For those wondering: that fast 'half the exponent' multiplication algorithm is not optimal in the number of matrix multiplications. For example, x^15 can be done in 5 multiplications, while 'half the exponent requires 6. See http://en.wikipedia.org/wiki/Addition_chain_exponentiation:

    x3 = x * x * x
    x6 = x3 * x3
    x12 = x6 * x6
    x15 = x12 * x3

vs

    x2 = x * x
    x4 = x2 * x2
    x8 = x4 * x4
    x12 = x4 * x8
    x14 = x12 * x2
    x15 = x14 * x