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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

        8589934591  8589934593      9603838835    4801919417       
o3 = {{{----------, ----------}, {- ----------, - ----------}}, {{-
        8589934592  8589934592      4294967296    2147483648       
     ------------------------------------------------------------------------
                                  17289631723                             
     --------------------------------------------------------------------,
     13479973333575319897333507543509815336818572211270286240551805124608 
     ------------------------------------------------------------------------
                                  14871302675                              
     --------------------------------------------------------------------},
     13479973333575319897333507543509815336818572211270286240551805124608  
     ------------------------------------------------------------------------
        9603838835    4801919417      8589934591  8589934593    4801919417 
     {- ----------, - ----------}}, {{----------, ----------}, {----------,
        4294967296    2147483648      8589934592  8589934592    2147483648 
     ------------------------------------------------------------------------
     9603838835                             4782210831                     
     ----------}}, {{- ---------------------------------------------------,
     4294967296        187072209578355573530071658587684226515959365500928 
     ------------------------------------------------------------------------
                         3782910073                        4801919417 
     --------------------------------------------------}, {----------,
     93536104789177786765035829293842113257979682750464    2147483648 
     ------------------------------------------------------------------------
     9603838835
     ----------}}}
     4294967296

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

            19207677669     
o4 = {{1, - -----------}, {-
             8589934592     
     ------------------------------------------------------------------------
                                  302291131                                
     -------------------------------------------------------------------, -
     3369993333393829974333376885877453834204643052817571560137951281152   
     ------------------------------------------------------------------------
     19207677669       19207677669  
     -----------}, {1, -----------},
      8589934592        8589934592  
     ------------------------------------------------------------------------
                           2783609315                      19207677669
     {---------------------------------------------------, -----------}}
      374144419156711147060143317175368453031918731001856   8589934592

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

o5 = {{[1,1], [-2.23607,-2.23607]}, {[-1.28262e-57,1.10321e-57],
     ------------------------------------------------------------------------
     [-2.23607,-2.23607]}, {[1,1], [2.23607,2.23607]},
     ------------------------------------------------------------------------
     {[-2.55634e-41,4.04433e-41], [2.23607,2.23607]}}

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

o6 = {{[.999512,1.00049], [-2.23633,-2.23535]}, {[-1.28287e-57,1.10334e-57],
     ------------------------------------------------------------------------
     [-2.23633,-2.23535]}, {[.999512,1.00049], [2.23535,2.23633]},
     ------------------------------------------------------------------------
     {[-2.55821e-41,4.04471e-41], [2.23535,2.23633]}}

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

o7 = {{1, -2.23607}, {-8.97008e-59, -2.23607}, {1, 2.23607}, {7.43993e-42,
     ------------------------------------------------------------------------
     2.23607}}

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

o8 = {{1, -2.23584}, {-8.9767e-59, -2.23584}, {1, 2.23584}, {7.43249e-42,
     ------------------------------------------------------------------------
     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 = {{[1,1], [-2.23607,-2.23607]}, {[-1.28262e-57,1.10321e-57],
      -----------------------------------------------------------------------
      [-2.23607,-2.23607]}, {[1,1], [2.23607,2.23607]},
      -----------------------------------------------------------------------
      {[-2.55634e-41,4.04433e-41], [2.23607,2.23607]}}

o10 : List

Ways to use msolveRealSolutions:

For the programmer

The object msolveRealSolutions is a method function with options.