Midpoint criteria for solving Pell's equation using the nearest square continued fraction

We derive midpoint criteria for solving Pell's equation x-Dy = ±1, using the nearest square continued fraction expansion of √D. The period of the expansion is on average 70% that of the regular continued fraction. We derive similar criteria for the diophantine equation x - xy - (D-1) 4 y = ±1, where D ≡ 1 (mod 4). We also present some numerical results and conclude with a comparison of the computational performance of the regular, nearest square and nearest integer continued fraction algorithms