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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

                              8810015623                     
o3 = {{{- --------------------------------------------------,
          23384026197294446691258957323460528314494920687616 
     ------------------------------------------------------------------------
                         23050832683                       4801919417 
     --------------------------------------------------}, {----------,
     23384026197294446691258957323460528314494920687616    2147483648 
     ------------------------------------------------------------------------
     9603838835      8589934591  8589934593    4801919417  9603838835       
     ----------}}, {{----------, ----------}, {----------, ----------}}, {{-
     4294967296      8589934592  8589934592    2147483648  4294967296       
     ------------------------------------------------------------------------
                         10134790527                     
     ---------------------------------------------------,
     374144419156711147060143317175368453031918731001856 
     ------------------------------------------------------------------------
                         1378591365                          9603838835   
     --------------------------------------------------}, {- ----------, -
     46768052394588893382517914646921056628989841375232      4294967296   
     ------------------------------------------------------------------------
     4801919417      8589934591  8589934593      9603838835    4801919417
     ----------}}, {{----------, ----------}, {- ----------, - ----------}}}
     2147483648      8589934592  8589934592      4294967296    2147483648

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

                           3560204265                      19207677669      
o4 = {{--------------------------------------------------, -----------}, {1,
       11692013098647223345629478661730264157247460343808   8589934592      
     ------------------------------------------------------------------------
     19207677669                         893940393                        
     -----------}, {---------------------------------------------------, -
      8589934592    748288838313422294120286634350736906063837462003712   
     ------------------------------------------------------------------------
     19207677669         19207677669
     -----------}, {1, - -----------}}
      8589934592          8589934592

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

o5 = {{[-3.76754e-40,9.85751e-40], [2.23607,2.23607]}, {[1,1],
     ------------------------------------------------------------------------
     [2.23607,2.23607]}, {[-2.70879e-41,2.94772e-41], [-2.23607,-2.23607]},
     ------------------------------------------------------------------------
     {[1,1], [-2.23607,-2.23607]}}

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

o6 = {{[-3.76848e-40,9.85797e-40], [2.23535,2.23633]}, {[.999512,1.00049],
     ------------------------------------------------------------------------
     [2.23535,2.23633]}, {[-2.70955e-41,2.94833e-41], [-2.23633,-2.23535]},
     ------------------------------------------------------------------------
     {[.999512,1.00049], [-2.23633,-2.23535]}}

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

o7 = {{3.04499e-40, 2.23607}, {1, 2.23607}, {1.19465e-42, -2.23607}, {1,
     ------------------------------------------------------------------------
     -2.23607}}

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

o8 = {{3.04474e-40, 2.23584}, {1, 2.23584}, {1.19391e-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 = {{[-3.76754e-40,9.85751e-40], [2.23607,2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [2.23607,2.23607]}, {[-2.70879e-41,2.94772e-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.