Macaulay2 » Documentation
Packages » Msolve :: msolveRealSolutions
next | previous | forward | backward | up | index | toc

msolveRealSolutions -- compute all real solutions to a zero dimensional system using symbolic methods

Description

This functions uses the msolve package to compute the real solutions to a zero dimensional polynomial ideal with either integer or rational coefficients.

The second input is optional, and indicates the alternative ways to provide output either using an exact rational interval QQi, a real interval RRi, or by taking a rational or real approximation of the midpoint of the intervals.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing
i2 : I = ideal {(x-1)*x, y^2-5}

             2       2
o2 = ideal (x  - x, y  - 5)

o2 : Ideal of R
i3 : rationalIntervalSols = msolveRealSolutions I

                               6783279059                      
o3 = {{{- ----------------------------------------------------,
          1496577676626844588240573268701473812127674924007424 
     ------------------------------------------------------------------------
                          2258277111                        4801919417 
     ---------------------------------------------------}, {----------,
     374144419156711147060143317175368453031918731001856    2147483648 
     ------------------------------------------------------------------------
     9603838835      8589934591  8589934593    4801919417  9603838835       
     ----------}}, {{----------, ----------}, {----------, ----------}}, {{-
     4294967296      8589934592  8589934592    2147483648  4294967296       
     ------------------------------------------------------------------------
                                  10644840461                             
     --------------------------------------------------------------------,
     13479973333575319897333507543509815336818572211270286240551805124608 
     ------------------------------------------------------------------------
                                  7335206409                                 
     -------------------------------------------------------------------}, {-
     6739986666787659948666753771754907668409286105635143120275902562304     
     ------------------------------------------------------------------------
     9603838835    4801919417      8589934591  8589934593      9603838835   
     ----------, - ----------}}, {{----------, ----------}, {- ----------, -
     4294967296    2147483648      8589934592  8589934592      4294967296   
     ------------------------------------------------------------------------
     4801919417
     ----------}}}
     2147483648

o3 : List
i4 : rationalApproxSols = msolveRealSolutions(I, QQ)

                            2249829385                       19207677669  
o4 = {{----------------------------------------------------, -----------},
       2993155353253689176481146537402947624255349848014848   8589934592  
     ------------------------------------------------------------------------
         19207677669                                 4025572357              
     {1, -----------}, {-----------------------------------------------------
          8589934592    26959946667150639794667015087019630673637144422540572
     ------------------------------------------------------------------------
                        19207677669         19207677669
     ---------------, - -----------}, {1, - -----------}}
     481103610249216     8589934592          8589934592

o4 : List
i5 : floatIntervalSols = msolveRealSolutions(I, RRi)

o5 = {{[-4.53253e-42,6.03584e-42], [2.23607,2.23607]}, {[1,1],
     ------------------------------------------------------------------------
     [2.23607,2.23607]}, {[-7.89678e-58,1.08831e-57], [-2.23607,-2.23607]},
     ------------------------------------------------------------------------
     {[1,1], [-2.23607,-2.23607]}}

o5 : List
i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)

o6 = {{[-4.5332e-42,6.03679e-42], [2.23535,2.23633]}, {[.999512,1.00049],
     ------------------------------------------------------------------------
     [2.23535,2.23633]}, {[-7.89701e-58,1.08841e-57], [-2.23633,-2.23535]},
     ------------------------------------------------------------------------
     {[.999512,1.00049], [-2.23633,-2.23535]}}

o6 : List
i7 : floatApproxSols = msolveRealSolutions(I, RR)

o7 = {{7.51658e-43, 2.23607}, {1, 2.23607}, {1.49317e-58, -2.23607}, {1,
     ------------------------------------------------------------------------
     -2.23607}}

o7 : List
i8 : floatApproxSols = msolveRealSolutions(I, RR_10)

o8 = {{7.51797e-43, 2.23584}, {1, 2.23584}, {1.49352e-58, -2.23584}, {1,
     ------------------------------------------------------------------------
     -2.23584}}

o8 : List

Note in cases where solutions have multiplicity this is not reflected in the output. While the solver does not return multiplicities, it reliably outputs the verified isolating intervals for multiple solutions.

i9 : I = ideal {(x-1)*x^3, (y^2-5)^2}

             4    3   4      2
o9 = ideal (x  - x , y  - 10y  + 25)

o9 : Ideal of R
i10 : floatApproxSols = msolveRealSolutions(I, RRi)

o10 = {{[-4.53253e-42,6.03584e-42], [2.23607,2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [2.23607,2.23607]}, {[-7.89678e-58,1.08831e-57], [-2.23607,-2.23607]},
      -----------------------------------------------------------------------
      {[1,1], [-2.23607,-2.23607]}}

o10 : List

Ways to use msolveRealSolutions:

  • msolveRealSolutions(Ideal)
  • msolveRealSolutions(Ideal,Ring)
  • msolveRealSolutions(Ideal,RingFamily)

For the programmer

The object msolveRealSolutions is a method function with options.


The source of this document is in Msolve.m2:636:0.