Example 7.2 (A Sequence Alignment Problem)
This worksheet requires the DiagonalAsymptotics/Kronecker package available at http://diagasympt.gforge.inria.fr 

>	# Import Kronecker package read "Kronecker.mpl"; # Define the rational function F := (x^2*y^2 - x*y + 1)/(1 - (x + y + x*y - x*y^2 - x^2*y + x^2*y^3 + x^3*y^2)); # Compute dominant asymptotics of the main power series diagonal A, U := DiagonalAsymptotics(numer(F),denom(F),[x,y],u,n):

 

(1)

 

>	# Asymptotics are represented by the symbolic expression A;

 

(2)

 

>	# When u is the unique root of the polynomial op(1,U[1]);

 

(3)

 

>	# Whose decimal expression begins evalf(op(2,U[1]),10);

 

(4)

 

>	# As usual, random choices are used to output this result. Rerun the code to get # different expressions which yield the same asymptotic behaviour. # Because u is an algebraic number of degree 5, we give the leading asymptotic with numeric approximations evalf(evala(Normal(subs(u=U[1],A))));

 

(5)

 

>