Algebraic systems

AXIOM SESSIONscale=2.5]

N := NonNegativeInteger
 

   (3)  NonNegativeInteger
                                                                 Type: Domain
                                                                  Time: 0 sec

P := Polynomial Integer
 

   (4)  Polynomial Integer
                                                                 Type: Domain
                                                                  Time: 0 sec

epsilon := 1/10^21 :: Float
 

   (5)  0.1 E -20
                                                                  Type: Float
                                                                  Time: 0 sec

syst(n:N): List P == [x^n+y+z-1, x+y^n+z-1, x+y+z^n-1]
 
                                                                   Type: Void
                                                                  Time: 0 sec


completeExactSolve(syst(1))
 

   (7)  [[z= - %D - %C + 1,y= %C,x= %D]]
                         Type: List List Equation Fraction Polynomial Integer
                                                   Time: 0.03 (IN) = 0.03 sec


completeExactSolve(syst(2))
 

   (8)
   [[z= 0,y= 0,x= 1], [z= 0,y= 1,x= 0], [z= 1,y= 0,x= 0],
                2
    [z= x,y= x,x  + 2x - 1= 0]]
                         Type: List List Equation Fraction Polynomial Integer
                                                   Time: 0.09 (EV) = 0.09 sec

realApproximateSolve(syst(2),[x,y,z],epsilon)
 

   (9)
   [[- 2.4142135623 730950488,- 2.4142135623 730950488,- 2.4142135623 730950488]
     ,
    [0.4142135623 730950488,0.4142135623 730950488,0.4142135623 730950488],
    [1.0,0.0,0.0], [0.0,1.0,0.0], [0.0,0.0,1.0]]
                                                        Type: List List Float
                                       Time: 0.10 (EV) + 0.02 (OT) = 0.12 sec


completeExactSolve(syst(3))
 

   (10)
   [[z= - 1,y= 1,x= 1], [z= 0,y= 0,x= 1], [z= 1,y= - 1,x= 1], [z= 0,y= 1,x= 0],
    [z= 1,y= 0,x= 0], [z= 1,y= 1,x= - 1],
                 2          2         3
    [z= - y - x,y  + x y + x  - 1= 0,x  - x - 1= 0],
                                       3           3
                3                   - x  + 1    - x  + 1  4    2
    [z= x,y= x,x  + 2x - 1= 0], [z= --------,y= --------,x  - x  - 2x + 3= 0],
                                        2           2
                3          4    3     2
    [z= x,y= - x  - x + 1,x  + x  + 2x  + 1= 0],
           3               4    3     2
    [z= - x  - x + 1,y= x,x  + x  + 2x  + 1= 0]]
                         Type: List List Equation Fraction Polynomial Integer
               Time: 0.01 (IN) + 1.58 (EV) + 0.01 (OT) + 0.13 (GC) = 1.73 sec

realApproximateSolve(syst(3),[x,y,z],epsilon)
 

   (11)
   [[0.4533976515 1640376765,0.4533976515 1640376765,0.4533976515 1640376765],
    [1.0,1.0,- 1.0], [1.0,0.0,0.0], [0.0,1.0,0.0], [1.0,- 1.0,1.0],
    [0.0,0.0,1.0], [- 1.0,1.0,1.0]]
                                                        Type: List List Float
                           Time: 1.56 (EV) + 0.01 (OT) + 0.12 (GC) = 1.69 sec


completeExactSolve(syst(4))
 

   (12)
   [[z= 0,y= 0,x= 1], [z= 0,y= 1,x= 0], [z= 1,y= 0,x= 0],
                4
    [z= x,y= x,x  + 2x - 1= 0],
           4               11     8     7     5     4     3
    [z= - x  - x + 1,y= x,x   + 2x  - 3x  + 2x  - 4x  + 3x  - 2x + 2= 0],
                4          11     8     7     5     4     3
    [z= x,y= - x  - x + 1,x   + 2x  - 3x  + 2x  - 4x  + 3x  - 2x + 2= 0],

            4            4
         - x  + 1     - x  + 1
     [z= --------, y= --------,
             2            2
       11    10    9    8     7     6     4     3    2
      x   + x   + x  - x  - 4x  - 4x  + 4x  + 7x  - x  - 5x - 7= 0]
     ,

                4       2     4          8    5     4    2
     [z= - y - x  + 1, y  + (x  - 1)y + x  - x  - 2x  + x  + x + 1= 0,
       12     9     8     6     5     4     2
      x   - 2x  - 3x  + 2x  + 4x  + 3x  - 2x  - 2x - 2= 0]
     ]
                         Type: List List Equation Fraction Polynomial Integer
              Time: 0.01 (IN) + 9.63 (EV) + 0.03 (OT) + 0.96 (GC) = 10.63 sec

realApproximateSolve(syst(4),[x,y,z],epsilon)
 

   (13)
   [[- 0.8755550011 8456161483,- 0.8755550011 8456161483,1.2878847152 284854786]
     ,
    [- 0.8755550011 8456161482,1.2878847152 284854786,- 0.8755550011 8456161482]
     ,
    [1.2878847152 284854786,- 0.8755550011 8456161482,- 0.8755550011 8456161482]
     ,
    [- 1.3953369944 670730188,- 1.3953369944 670730188,- 1.3953369944 670730188]
     ,
    [0.4746266175 6260555033,0.4746266175 6260555033,0.4746266175 6260555033],
    [1.0,0.0,0.0], [0.0,1.0,0.0], [0.0,0.0,1.0]]
                                                        Type: List List Float
                          Time: 9.63 (EV) + 0.01 (OT) + 0.83 (GC) = 10.47 sec


-- completeExactSolve(syst(5))    MORE THAN FIVE MUNITES !!!!

Marc Moreno Maza
2008-01-07