refine(List,List) -- refine numerical solutions to a system of polynomial equations

Synopsis

Function: refine
Usage:

solsR = refine(T,sols)
Inputs:
- T, a list, contains the polynomials of the system (may be of type PolySystem)
- sols, a list, contains (a) solution(s) (points or lists of coordinates of points)
Optional inputs:
- Bits => ..., default value null, options for core functions of Numerical Algebraic Geometry
- ErrorTolerance => ..., default value null, options for core functions of Numerical Algebraic Geometry
- Iterations => ..., default value null, options for core functions of Numerical Algebraic Geometry
- ResidualTolerance => ..., default value null, options for core functions of Numerical Algebraic Geometry
- SingularConditionNumber => ..., default value null, set/get the default parameters for continuation algorithms
- Software => ..., default value null, specify internal or external software
Outputs:
- solsR, a list, contains refined solutions (as points)

Description

Uses Newton's method to correct the given solutions so that the resulting approximation has its estimated relative error bounded by min(ErrorTolerance,2^(-Bits)). The number of iterations made is at most Iterations.

i1 : R = CC[x];

i2 : F = polySystem {x^2-2};

i3 : P := refine(F, point{{1.5+0.001*ii}}, Bits=>1000)

o3 = {1.41421}

o3 : Point

i4 : first coordinates P

o4 = 1.4142135623730950488016887242096980785696718753769480731766797379907324
     784621070388503875343276415727350138462309122970249248360558507372126441
     214970999358314132226659275055927557999505011527820605714701095599716059
     702745345968620147285174186408891986095523292304843087143214508397626036
     2799525140799

o4 : CC (of precision 1002)

i5 : R = CC[x,y];

i6 : T = {x^2+y^2-1, x*y};

i7 : sols = { {1.1,-0.1}, {0.1,1.2} };

i8 : refine(T, sols, Software=>M2, ErrorTolerance=>.001, Iterations=>10)

o8 = {{1, -2.17629e-17}, {7.01931e-15, 1}}

o8 : List

In case of a singular (multiplicity>1) solution, while solveSystem and track return the end of the homotopy paths marked as a 'failure', it is possible to improve the quality of approximation with refine. The resulting point will be marked as singular:

i9 : R = CC[x,y];

i10 : S = {x^2-1,y^2-1};

i11 : T = {x^2+y^2-1, (x-y)^2};

i12 : solsS = {(1,1),(-1,-1)};

i13 : solsT = track(S,T,solsS)

o13 = {[M,t=.999998], [M,t=.999998]}

o13 : List

i14 : solsT / coordinates

o14 = {{.70667, .707543}, {-.70667, -.707543}}

o14 : List

i15 : refSols = refine(T, solsT)

o15 = {(.707107, .707107), (-.707107, -.707107)}

o15 : List

i16 : refSols / status

o16 = {Singular, Singular}

o16 : List

The failure to complete the refinement procedure is indicated by warning messages and the resulting point is displayed as [R].

i17 : R = CC[x];

i18 : F = polySystem {x^2-2};

i19 : Q := refine(F, point{{1.5+0.001*ii}}, Bits=>1000, Iterations=>2)

o19 = [RF]

o19 : Point

i20 : peek Q

o20 = Point{cache => CacheTable{...4...}}
            Coordinates => {1.41422}

Caveat

There are 2 'safety' bits in the computation. If the condition of the system at the refined point is poor the number of correct bits may be much smaller than requested.

Ways to use this method: