rtoy
commented
4 months ago Imported from SourceForge on 2024-07-05 20:38:15 Created by willisbl on 2008-03-17 12:05:35 Original: https://sourceforge.net/p/maxima/bugs/1370/#32be
Logged In: YES user_id=895922 Originator: NO
Thanks for reporting this bug. The function algsys isn't nearly as good as we would like. The best I can offer is my favorite workaround (that might work); it's something like:
(%i92) load(topoly)$
(%i93) algebraic : true$
Use 'elim' to triangularize the equations---this can create spurious solutions:
(%i94) elim([F,G],[x,y]); (%o94) [[],[x^4+a^2*x^3-a*x^3+a^3*x^2-a^2*x^2+a^3*x,x^3*y^2+2*a*x^2*y^2+a^2*x*y^2+x^2*y+2*a*x*y+a^2*y+1]]
Use algsys to solve these equations:
(%i95) sol : algsys(first(rest(%)),[x,y])$
Try to reject the spurious solutions:
(%i96) true_sol : set()$ (%i97) for si in sol do if every(lambda([a], is(equal(a,0))), ratsimp(subst(si,[F,G]))) then true_sol : adjoin(si, true_sol); (%o97) done (%i98) true_sol;
(%o98) {[x=(a*sqrt(a^2-4*a)-a^2+2*a)/2,y=-sqrt(a^2-4*a)/(a^3-4*a^2)], [x=-(a*sqrt(a^2-4*a)+a^2-2*a)/2,y=sqrt(a^2-4*a)/(a^3-4*a^2)]}
Two solutions *might* be spurious and two checked out OK. You'll need to inspect the two rejected solutions to see if they are really spurious. And a = 0 and a = 4 might be special cases that need to be checked.
Of course, all this should be automatic; unfortunately, algsys isn't there yet.
Barton
Imported from SourceForge on 2024-07-05 20:38:14 Created by satoshi_adachi on 2008-03-17 09:40:58 Original: https://sourceforge.net/p/maxima/bugs/1370
Dear Developers of Maxima,
I met a set of two albebraic equations F = 0 and G = 0 of two variables x and y such that (i) solve([F=0,G=0],[x,y]) returns no solution [] (this is wrong), whereas (ii) solve([F=0,F-G=0],[x,y]) returns true solutions. This is strange, since the first set of equations F=0 and G=0 should have the same solutions of the second set of equations F=0 and F-G=0.
(1) The Maxima program to solve F=0 and F-G=0 is ------------------------------------------------------------------------------- /* * solve_system_of_algebraic_equations_OK.maxima: * * F is a polynomial of x and y, which is degree 3 in x and 2 in y. * G is a polynomial of x and y, which is degree 3 in x and 2 in y. * * solve() and algsys() can solve the system equations F=0 and F-G=0. */
display2d:false;
F:2*x^3*y^2+2*a*x^2*y^2+3*x^2*y+2*a*x*y+x+1;
G:x^3*y^2+2*a*x^2*y^2+a^2*x*y^2+x^2*y+2*a*x*y+a^2*y+1;
solutions:solve([F=0,F-G=0],[x,y]); /* OK: solutions are found. */ -------------------------------------------------------------------------------
The result is ------------------------------------------------------------------------------- Maxima 5.14.0cvs http://maxima.sourceforge.net Using Lisp GNU Common Lisp (GCL) GCL 2.6.7 (aka GCL) Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report() provides bug reporting information. (%i1) batch(solve_system_of_algebraic_equations_OK.maxima)
batching #p/Volumes/HFS+2/home/adachi/work/285/solve_system_of_algebraic_equations_OK.maxima (%i2) display2d : false (%o2) false (%i3) F:1+x+2*a*x*y+3*x^2*y+2*a*x^2*y^2+2*x^3*y^2 (%o3) 2*x^3*y^2+2*a*x^2*y^2+3*x^2*y+2*a*x*y+x+1 (%i4) G:1+a^2*y+2*a*x*y+x^2*y+a^2*x*y^2+2*a*x^2*y^2+x^3*y^2 (%o4) x^3*y^2+2*a*x^2*y^2+a^2*x*y^2+x^2*y+2*a*x*y+a^2*y+1 (%i5) solutions:solve([F = 0,F-G = 0],[x,y]) (%o5) [[x = (sqrt(a)*(a^2-4*a)+sqrt(a-4)*(2*a-a^2))/(2*sqrt(a-4)), y = -1/(a*sqrt(a^2-4*a))], [x = -(sqrt(a-4)*(a^2-2*a)+sqrt(a)*(a^2-4*a))/(2*sqrt(a-4)), y = 1/(a*sqrt(a^2-4*a))]] (%o6) "solve_system_of_algebraic_equations_OK.maxima" ------------------------------------------------------------------------------
