optimise my solution

Question

I just solve this but want know more efficient way to do matrix multiplication

M :
------
1 1 0
0 0 5
3 2 0 

f[n] = M^n

I have implemented using Exponentiation_by_squaring

Is there more efficient then this ?

Answer 1

Since N can be as large as 10^12 it should have been clear that iterating from 1 to N is not the desired solution. The key insight is that the recursion can be rewritten as V(i) = M * V(i-1), where

 V(i) = [RR(i), MM(i), PP(i)] (a column vector)

so

 V(0) = [ 3 1 0 ]

and

 M = | 1 0 3 |
     | 1 0 2 |
     | 0 5 0 |

Now V(N) = M^N * V(0)

We can calculate M^N in log(N) time by repeatedly squaring:

M^2 = M * M
M^4 = M^2 * M^2
...

Perform all calculations mod 100000006 to avoid accumulating large numbers.

To arrive at this solution, it helps to have a basic familiarity with linear algebra.