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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

                              3967775923                     
o3 = {{{- --------------------------------------------------,
          11692013098647223345629478661730264157247460343808 
     ------------------------------------------------------------------------
                         15686768283                         9603838835   
     --------------------------------------------------}, {- ----------, -
     23384026197294446691258957323460528314494920687616      4294967296   
     ------------------------------------------------------------------------
     4801919417                            3542882645                    
     ----------}}, {{- -------------------------------------------------,
     2147483648        2923003274661805836407369665432566039311865085952 
     ------------------------------------------------------------------------
                         4363671513                       4801919417 
     -------------------------------------------------}, {----------,
     5846006549323611672814739330865132078623730171904    2147483648 
     ------------------------------------------------------------------------
     9603838835      8589934591  8589934593      9603838835    4801919417   
     ----------}}, {{----------, ----------}, {- ----------, - ----------}},
     4294967296      8589934592  8589934592      4294967296    2147483648   
     ------------------------------------------------------------------------
       8589934591  8589934593    4801919417  9603838835
     {{----------, ----------}, {----------, ----------}}}
       8589934592  8589934592    2147483648  4294967296

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

                           7751216437                        19207677669     
o4 = {{--------------------------------------------------, - -----------}, {-
       46768052394588893382517914646921056628989841375232     8589934592     
     ------------------------------------------------------------------------
                         2722093777                      19207677669        
     --------------------------------------------------, -----------}, {1, -
     11692013098647223345629478661730264157247460343808   8589934592        
     ------------------------------------------------------------------------
     19207677669       19207677669
     -----------}, {1, -----------}}
      8589934592        8589934592

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

o5 = {{[-3.39358e-40,6.70833e-40], [-2.23607,-2.23607]},
     ------------------------------------------------------------------------
     {[-1.21207e-39,7.46436e-40], [2.23607,2.23607]}, {[1,1],
     ------------------------------------------------------------------------
     [-2.23607,-2.23607]}, {[1,1], [2.23607,2.23607]}}

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

o6 = {{[-3.39361e-40,6.71009e-40], [-2.23633,-2.23535]},
     ------------------------------------------------------------------------
     {[-1.21216e-39,7.46522e-40], [2.23535,2.23633]}, {[.999512,1.00049],
     ------------------------------------------------------------------------
     [-2.23633,-2.23535]}, {[.999512,1.00049], [2.23535,2.23633]}}

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

o7 = {{1.65737e-40, -2.23607}, {-2.32817e-40, 2.23607}, {1, -2.23607}, {1,
     ------------------------------------------------------------------------
     2.23607}}

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

o8 = {{1.65824e-40, -2.23584}, {-2.32817e-40, 2.23584}, {1, -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 = {{[-3.39358e-40,6.70833e-40], [-2.23607,-2.23607]},
      -----------------------------------------------------------------------
      {[-1.21207e-39,7.46436e-40], [2.23607,2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [-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.