
Updating the current discrete-time state vector [x
1,A
(i), x
2,A
(i),..., x
n,A
(i)] to the new vector
[x
1,A
(i þ1), x
2,A
(i þ1),..., x
n,A
(i þ1)] with RK-m integration consists of determining, in the
proper sequence, the derivatives k
j,p
, j ¼1, 2, . . . , m and p ¼1, 2, 3, . . . , n. By the proper sequence,
we mean k
1,1
, k
1,2
,...,k
1,n
, followed by k
2,1
, k
2,2
,...,k
2,n
up through by k
m,1
, k
m,2
,...,k
m,n
.
To illustrate, suppose we are dealing with a third-order (n ¼3) system and choose to implement a
fourth-order (m ¼4) RK-4 integrator to update the discr ete-time state. The three derivative functions
are each calculated four times in the following order:
k
1,1 ...