[Studies in Computational Intelligence 481] Artur Babiarz, Robert Bieda, Karol Jędrasiak, Aleksander Nawrat (auth.), Aleksander Nawrat, Zygmunt Kuś (eds.) - Vision Based Systemsfor UAV Applications (2013, Sprin

324 A. Czornik, A. Nawrat, and M. Niezabitowski
Finally,
b(n)= 1 2
s
∑
i, j,k=1
∣
∣a (i)
∣ ∣∣
jk (n)+a(i) kj (n) =
1
[∣ ∣ ∣ ∣ ∣ ∣ ∣∣a (1)
∣∣ ∣∣a
11
2
(n)+a(1) 11 (n) (1)
∣∣ ∣∣a +
12 (n)+a(1) 21 (n) (1)
∣∣+
+
21 (n)+a(1) 12 (n) ∣ ∣ ∣ ∣ ∣
∣
∣a (1)
∣∣ ∣∣a
22 (n)+a(1) 22 (n) (2)
∣∣ ∣∣a +
11 (n)+a(2) 11 (n) (2)
∣∣+
+
12 (n)+a(2) 21 (n) ∣ ∣ ∣] ∣
∣a (2)
∣∣ ∣∣a
21 (n)+a(2) 12 (n) (2)
∣∣ +
22 (n)+a(2) 22 (n) =
1 [∣ ∣e 2 − 1 + e 2 − 1 ∣ ∣ + 2|(e − 1) f +(e + 1) f | + f 2 + f 2∣ ∣ +
2
+ ∣ ∣g 2 + g 2∣ ∣ + 2|g(h + 1)+g(h − 1)| + ∣ ∣h 2 − 1 + h 2 − 1 ∣ ] =
1 [ ∣
2 e 2 − 1 ∣ ∣+ 4|ef| + 2 ∣ ∣f 2∣ ∣ + 2 ∣ ∣g 2∣ ∣ + 4|gh|+ 2 ∣ ∣h 2 − 1 ∣ ] =
2
∣ e 2 − 1 ∣ + 2|ef| + f 2 + g 2 + 2|gh| + 2 ∣ h 2 − 1 ∣ For any a,b,c,d we have
Adding 1 we obtain
b(p)= ∣ ∣ e 2 − 1 ∣ ∣ + 2|ef| + f 2 + g 2 + 2|gh| + 2 ∣ ∣ h 2 − 1 ∣ ∣
b(p)+1 = ∣ e 2 − 1 ∣ + 2|ef| + f 2 + g 2 + 2|gh|+ 2 ∣ h 2 − 1 ∣ + 1
(√
) n
‖A n ‖≤ |e 2 − 1|+ 2|ef| + f 2 + g 2 + 2|gh| + 2|h 2 − 1|+ 1
( √ ) n
‖A n 2
‖≤ b(p)+1
Assuming a,d > 1,b,c > 0 our estimate can be simplified as follows
b(p)=(e + f ) 2 +(g + h) 2 − 2
b(p)+1 =(e + f ) 2 +(g + h) 2 − 1
(√
‖A n ‖≤ (e + f ) 2 +(g + h) 2 − 1
( √ ) n
‖A n 2
‖≤ b(p)+1
) n