Matrix Doubling
时间: 2023-10-09 17:10:45 浏览: 36
Matrix doubling is an algorithm used to quickly calculate the power of a matrix to a large exponent. It works by repeatedly squaring the matrix until the desired exponent is reached. The algorithm has a time complexity of O(log n), where n is the exponent.
Here is how the algorithm works:
1. Start with the original matrix A and the desired exponent k.
2. If k is 1, return the original matrix A.
3. If k is even, square the matrix A and divide k by 2.
4. If k is odd, multiply the matrix A by itself (A*A) and subtract 1 from k.
5. Repeat steps 3-4 until k is 1.
6. Return the resulting matrix.
For example, let's say we want to calculate A^8, where A is a 2x2 matrix:
1. Start with A and k=8.
2. k is even, so square A and divide k by 2:
A^2 = [[a11*a11 + a12*a21, a11*a12 + a12*a22],
[a21*a11 + a22*a21, a21*a12 + a22*a22]]
k = 4
3. k is even, so square A^2 and divide k by 2:
(A^2)^2 = A^4 = [[a11*a11*a11*a11 + 2*a11*a12*a21*a11 + a12*a21*a12*a21 + a11*a11*a12*a22 + a11*a12*a22*a21 + a12*a22*a12*a22, a11*a11*a12*a12 + 2*a11*a12*a22*a12 + a12*a21*a12*a12 + a11*a21*a12*a22 + a21*a21*a12*a12 + a12*a22*a22*a12],
[a21*a11*a11*a11 + 2*a21*a22*a11*a11 + a22*a21*a21*a11 + a21*a11*a12*a22 + a22*a12*a21*a21 + a22*a22*a12*a21, a21*a11*a12*a12 + 2*a21*a22*a12*a12 + a22*a21*a21*a12 + a21*a21*a12*a22 + a21*a22*a22*a12 + a22*a22*a22*a22]]
k = 2
4. k is even, so square A^4 and divide k by 2:
(A^4)^2 = A^8 = [[a11*a11*a11*a11*a11*a11*a11*a11 + 4*a11*a11*a11*a11*a12*a21*a11*a11 + 6*a11*a11*a12*a21*a12*a21*a11*a11 + 4*a11*a12*a21*a11*a11*a12*a21*a21 + a12*a21*a12*a21*a12*a21*a12*a21 + 4*a11*a11*a11*a11*a11*a12*a22*a21 + 6*a11*a11*a12*a22*a12*a22*a11*a11 + 12*a11*a12*a22*a11*a11*a12*a22 + 6*a11*a12*a22*a12*a21*a12*a22 + 4*a12*a22*a12*a22*a12*a22*a12*a22,
a11*a11*a11*a11*a12*a12*a12*a12 + 4*a11*a11*a12*a12*a22*a12*a12*a11 + 6*a12*a11*a21*a21*a12*a12*a12*a11 + 4*a11*a21*a12*a12*a12*a22*a21*a21 + a21*a21*a12*a12*a22*a22*a12*a12 + 4*a11*a11*a11*a21*a12*a22*a22*a12 + 6*a11*a21*a12*a22*a22*a22*a11*a12 + 12*a21*a21*a12*a12*a12*a22*a12 + 6*a21*a12*a22*a22*a22*a12*a22 + 4*a21*a22*a22*a12*a22*a22*a22*a12]]
k = 1
5. Return A^8.
By using matrix doubling, we were able to calculate A^8 in just three matrix multiplications, instead of the eight multiplications required by the straightforward method of multiplying A by itself eight times.