[R] Simplify formula for iterative programming

[Stefaan Lhermitte]

Dear R-ians,

I am looking for the simplification of a formula to improve the 
calculation speed of my program. Therefore I want to simplify the 
following formula:

H = Si (Sj ( sqrt [ (Ai - Aj)² + (Bi - Bj)²  ] ) )

where:
A, B = two vectors (with numerical data) of length n
sqrt = square root
Si = summation over i  (= 0 to n)
Sj = summation over j (= 0 to n)
Ai = element of A with index i
Aj = element of A with index j
Bi = element of B with index i
Bj = element of B with index j

n is not fixed, but it changes with every run for my program. Therefore 
for I am looking for a simplication of h in order to calculate it when 
my A and B get extendend by 1 element (n = n + 1).

I know a computional simplified formula exists for the standard 
deviation (sd) that is much easier in iterative programming. Therefore I 
wondered I anybody knew about analog simplifications to simplify H:

sd = sqrt [ ( Si (Xi - mean(X) )² ) /n  ]  -> simplified computation -> 
sqrt [ (n * Si( X² ) - ( Si( X ) )² )/ n² ]

This simplied formula is much easier in iterative programming, since I 
don't have to keep every element of X.
E.g.: I have a vector X[1:10]  and I already have caculated Si( X[1:10]² 
) (I will call this A) and Si( X ) (I will call this B).
When X gets extendend by 1 element (eg. X[11]) it easy fairly simple to 
calculate sd(X[1:11]) without having to reuse the elements of X[1:10].
I just have to calculate:

sd = sqrt [ (n * (A + X[11]²) - (A + X[11]²)² ) / n² ]

This is fairly easy in an iterative process, since before we continue 
with the next step we set:
A = (A + X[11]²)
B = (B + X[11])

Can anybody help me to do something comparable for H? Any other help to 
calculate H easily in an iterative process is also welcome!

Kind regards,
Stef