Calculating the nearest square continued fraction of a quadratic irrational

This program finds the nearest square continued fraction expansion of a quadratic irrational α = (u + t√d)/v, where d,t,u,v are integers, d >1, and nonsquare, t and v nonzero.
We first convert α to (P+√d)/Q, where Q divides d - P².
Our algorithm is based on papers by A.A. Krisnaswami Ayyangar who defined a reduced surd on page 27 as "the successor of the successor of a special surd".
We use a more direct equivalent version which allows us to detect an NSCF-reduced complete quotient directly.
Let (P + √d)/Q = ξ₀ be a standard surd (ie., Q divides d - P²) and gcd(P, Q, (d-P²)/Q)=1). Let a be the integer part of ξ₀. Then we can write

ξ₀ = a + Q'/(P + √d) or ξ₀ = a + 1- Q"/(P" + √d),

where (P' + √d)/Q' and (P" + √d)/Q" are also standard surds.
We choose a₀ to be a or a + 1, according as (i) |Q'| < |Q"| or |Q'| > |Q"|, if |Q'| ≠ |Q"| ; or (ii) if |Q'| = |Q"|, according as Q < 0 or Q > 0.

(This differs slightly in case (ii) from Solving the Pell equation, page 407, H.C. Williams, Proc. Millennial Conference on Number Theory, A.K. Peters, 2002, pp. 397-435.)

(We see that |P'² - d|/|P"² - d|=|Q'|/|Q"|. Hence |Q'| is ≤ or > |Q"| according as |P'² - d| is ≤ or > |P"² - d|.
So choosing the lesser of |Q'| and |Q"| is equivalent to choosing the nearest of the two squares P'² and P"² to d.)

Then (P + √d)/Q = ξ₀ = a₀ + ε₁/ ξ₁ and where |ε₁| = 1 and ξ₁ = (P₁ + √d)/ Q₁ > 1.

We proceed similarly with ξ₁ and so on. Then we get complete quotients ξ_n, where

ξ_n = a_n + ε_{n + 1}/ ξ_{n + 1} and ξ₀ = a₀ + ε₁/a₁ + ε₂/a₂ + · · ·

We have identities

P_n+1 + P_n = a_nQ_n,
P_n+1² + ε_n+1Q_nQ_n+1 = d.

The convergents A_n/B_n are defined by A_-2 = 0, A_-1 = 1, B_-2 = 1, B_-1 = 0 and for k ≥ -1,

A_k+1 = a_k+1A_k + ε_k+1A_k-1,
B_k+1 = a_k+1B_k + ε_k+1B_k-1.

We print the partial numerators (Teilzähler ) ε_n, the partial denominators (Teilnenner ) a_n, the convergents and complete quotients, with the period elements in bold font.

Last modified 29th July 2010
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