solveSystem -- a numerical blackbox solver

Synopsis

Usage:

solveSystem(S)

solveSystem(S,numThreads=>4)

solveSystem(S,computingPrecision=>2)

solveSystem(S,randomSeed=>12345)
Inputs:
- S, a list, contains a zero-dimensional system of polynomials with complex coefficients that contains at least as many equations as indeterminates
Optional inputs:
- computingPrecision => ..., default value 1, option to specify the working precision
- numThreads => ..., default value 0, option to set the number of threads
- randomSeed => ..., default value -1, option to set the seed of the random number generators
- Verbose => ..., default value false, option to specify whether additional output is wanted
Outputs:
- a list, containing the solutions of S, each of type Point.
Consequences:
- Writes the system to temporary file
- Adds slack variables if needed (i.e. if system is overdetermined)
- Invokes the command phc -b (launches the blackbox solver)
- Stores output of phc in temporary file
- Parses and outputs the solutions.

Description

Suppose we want numerical approximations of all complex isolated solutions to the following system:

i1 : R = CC[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : S = {x+y+z-1, x^2+y^2, x+y-z-3}

                      2    2
o2 = {x + y + z - 1, x  + y , x + y - z - 3}

o2 : List

We call PHCpack's blackbox solver:

i3 : L = solveSystem(S)

o3 = {{1-ii, 1+ii, -1}, {1+ii, 1-ii, -1}}

o3 : List

The method solveSystem prints the PHCpack input and output file names and returns two solutions. The solutions are of type Point, defined in NAGtypes. Each point p comes with cached diagnostics. For example, p.cache.LastT is the end value of the continuation parameter; if it equals 1, then the solver reached the end of the path properly.

i4 : oneSoln = L_0

o4 = oneSoln

o4 : Point

i5 : peek oneSoln

o5 = Point{cache => CacheTable{...3...}   }
           Coordinates => {1-ii, 1+ii, -1}

The method handles overdetermined systems by inserting slack variables.

i6 : system = {y-x^2, z-x^3, x+y+z-1, x+y+ x^3 -1}

         2         3                      3
o6 = {- x  + y, - x  + z, x + y + z - 1, x  + x + y - 1}

o6 : List

i7 : #system > numcols vars R --overdetermined system

o7 = true

i8 : solns = solveSystem(system);
*** after parseSolutions, ring has {x, y, z} ***

i9 : numSolns = #solns

o9 = 3

Caveat

The method solveSystem does not check the dimension of the system; it checks for number of equations instead. So solveSystem will return an error if there are less equations than unknowns even if the system is zero-dimensional. In addition, if the system is square but not zero-dimensional, the output is meaningless. Thus, it is suggested that the user checks the dimension of the system before using solveSystem.

Ways to use solveSystem :

solveSystem(List)

For the programmer