with assuming(l<0): print(solve(FOC[0], x, to_poly_solve=True)) > [x == (l*p_x)^(1/a)*e^(2*I*pi*z4353/a)] > > Interestingly: > > sage: solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l) > [[l == (1/p_y), x == (p_x/p_y)^(1/a)*e^(2*I*pi*z3540/a), y == > -(p_x*(p_x/p_y)^(1/a)*e^(2*I*pi*z3540/a) - R)/p_y]] > sage: solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l, algorithm="sympy") > [{x: (p_x/p_y)^(1/a), l: 1/p_y, y: -(p_x*(p_x/p_y)^(1/a) - R)/p_y}] > sage: solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l, algorithm="fricas") > [[l == (1/p_y), x == (p_x/p_y)^(1/a)*e^(2*I*pi*z3891/a), y == > -(p_x*(p_x/p_y)^(1/a)*e^(2*I*pi*z3891/a) - R)/p_y]] > sage: giac.solve(giac(FOC),giac([x,y,l])).sage() > [[(p_x/p_y)^(1/a), -(p_x*(p_x/p_y)^(1/a) - R)/p_y, 1/p_y]] > > HTH, > > Le dimanche 28 novembre 2021 à 22:13:12 UTC+1, cyrille piatecki a écrit : > >> On my computer the solution of >> >> var('a x y p_x p_y D Rev R l') >> assume(a,'real') >> assume(x,'real') >> assume(y,'real') >> assume(p_x,'real') >> assume(p_y,'real') >> assume(D,'real') >> assume(Rev,'real') >> assume(R,'real') >> assume(l,'real') >> assume(a<1) >> assume(a>0) >> assume(p_x>0) >> assume(p_y>0) >> assume(R>0) >> U =(1/(a+1))*x^(a+1)+y >> show(LatexExpr(r'\text{La fonction d}^\prime\text{utilité est }U(x,y) = >> '),U) >> D= x*p_x + y*p_y >> show(LatexExpr(r'\text{La Dépense } D = '),D) >> Rev= R >> show(LatexExpr(r'\text{Le Revenu } Rev = '),R) >> L=U+l*(Rev-D) >> show(LatexExpr(r'\text{Le lagrangien est } \mathcal{L}(x, y, λ) = '),L) >> FOC = [diff(L,x),diff(L,y),diff(L,l)] >> show(LatexExpr(r'\text{Les condition du premier ordre sont } >> \left\{\begin{array}{c}\mathcal{L}_x= 0\\\mathcal{L}_y= 0\\\mathcal{L}_λ= >> 0\end{array}\right. ')) >> show(LatexExpr(r'\text{soit }')) >> show(LatexExpr(r'\mathcal{L}_x= 0 \Longleftrightarrow '),FOC[0]==0) >> show(LatexExpr(r'\mathcal{L}_y= 0 \Longleftrightarrow '),FOC[1]==0) >> show(LatexExpr(r'\mathcal{L}_λ= 0 \Longleftrightarrow '),FOC[2]==0) >> sol=solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l) >> show(sol) >> >> Is nearly correct, but an extra complex exponential term multiplies $x$ >> and then modifies $y$. Even as an element I do not understand its form : >> >> $e^{(2iπz_{5797}a)}$ >> >> Could some one explain why ? >> >> > > > -- > > 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...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-support/e745f8a7-df76-4def-b3b9-09ac7684e9a3n%40googlegroups.com > > <https://groups.google.com/d/msgid/sage-support/e745f8a7-df76-4def-b3b9-09ac7684e9a3n%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > > -- 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. 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