Node:Lucas Numbers Algorithm, Previous:Fibonacci Numbers Algorithm, Up:Other Algorithms
mpz_lucnum2_ui derives a pair of Lucas numbers from a pair of Fibonacci
numbers with the following simple formulas.
L[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1]
mpz_lucnum_ui is only interested in L[n], and some work can be
saved. Trailing zero bits on n can be handled with a single square
each.
L[2k] = L[k]^2 - 2*(-1)^k
And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
numbers, similar to what mpz_fib_ui does.
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k