===============================================================================
(2) The Maxima program to solve F=0 and G=0 is ------------------------------------------------------------------------------- /* * solve_system_of_algebraic_equations_NG.maxima: * * F is a polynomial of x and y, which is degree 3 in x and 2 in y. * G is a polynomial of x and y, which is degree 3 in x and 2 in y. * * solve() and algsys() can NOT solve the system equations F=0 and G=0. */
display2d:false;
F:2*x^3*y^2+2*a*x^2*y^2+3*x^2*y+2*a*x*y+x+1;
G:x^3*y^2+2*a*x^2*y^2+a^2*x*y^2+x^2*y+2*a*x*y+a^2*y+1;
solutions:solve([F=0,G=0],[x,y]); /* NG: solutions are NOT found! */ -------------------------------------------------------------------------------
The result is ------------------------------------------------------------------------------- Maxima 5.14.0cvs http://maxima.sourceforge.net Using Lisp GNU Common Lisp (GCL) GCL 2.6.7 (aka GCL) Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report() provides bug reporting information. (%i1) batch(solve_system_of_algebraic_equations_NG.maxima)
batching #p/Volumes/HFS+2/home/adachi/work/285/solve_system_of_algebraic_equations_NG.maxima (%i2) display2d : false (%o2) false (%i3) F:1+x+2*a*x*y+3*x^2*y+2*a*x^2*y^2+2*x^3*y^2 (%o3) 2*x^3*y^2+2*a*x^2*y^2+3*x^2*y+2*a*x*y+x+1 (%i4) G:1+a^2*y+2*a*x*y+x^2*y+a^2*x*y^2+2*a*x^2*y^2+x^3*y^2 (%o4) x^3*y^2+2*a*x^2*y^2+a^2*x*y^2+x^2*y+2*a*x*y+a^2*y+1 (%i5) solutions:solve([F = 0,G = 0],[x,y]) (%o5) [] (%o6) "solve_system_of_algebraic_equations_NG.maxima" -------------------------------------------------------------------------------
===============================================================================
(3) I also tried algebraic:true to solve F=0 and G=0. The program is ------------------------------------------------------------------------------- /* * solve_system_of_algebraic_equations_with_algebraic_NG.maxima: * * F is a polynomial of x and y, which is degree 3 in x and 2 in y. * G is a polynomial of x and y, which is degree 3 in x and 2 in y. * * solve() and algsys() can NOT solve the system equations F=0 and G=0. */
display2d:false;
algebraic:true;
F:2*x^3*y^2+2*a*x^2*y^2+3*x^2*y+2*a*x*y+x+1;
G:x^3*y^2+2*a*x^2*y^2+a^2*x*y^2+x^2*y+2*a*x*y+a^2*y+1;
solutions:solve([F=0,G=0],[x,y]); /* NG: internal error! */ -------------------------------------------------------------------------------
The result is further strange for me as ------------------------------------------------------------------------------- Maxima 5.14.0cvs http://maxima.sourceforge.net Using Lisp GNU Common Lisp (GCL) GCL 2.6.7 (aka GCL) Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report() provides bug reporting information. (%i1) batch(solve_system_of_algebraic_equations_with_algebraic.maxima)
batching #p/Volumes/HFS+2/home/adachi/work/285/solve_system_of_algebraic_equations_with_algebraic.maxima (%i2) display2d : false (%o2) false (%i3) algebraic:true (%o3) true (%i4) F:1+x+2*a*x*y+3*x^2*y+2*a*x^2*y^2+2*x^3*y^2 (%o4) 2*x^3*y^2+2*a*x^2*y^2+3*x^2*y+2*a*x*y+x+1 (%i5) G:1+a^2*y+2*a*x*y+x^2*y+a^2*x*y^2+2*a*x^2*y^2+x^3*y^2 (%o5) x^3*y^2+2*a*x^2*y^2+a^2*x*y^2+x^2*y+2*a*x*y+a^2*y+1 (%i6) solutions:solve([F = 0,G = 0],[x,y]) Polynomial quotient is not exact -- an error. To debug this try debugmode(true); -------------------------------------------------------------------------------
Sincerely yours, Satoshi Adachi