Is there any fast method of matrix exponentiation?
You could factor the matrix into eigenvalues and eigenvectors. Then you get
M = V * D * V^-1
Where V is the eigenvector matrix and D is a diagonal matrix. To raise this to the Nth power, you get something like:
M^n = (V * D * V^-1) * (V * D * V^-1) * ... * (V * D * V^-1)
= V * D^n * V^-1
Because all the V and V^-1 terms cancel.
Since D is diagonal, you just have to raise a bunch of (real) numbers to the nth power, rather than full matrices. You can do that in logarithmic time in n.
Calculating eigenvalues and eigenvectors is r^3 (where r is the number of rows/columns of M). Depending on the relative sizes of r and n, this might be faster or not.
It's quite simple to use Euler fast power algorith. Use next algorith.
#define SIZE 10
//It's simple E matrix
// 1 0 ... 0
// 0 1 ... 0
// ....
// 0 0 ... 1
void one(long a[SIZE][SIZE])
{
for (int i = 0; i < SIZE; i++)
for (int j = 0; j < SIZE; j++)
a[i][j] = (i == j);
}
//Multiply matrix a to matrix b and print result into a
void mul(long a[SIZE][SIZE], long b[SIZE][SIZE])
{
long res[SIZE][SIZE] = {{0}};
for (int i = 0; i < SIZE; i++)
for (int j = 0; j < SIZE; j++)
for (int k = 0; k < SIZE; k++)
{
res[i][j] += a[i][k] * b[k][j];
}
for (int i = 0; i < SIZE; i++)
for (int j = 0; j < SIZE; j++)
a[i][j] = res[i][j];
}
//Caluclate a^n and print result into matrix res
void pow(long a[SIZE][SIZE], long n, long res[SIZE][SIZE])
{
one(res);
while (n > 0) {
if (n % 2 == 0)
{
mul(a, a);
n /= 2;
}
else {
mul(res, a);
n--;
}
}
}
Below please find equivalent for numbers:
long power(long num, long pow)
{
if (pow == 0) return 1;
if (pow % 2 == 0)
return power(num*num, pow / 2);
else
return power(num, pow - 1) * num;
}