#Sowmya Srinivasan hw9.txt 20 February 2014

#read C6.txt 

#ApplyUmbra(P,x,U,m) inputs a polynomial P in variable x and an expression 
#U in discrete variable m, and outputs, for each term a[i]*x^i in P, U(i)

with(PolynomialTools) : 

ApplyUmbra:=proc(P,x,U,m) local L,i,s   : 

L:=CoefficientList(P,x)  : 

for i from 0 to nops(L)-1 do 

s[i]:=L[i+1]*subs(m=i,U) : 

od : 

add(s[i],i=0..nops(L)-1): 
 
end: 


#PnG(n,x,U,m) inputs a nonnegative integer n, variable name x, 
#expression U in discrete variable m, and outputs the monic polynomial P 
#of degree n st U applied to P*x^i=0 for i=0..n-1 

PnG:=proc(n,x,U,m) local P,i,a,eqs,vars : 

P:=x^n+add(a[i]*x^i,i=0..n-1) : 

vars:={seq(a[i],i=0..n-1)}  : 

eqs:={seq( ApplyUmbra(P*x^i,x,U,m)=0,i=0..n-1)  } : 

sort(subs(solve(eqs,vars),P)): 

end: 
#I checked that PnG(n,x,1/(m+1),m) agrees with Pn(n,x) for n=0..10 . 


#GuessCoeff(n,i,U,m)

GuessCoeff:=proc(n,i,U,m) local L,j,x  : 

L:=[seq(coeff(PnG(j,x,U,m),x^(j-i)),j=1..2*i+10)] : 

GuessRF(L,n) : 

end: 
#I don't know how to fix my GuessCoeff function. In particular I'm not sure 
#if the way I defined the list L is correct.