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];
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i2 : F = polySystem {x^2-2};
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i3 : P := refine(F, point{{1.5+0.001*ii}}, Bits=>1000)
o3 = {1.41421}
o3 : Point
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i4 : first coordinates P
o4 = 1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492
4836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889
19860955232923048430871432145083976260362799525140799
o4 : CC (of precision 1002)
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i5 : R = CC[x,y];
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i6 : T = {x^2+y^2-1, x*y};
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i7 : sols = { {1.1,-0.1}, {0.1,1.2} };
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i8 : refine(T, sols, Software=>M2, ErrorTolerance=>.001, Iterations=>10)
o8 = {{1, -2.17629e-17}, {7.01931e-15, 1}}
o8 : List
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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];
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i10 : S = {x^2-1,y^2-1};
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i11 : T = {x^2+y^2-1, (x-y)^2};
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i12 : solsS = {(1,1),(-1,-1)};
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i13 : solsT = track(S,T,solsS)
o13 = {[M,t=.999998], [M,t=.999998]}
o13 : List
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i14 : solsT / coordinates
o14 = {{.70667, .707543}, {-.70667, -.707543}}
o14 : List
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i15 : refSols = refine(T, solsT)
o15 = {(.707107, .707107), (-.707107, -.707107)}
o15 : List
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i16 : refSols / status
o16 = {Singular, Singular}
o16 : List
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The failure to complete the refinement procedure is indicated by warning messages and the resulting point is displayed as
[R].
i17 : R = CC[x];
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i18 : F = polySystem {x^2-2};
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i19 : Q := refine(F, point{{1.5+0.001*ii}}, Bits=>1000, Iterations=>2)
o19 = [RF]
o19 : Point
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i20 : peek Q
o20 = Point{ConditionNumber => 1 }
Coordinates => {1.41422}
ErrorBoundEstimate => .00245131
SolutionStatus => RefinementFailure
SolutionSystem => {-2} | x2-2 |
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