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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

        8589934591  8589934593      9603838835    4801919417      8589934591 
o3 = {{{----------, ----------}, {- ----------, - ----------}}, {{----------,
        8589934592  8589934592      4294967296    2147483648      8589934592 
     ------------------------------------------------------------------------
     8589934593    4801919417  9603838835       
     ----------}, {----------, ----------}}, {{-
     8589934592    2147483648  4294967296       
     ------------------------------------------------------------------------
                          3694478609                      
     ----------------------------------------------------,
     1496577676626844588240573268701473812127674924007424 
     ------------------------------------------------------------------------
                          39285932913                          9603838835   
     ----------------------------------------------------}, {- ----------, -
     2993155353253689176481146537402947624255349848014848      4294967296   
     ------------------------------------------------------------------------
     4801919417                             12346786025                     
     ----------}}, {{- ----------------------------------------------------,
     2147483648        5986310706507378352962293074805895248510699696029696 
     ------------------------------------------------------------------------
                           6645529979                         4801919417 
     -----------------------------------------------------}, {----------,
     11972621413014756705924586149611790497021399392059392    2147483648 
     ------------------------------------------------------------------------
     9603838835
     ----------}}}
     4294967296

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

            19207677669       19207677669  
o4 = {{1, - -----------}, {1, -----------},
             8589934592        8589934592  
     ------------------------------------------------------------------------
                           31896975695                        19207677669  
     {----------------------------------------------------, - -----------},
      5986310706507378352962293074805895248510699696029696     8589934592  
     ------------------------------------------------------------------------
                             18048042071                       19207677669
     {- -----------------------------------------------------, -----------}}
        23945242826029513411849172299223580994042798784118784   8589934592

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

o5 = {{[1,1], [2.23607,2.23607]}, {[-1.1424e-41,1.12702e-41],
     ------------------------------------------------------------------------
     [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
     ------------------------------------------------------------------------
     {[-1.9607e-40,1.58153e-40], [-2.23607,-2.23607]}}

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

o6 = {{[-6.37276e-59,8.18132e-59], [2.23535,2.23633]}, {[.999512,1.00049],
     ------------------------------------------------------------------------
     [2.23535,2.23633]}, {[-9.77636e-58,9.17272e-58], [-2.23633,-2.23535]},
     ------------------------------------------------------------------------
     {[.999512,1.00049], [-2.23633,-2.23535]}}

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

o7 = {{1, -2.23607}, {-1.43368e-40, -2.23607}, {1, 2.23607}, {-5.93459e-42,
     ------------------------------------------------------------------------
     2.23607}}

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

o8 = {{1, 2.23584}, {-2.68212e-58, 2.23584}, {1, -2.23584}, {5.84186e-59,
     ------------------------------------------------------------------------
     -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 = {{[-1.77739e-39,8.46482e-40], [2.23607,2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [2.23607,2.23607]}, {[-1.05389e-57,7.711e-58], [-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 /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Msolve.m2:636:0.