Help:=proc(): print(`GuessPol(L,n), GuessRat(L,n)`):end:


#============================================================
#GuessPol1(L,d,n): guesses a polynomial of degree d in n for
# the list L, such that L[i]=P[i] for i>=s0 for example, try: 
#GuessPol1([(seq(i,i=1..10),1,n,1);
GuessPol1:=proc(L,d,n) local P,i,a,eq,var:
if d>=nops(L)-3 then
 ERROR(`the list is too small`):
fi:

P:=add(a[i]*n^i,i=0..d):
var:={seq(a[i],i=0..d)}:
eq:={seq(subs(n=i,P)-L[i],i=1..d+4)}:

var:=solve(eq,var):

if var=NULL then
 RETURN(FAIL):
fi:

subs(var,P):

end:


#============================================================
#GuessPol(L,n): guesses a polynomial of degree d in n for
# the list L, such that L[i]=P[i] for i>=1 for example, try: 
#GuessPol([(seq(i,i=1..10),n);
GuessPol:=proc(L,n) local d,gu:

for d from 0 to nops(L)-4 do
 gu:=GuessPol1(L,d,n):
 if gu<>FAIL then
    RETURN(gu):
 fi:
od:

FAIL:

end:

#################



GuessRat := proc(L,x)
local d,PoQ;

for d from 1 to (nops(L)-5)/2 do
	PoQ := GuessRat1(L,x,d);
	if not PoQ = FAIL then
		RETURN(PoQ);
	fi;
od;

RETURN(FAIL);

end:



#============================================================
#Recall that a rational function f(x) is a quotient of 
#polynomials f(x)=P(x)/Q(x). Write a program GuessRat1(L,x,d) 
#that inputs a list L, a variable x, and a non.neg. 
#integer d such that 2*d+5 <= nops(L), and outputs a 
#rational function P(x)/Q(x) with degree(P,x), 
#degree(Q,x) <= d, such that P(i)/Q(i)=L[i] for all 
#i from 1 to nops(L), if one exist, and FAIL otherwise. 

GuessRat1 := proc(L,x,d)
local P,Q,var,eq,i,Ans,a,b,j;

if 2*d+5>nops(L) then
	print(`List not big enough`);
	RETURN(FAIL);
fi;

P := add(a[i]*x^i, i=0..d);
Q := add(b[i]*x^i, i=0..d);

var := {seq(a[i], i=0..d),seq(b[j], j=0..d)};

eq := {};

for i from 1 to nops(L) do
	if not subs(x=i,Q)=0 then
		eq := eq union {subs(x=i,P/Q)-L[i]};
	else
		print(`denominator equals 0 for i=`,i);
		RETURN(FAIL);
	fi;
od;

Ans := solve(eq,var);

if Ans=NULL then
	RETURN(FAIL);
fi;

RETURN(factor(subs(Ans,P/Q)));

end: