# randomInitialPair -- a random initial pair

## Synopsis

• Usage:
(S,sol) = randomInitialPair T
• Inputs:
• T, contains homogeneous polynomials
• Outputs:
• S, contains homogeneous polynomials
• sol, contains one solution of S

## Description

Generates a start system S that has an equal chance of reaching any of the solutions of the target system T. 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) = randomInitialPair T 2 o2 = ({(.244218 - .220355*ii)x1 + (.539808 + .144395*ii)x1*x2 + (- .221394 - ------------------------------------------------------------------------ 2 .0613168*ii)x2 + (- .259397 - .00345748*ii)x1*x3 + (- .178951 - ------------------------------------------------------------------------ 2 .107379*ii)x2*x3 + (- .0128909 - .144621*ii)x3 , (.359958 - ------------------------------------------------------------------------ 3 2 .0342736*ii)x1 + (.221311 + .556399*ii)x1 x2 + (- .133948 + ------------------------------------------------------------------------ 2 3 .240825*ii)x1*x2 + (- .299729 - .0133971*ii)x2 + (- .119922 - ------------------------------------------------------------------------ 2 .0166779*ii)x1 x3 + (.132036 + .509053*ii)x1*x2*x3 + (- .0926582 + ------------------------------------------------------------------------ 2 2 .0968547*ii)x2 x3 + (- .710775 + .256295*ii)x1*x3 + (- .104174 + ------------------------------------------------------------------------ 2 3 .202058*ii)x2*x3 + (.170022 - .20222*ii)x3 }, {{-.710384+.284524*ii, ------------------------------------------------------------------------ .183054-.427437*ii, .40779-.178596*ii}}) o2 : Sequence i3 : M = track(S,T,solsS,gamma=>0.6+0.8*ii,Software=>M2) o3 = {{-.526105-.010808*ii, -.10453-.322525*ii, .60848-.487749*ii}} o3 : List