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

Synopsis

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

                                 3604677209                       
o3 = {{{- -------------------------------------------------------,
          6129982163463555433433388108601236734474956488734408704 
     ------------------------------------------------------------------------
                           13166147471                          4801919417 
     -------------------------------------------------------}, {----------,
     6129982163463555433433388108601236734474956488734408704    2147483648 
     ------------------------------------------------------------------------
     9603838835      8589934591  8589934593    4801919417  9603838835       
     ----------}}, {{----------, ----------}, {----------, ----------}}, {{-
     4294967296      8589934592  8589934592    2147483648  4294967296       
     ------------------------------------------------------------------------
                                  598658939                             
     ------------------------------------------------------------------,
     842498333348457493583344221469363458551160763204392890034487820288 
     ------------------------------------------------------------------------
                                  3543893333                                 
     -------------------------------------------------------------------}, {-
     3369993333393829974333376885877453834204643052817571560137951281152     
     ------------------------------------------------------------------------
     9603838835    4801919417      8589934591  8589934593      9603838835   
     ----------, - ----------}}, {{----------, ----------}, {- ----------, -
     4294967296    2147483648      8589934592  8589934592      4294967296   
     ------------------------------------------------------------------------
     4801919417
     ----------}}}
     2147483648

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

                              4780735131                        19207677669  
o4 = {{-------------------------------------------------------, -----------},
       6129982163463555433433388108601236734474956488734408704   8589934592  
     ------------------------------------------------------------------------
         19207677669                                 1149257577              
     {1, -----------}, {-----------------------------------------------------
          8589934592    67399866667876599486667537717549076684092861056351431
     ------------------------------------------------------------------------
                       19207677669         19207677669
     --------------, - -----------}, {1, - -----------}}
     20275902562304     8589934592          8589934592

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

o5 = {{[-5.8804e-46,2.14783e-45], [2.23607,2.23607]}, {[1,1],
     ------------------------------------------------------------------------
     [2.23607,2.23607]}, {[-7.10576e-58,1.0516e-57], [-2.23607,-2.23607]},
     ------------------------------------------------------------------------
     {[1,1], [-2.23607,-2.23607]}}

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

o6 = {{[-5.88094e-46,2.14813e-45], [2.23535,2.23633]}, {[.999512,1.00049],
     ------------------------------------------------------------------------
     [2.23535,2.23633]}, {[-7.10668e-58,1.05169e-57], [-2.23633,-2.23535]},
     ------------------------------------------------------------------------
     {[.999512,1.00049], [-2.23633,-2.23535]}}

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

o7 = {{7.79894e-46, 2.23607}, {1, 2.23607}, {1.70513e-58, -2.23607}, {1,
     ------------------------------------------------------------------------
     -2.23607}}

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

o8 = {{7.8002e-46, 2.23584}, {1, 2.23584}, {1.70511e-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 = {{[-5.8804e-46,2.14783e-45], [2.23607,2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [2.23607,2.23607]}, {[-7.10576e-58,1.0516e-57], [-2.23607,-2.23607]},
      -----------------------------------------------------------------------
      {[1,1], [-2.23607,-2.23607]}}

o10 : List

Ways to use msolveRealSolutions:

For the programmer

The object msolveRealSolutions is a method function with options.