feuerstein
Jan21-11, 02:06 AM
Hi,
I am new to this forum and apologize for bothering you with my question on recurrence/difference equations. Unfortunately I was unable to find an answer in textbooks etc., so I would really appreciate a short answer or a reference to relevant literature. The problem is as follows:
Say we have an order-d linear recurrence/difference equation:
a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_d a_{n-d}
We are interested in stability, i.e. whether the sequence a_n has a
limit as n goes to infinity. According to standard theorems convergence
occurs if and only if all roots of the corresponding characteristic
equation have absolute value less than 1.
My problem is that I do not quite have the above recurrence, but a
recurrence that "converges" to the above. More precisely, my recurrence is
b_n = c_1 b_{n-1} + c_2 b_{n-2} + ... + c_d b_{n-d} + \alpha(n),
where \alpha(n) is a sequence that converges to 0.
The questions are:
- Is b_n stable under the same conditions as for a_n?
- Are the limits of a_n and b_n the same (if any)?
Best regards,
Feuerstein
I am new to this forum and apologize for bothering you with my question on recurrence/difference equations. Unfortunately I was unable to find an answer in textbooks etc., so I would really appreciate a short answer or a reference to relevant literature. The problem is as follows:
Say we have an order-d linear recurrence/difference equation:
a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_d a_{n-d}
We are interested in stability, i.e. whether the sequence a_n has a
limit as n goes to infinity. According to standard theorems convergence
occurs if and only if all roots of the corresponding characteristic
equation have absolute value less than 1.
My problem is that I do not quite have the above recurrence, but a
recurrence that "converges" to the above. More precisely, my recurrence is
b_n = c_1 b_{n-1} + c_2 b_{n-2} + ... + c_d b_{n-d} + \alpha(n),
where \alpha(n) is a sequence that converges to 0.
The questions are:
- Is b_n stable under the same conditions as for a_n?
- Are the limits of a_n and b_n the same (if any)?
Best regards,
Feuerstein