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## cs2223, D97/98 Classifying Recurrence Relations

There are only a few types of recurrence relations which can be solved
in closed form - which means any term in the sequence can be evaluated by
plugging numbers into an equation instead of having to calculate the entire
sequence. Classifying the recurrence relation helps to decide which, if
any, techniques can be used to solve it.

Here is a generic recurrence relation, consisting of a sequence of terms
*A*_{n}.

The right side *f(n)*, is some function of *n*, the number
of *A*_{n} in the sequence. In the standard form, the right
side of a recurrence relation can contain any function of *n*, but
must not contain any of the *A*_{n}.

There are four characteristics of the recurrence relation to note.

#### Homogeneous or non-homogeneous

A recurrence relation is homogeneous if the right side is zero. The above
recurrence relation is non-homogeneous, but this one is homogeneous.

#### Linear or non-linear

A recurrence relation is linear if there are no products or powers of
the sequence elements. The above recurrence relations are non-linear. These
recurrence relations are linear.

These recurrence relations are non-linear.

#### Order of the Recurrence Relation

The order of the recurrence relation is the number of steps along the
sequence from the first to the last members in the relation.

The last recurrence relation does not have a constant order. That complicates
the solution.

#### Constant versus non-constant coefficients

We make a distinction between recurrence relations in which the coefficients
which multiply the sequential terms are constant, such as these

and those recurrence relations in which the coefficients are not constant
- they vary with *n*, such as these.

#### So what?

There are two classes of recurrence relations which are always solvable
so it is important to recognize them.

- First-order, non-homogeneous, linear recurrence relations in which
the coefficients are never zero.
- Any constant order linear recurrence relation with constant coefficients
which are homogeneous or whose right sides can be expressed as the product
of polynomials in n and constants to the n-th power.

The solution techniques for these two classes of recurrence relation
are discussed in detail in the Recurrence page.
Other recurrence relations are generally harder to solve, although a few
special cases have been solved.

Contents ©1994-1998, Norman Wittels

Updated 19Mar98