msolveRealSolutions -- compute all real solutions to a zero dimensional system using symbolic methods

Synopsis

Usage:

msolveRealSolutions(I)

msolveRealSolutions(I, K)
Inputs:
- I, an ideal, which is zero dimensional, in a polynomial ring with coefficients over QQ
- K, a ring or an instance of the type RingFamily, the field to find answers in, which must be one of QQi (default), QQ, RR, or RRi (possibly with specified precision)
Optional inputs:
- Threads => an integer, default value 5, number of processor threads to use;
- Verbosity => an integer, default value 0, level of verbosity between 0, 1, and 2;
Outputs:
- a list, of lists; each entry in the list consists of a list representing the coordinates of a solution. By default each solution coordinate value is also represented by a rational interval consisting of a two element list of rational numbers, {a, b}, this means that that coordinate of the solution has a value greater than or equal to a and less than or equal to b. This interval is computed symbolically and its correctness is guaranteed by exact 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

                              7478009799                     
o3 = {{{- --------------------------------------------------,
          93536104789177786765035829293842113257979682750464 
     ------------------------------------------------------------------------
                         3941335657                        4801919417 
     --------------------------------------------------}, {----------,
     23384026197294446691258957323460528314494920687616    2147483648 
     ------------------------------------------------------------------------
     9603838835      8589934591  8589934593    4801919417  9603838835       
     ----------}}, {{----------, ----------}, {----------, ----------}}, {{-
     4294967296      8589934592  8589934592    2147483648  4294967296       
     ------------------------------------------------------------------------
                         8335431791                    
     -------------------------------------------------,
     5846006549323611672814739330865132078623730171904 
     ------------------------------------------------------------------------
                         7358164891                         9603838835   
     -------------------------------------------------}, {- ----------, -
     5846006549323611672814739330865132078623730171904      4294967296   
     ------------------------------------------------------------------------
     4801919417      8589934591  8589934593      9603838835    4801919417
     ----------}}, {{----------, ----------}, {- ----------, - ----------}}}
     2147483648      8589934592  8589934592      4294967296    2147483648

o3 : List

i4 : rationalApproxSols = msolveRealSolutions(I, QQ)

                            8287332829                      19207677669      
o4 = {{---------------------------------------------------, -----------}, {1,
       187072209578355573530071658587684226515959365500928   8589934592      
     ------------------------------------------------------------------------
     19207677669                          244316725                       
     -----------}, {- -------------------------------------------------, -
      8589934592      2923003274661805836407369665432566039311865085952   
     ------------------------------------------------------------------------
     19207677669         19207677669
     -----------}, {1, - -----------}}
      8589934592          8589934592

o4 : List

i5 : floatIntervalSols = msolveRealSolutions(I, RRi)

o5 = {{[-7.99478e-41,1.68548e-40], [2.23607,2.23607]}, {[1,1],
     ------------------------------------------------------------------------
     [2.23607,2.23607]}, {[-1.42583e-39,1.25867e-39], [-2.23607,-2.23607]},
     ------------------------------------------------------------------------
     {[1,1], [-2.23607,-2.23607]}}

o5 : List

i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)

o6 = {{[-7.99525e-41,1.68559e-40], [2.23535,2.23633]}, {[.999512,1.00049],
     ------------------------------------------------------------------------
     [2.23535,2.23633]}, {[-1.42632e-39,1.25915e-39], [-2.23633,-2.23535]},
     ------------------------------------------------------------------------
     {[.999512,1.00049], [-2.23633,-2.23535]}}

o6 : List

i7 : floatApproxSols = msolveRealSolutions(I, RR)

o7 = {{4.43002e-41, 2.23607}, {1, 2.23607}, {-8.35841e-41, -2.23607}, {1,
     ------------------------------------------------------------------------
     -2.23607}}

o7 : List

i8 : floatApproxSols = msolveRealSolutions(I, RR_10)

o8 = {{4.43035e-41, 2.23584}, {1, 2.23584}, {-8.35847e-41, -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 = {{[-7.99478e-41,1.68548e-40], [2.23607,2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [2.23607,2.23607]}, {[-1.42583e-39,1.25867e-39], [-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.