# goodInitialPair(List) -- make an initial pair conjectured to be good by Shub and Smale

## Synopsis

• Function: goodInitialPair
• Usage:
(S,sol) = goodInitialPair T
• Inputs:
• T, a list, contains homogeneous polynomials
• Optional inputs:
• GeneralPosition => ..., default value false, make a random unitary change of coordinates
• Outputs:
• S, contains homogeneous polynomials
• sol, contains one solution of S

## Description

Generates a start system S that is conjectured to have good complexity when used in linear homotopy with target system T leading to one solution. For more details see: C. Beltran and A. Leykin, "Certified numerical homotopy tracking", Experimental Mathematics 21(1): 69-83 (2012)

 i1 : T = randomSd {2,3}; i2 : (S,solsS) = goodInitialPair T 2 o2 = ({1.41421x1*x3, 1.73205x2*x3 }, {{0, 0, 1}}) o2 : Sequence i3 : M = track(S,T,solsS,gamma=>0.6+0.8*ii,Software=>M2) o3 = {{.751732+.463283*ii, .284351-.0717164*ii, .348144-.114302*ii}} o3 : List

## See also

• Certified -- a value for the option Predictor that triggers certified tracking
• track -- track a linear segment homotopy given start and target system