OK, here's a programming/physics puzzle in, let's say Java (although it doesn't really matter - but for consistency, we will use Java).
I have an array of doubles where each element represents the successive derivates of an object's position. So:
d[0] // position
d[1] // velocity
d[2] // accelleration
etc ...
There will be at least 1 and potentially any number of elements (but in practice, probably not more than 5). All elements beyond the end of the array are assumed to be zero.
We also have a double t for time.
So if d.length were 3 then very specifically:
d[0] += d[1] * t + d[2] * t * t / 2
d[1] += d[2] * t
But how can we write an iterative solution (no recursion!) that will update all of the elements of the array for time t for any length of a?
Votes should be awarded for efficiency and conciseness.
EDIT: The laws of motion are on the physics pages of this site, but basically each term has the form:
power(t,n) / factorial(n)
So the next term of the first equation would be d[3] * t*t*t / 6
If you are not familiar with these equations, please don't ask me to explain.
A test case:
final static double[] D = { 5.0, 4.0, 3.0, 2.0, 1.0 };
double t = 10.0
should give:
[945.0, 300.66666666666663, 73.0, 12.0, 1.0]
Second element may vary due to precision errors.
I think something like this works.
int length = d.length;
for (int i = 0; i < length - 1; i++) {
double temp = 0;
for (int j = length - 1; j > i; j--)
temp = (temp + d[j]) * t / (j - i);
d[i] += temp;
}