#3150 solve fails on system of two quadratic equations

[Robert Dodier]

From the wxMaxima issue tracker #690.

solve can't solve a system of two quadratic equations with no free parameters. This example should have two real solutions:

(%i1) solve([x^2+y^2=1, (x-2)^2+(y-3)^2=9],[x,y]);
(%o1)                                 []

although solve can solve some examples like that:

(%i26) solve([x^2+y^2=1, x^2+(y-3)^2=9],[x,y]);
                       sqrt(35)      1        sqrt(35)      1
(%o26)         [[x = - --------, y = -], [x = --------, y = -]]
                          6          6           6          6
(%i27) solve([x^2+y^2=1, (x-1)^2+(y-3)^2=9],[x,y]);
                              4      3
(%o27)                [[x = - -, y = -], [x = 1, y = 0]]
                              5      5

For the record, Solver (loaded by load(solver);) finds a solution with x in terms of y and to_poly_solve can then find y:

(%i1) load (solver) $
(%i2) Solver ([x^2+y^2=1, (x-2)^2+(y-3)^2=9], [x,y]);
                        2                2
(%o2) [[x = - sqrt(1 - y ), [4 sqrt(1 - y ) - 6 y + 5]], 
                                           2                   2
                            [x = sqrt(1 - y ), [(- 4 sqrt(1 - y )) - 6 y + 5]]]
(%i3) load (to_poly_solve) $
(%i4) map (lambda ([e], to_poly_solve (e, y)), map (second, %o2));
                           5/2                          5/2
                        2 3    + 45                  2 3    - 45
(%o4)      [%union([y = -----------]), %union([y = - -----------])]
                            78                           78

[David Billinghurst]

Duplicate of https://sourceforge.net/p/maxima/bugs/3208/

[Aleksas]

(%i3) eq1:x^2+y^2=1$
eq2:(x-2)^2+(y-3)^2=9$
solve([eq1,eq2-eq1],[x,y]);
(%o3)
[[x=-(3^(5/2)-10)/26,y=(23^(5/2)+45)/78],[x=(3^(5/2)+10)/26,y=-(23^(5/2)-45)/78]]

best
Aleksas D

2016-09-22 5:00 GMT+03:00 David Billinghurst billingd@users.sf.net:

Duplicate of https://sourceforge.net/p/maxima/bugs/3208/

• [bugs:#3150] https://sourceforge.net/p/maxima/bugs/3150/ solve fails
  on system of two quadratic equations*

Status: open
Group: None
Labels: solve
Created: Fri Apr 29, 2016 09:41 PM UTC by Robert Dodier
Last Updated: Tue May 10, 2016 10:58 PM UTC
Owner: nobody

From the wxMaxima issue tracker #690
https://github.com/andrejv/wxmaxima/issues/690.

solve can't solve a system of two quadratic equations with no free
parameters. This example should have two real solutions:

(%i1) solve([x^2+y^2=1, (x-2)^2+(y-3)^2=9],[x,y]);
(%o1) []

although solve can solve some examples like that:

(%i26) solve([x^2+y^2=1, x^2+(y-3)^2=9],[x,y]);
sqrt(35) 1 sqrt(35) 1
(%o26) [[x = - --------, y = -], [x = --------, y = -]]
6 6 6 6
(%i27) solve([x^2+y^2=1, (x-1)^2+(y-3)^2=9],[x,y]);
4 3
(%o27) [[x = - -, y = -], [x = 1, y = 0]]
5 5

For the record, Solver (loaded by load(solver);) finds a solution with x
in terms of y and to_poly_solve can then find y:

(%i1) load (solver) $
(%i2) Solver ([x^2+y^2=1, (x-2)^2+(y-3)^2=9], [x,y]);
2 2
(%o2) [[x = - sqrt(1 - y ), [4 sqrt(1 - y ) - 6 y + 5]],
2 2
[x = sqrt(1 - y ), [(- 4 sqrt(1 - y )) - 6 y + 5]]]
(%i3) load (to_poly_solve) $
(%i4) map (lambda ([e], to_poly_solve (e, y)), map (second, %o2));
5/2 5/2
2 3 + 45 2 3 - 45
(%o4) [%union([y = -----------]), %union([y = - -----------])]
78 78

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[David Billinghurst]

Fixed by commit [2986fc] See [bugs:#3208] for details

(%i1) solve([x^2+y^2=1, (x-2)^2+(y-3)^2=9],[x,y]);

               5/2                   3/2
              3    - 10        (- 2 3   ) - 15
(%o1) [[x = - ---------, y = - ---------------],
                 26                  26
                                                  5/2                3/2
                                                 3    + 10        2 3    - 15
                                            [x = ---------, y = - -----------]]
                                                    26                26