Hello,
I cannot solve the following problem:
What can we say about the asymptotic behavior, depending on the initial conditions, of a recurrence relation that has a inhomogeneous part (log n)^(log n) and has roots 1,2,3 ?
Hi aa2010,
I am assuming that you are asking about the asymptotic nature of the difference equation
for
under the initial conditions
, and .
Here, is the forward difference operator, i.e., and .
For such problems, you need to compute the fundamental solutions, i.e.,
the solutions of the following initial value problems
: for with , and
: for with , and
: for with , and .
Then, by the variation of constants formula, we can write the solution as follows
for .
Here, we have
for
for
and
for .
Assume that , then for , which yields
.........
for all .
Thus, if (the coefficients of the dominant term ), then as .
Obtaining a complete result for your problem requires a deep inspection since the forcing term is not a nice one.
As I said, better inequalities will give you better results.
Note that for , we have ,
then the contribution of the forcing term to the solution can not be less than
for all .
As ,
,
which implies that yields as .