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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

                                2296185875                      
o3 = {{{- -----------------------------------------------------,
          95780971304118053647396689196894323976171195136475136 
     ------------------------------------------------------------------------
                           10294793213                           9603838835 
     ------------------------------------------------------}, {- ----------,
     383123885216472214589586756787577295904684780545900544      4294967296 
     ------------------------------------------------------------------------
       4801919417      8589934591  8589934593      9603838835   
     - ----------}}, {{----------, ----------}, {- ----------, -
       2147483648      8589934592  8589934592      4294967296   
     ------------------------------------------------------------------------
     4801919417                             5266333839                     
     ----------}}, {{- ---------------------------------------------------,
     2147483648        187072209578355573530071658587684226515959365500928 
     ------------------------------------------------------------------------
                          5829290567                        4801919417 
     ---------------------------------------------------}, {----------,
     187072209578355573530071658587684226515959365500928    2147483648 
     ------------------------------------------------------------------------
     9603838835      8589934591  8589934593    4801919417  9603838835
     ----------}}, {{----------, ----------}, {----------, ----------}}}
     4294967296      8589934592  8589934592    2147483648  4294967296

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

                             1110049713                         
o4 = {{------------------------------------------------------, -
       766247770432944429179173513575154591809369561091801088   
     ------------------------------------------------------------------------
     19207677669         19207677669  
     -----------}, {1, - -----------},
      8589934592          8589934592  
     ------------------------------------------------------------------------
                           70369591                       19207677669      
     {--------------------------------------------------, -----------}, {1,
      46768052394588893382517914646921056628989841375232   8589934592      
     ------------------------------------------------------------------------
     19207677669
     -----------}}
      8589934592

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

o5 = {{[-2.39733e-44,2.68707e-44], [-2.23607,-2.23607]}, {[1,1],
     ------------------------------------------------------------------------
     [-2.23607,-2.23607]}, {[-2.81513e-41,3.11606e-41], [2.23607,2.23607]},
     ------------------------------------------------------------------------
     {[1,1], [2.23607,2.23607]}}

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

o6 = {{[-2.39753e-44,2.68765e-44], [-2.23633,-2.23535]}, {[.999512,1.00049],
     ------------------------------------------------------------------------
     [-2.23633,-2.23535]}, {[-2.81605e-41,3.11649e-41], [2.23535,2.23633]},
     ------------------------------------------------------------------------
     {[.999512,1.00049], [2.23535,2.23633]}}

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

o7 = {{1.44868e-45, -2.23607}, {1, -2.23607}, {1.50465e-42, 2.23607}, {1,
     ------------------------------------------------------------------------
     2.23607}}

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

o8 = {{1.45056e-45, -2.23584}, {1, -2.23584}, {1.50219e-42, 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 = {{[-2.39733e-44,2.68707e-44], [-2.23607,-2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [-2.23607,-2.23607]}, {[-2.81513e-41,3.11606e-41], [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.