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

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

                    3394316794873318371           
o3 = {{{- ---------------------------------------,
          340282366920938463463374607431768211456 
     ------------------------------------------------------------------------
               9090979498917902541              41248173712355948585 
     ---------------------------------------}, {--------------------,
     340282366920938463463374607431768211456    18446744073709551616 
     ------------------------------------------------------------------------
     20624086856177974295      18446744073709551615  18446744073709551617  
     --------------------}}, {{--------------------, --------------------},
      9223372036854775808      18446744073709551616  18446744073709551616  
     ------------------------------------------------------------------------
      41248173712355948587  10312043428088987147       
     {--------------------, --------------------}}, {{-
      18446744073709551616   4611686018427387904       
     ------------------------------------------------------------------------
               27943880522800669215          
     ---------------------------------------,
     340282366920938463463374607431768211456 
     ------------------------------------------------------------------------
               22702554030152245153               41248173712355948597   
     ---------------------------------------}, {- --------------------, -
     340282366920938463463374607431768211456      18446744073709551616   
     ------------------------------------------------------------------------
     41248173712355948577      9223372036854775807  9223372036854775809     
     --------------------}}, {{-------------------, -------------------}, {-
     18446744073709551616      9223372036854775808  9223372036854775808     
     ------------------------------------------------------------------------
     41248173712355948591    41248173712355948585
     --------------------, - --------------------}}}
     18446744073709551616    18446744073709551616

o3 : List

i4 : rationalApproxSols = msolveRealSolutions(I, QQ)

                 2848331352022292085            82496347424711897175      
o4 = {{---------------------------------------, --------------------}, {1,
       340282366920938463463374607431768211456  36893488147419103232      
     ------------------------------------------------------------------------
     82496347424711897175                2620663246324212031             
     --------------------}, {- ---------------------------------------, -
     36893488147419103232      340282366920938463463374607431768211456   
     ------------------------------------------------------------------------
     41248173712355948587         10312043428088987147
     --------------------}, {1, - --------------------}}
     18446744073709551616          4611686018427387904

o4 : List

i5 : floatIntervalSols = msolveRealSolutions(I, RRi)

o5 = {{[1,1], [2.23607,2.23607]}, {[-8.59181e-22,1.1497e-21],
     ------------------------------------------------------------------------
     [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
     ------------------------------------------------------------------------
     {[-2.79363e-21,6.6574e-21], [-2.23607,-2.23607]}}

o5 : List

i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)

o6 = {{[.999999,1], [2.23607,2.23607]}, {[-2.3434e-7,3.31111e-7],
     ------------------------------------------------------------------------
     [2.23606,2.23607]}, {[.999998,1], [-2.23607,-2.23606]},
     ------------------------------------------------------------------------
     {[-5.92081e-9,4.47108e-9], [-2.23607,-2.23607]}}

o6 : List

i7 : floatApproxSols = msolveRealSolutions(I, RR)

o7 = {{1, -2.23607}, {1.03407e-19, -2.23607}, {1, 2.23607}, {2.40267e-22,
     ------------------------------------------------------------------------
     2.23607}}

o7 : List

i8 : floatApproxSols = msolveRealSolutions(I, RR_10)

o8 = {{-1.92693e-9, 2.23607}, {1, 2.23607}, {-2.2555e-8, -2.23607}, {1,
     ------------------------------------------------------------------------
     -2.23607}}

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 = {{[-3.61574e-21,5.06446e-21], [2.23607,2.23607]}, {[1,1],
      -----------------------------------------------------------------------
      [2.23607,2.23607]}, {[-6.42531e-29,4.90632e-29], [-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/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Msolve.m2:644:0.