On Friday, May 24, 2019 at 5:29:19 PM UTC-5, MM wrote:   
   > Hello all.   
   > Straight to the point, stemming from a problem concerning a phi^3   
   > potential.   
   >   
   > How to find out the series expansion for s = s(t) given by the implicit   
   > equation:   
   >   
   > 18 t + s(1+s) (1+2s) =0   
   >   
   > There actually exists a closed expression for series   
   > s(t) = sum_k s_k t^k   
   >   
   > Anyone has some ideas? Thanks.   
      
   Start (P,Q,R,S,s) = (18t,1,3,2,0) At each point update (P,Q,R,S,s) to   
   (P+Qd+Rdd+Sddd, Q+2Rd+3Sdd, R+3Sd, S, s+d), where d = -P/Q, taken as a   
   polynomial quotient without remainder term.   
      
   That's the power-series version of Newton's Method.   
      
   At each point of the iteration P = 18t + s(1+s)(1+2s), Q = 1+6s(1+s),   
   R = 3(1+2s), S = 2. Use induction to prove that the update formula is   
   consistent and use the result deg(s+d) > deg(s) to show that it   
   converges in the sense that it produces the leading N terms (for any N =   
   0, 1, 2, 3, ...) after a finite number of steps.   
      
   And get ready to use and need extended precision math packages to do the   
   coefficients and/or efficient computation of the polynomial. You might   
   be able to adapt the Toom method used to multiply large numbers to   
   multiplying large polynomials and/or numbers (the coefficients   
   themselves may become large).   
      
   There is no reference to the method that I am aware of. I just made it up   
   right here and now.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)