Continued fraction convergent

Truncating a simple continued fraction gives a convergent; sufficiently accurate rational approximations with bounded denominator occur among these convergents.

Best approximation of the second kind

A rational

p / q

is a best approximation of the second kind to

α

when

∣ q α - p ∣

is no larger than

∣ b α - a ∣

for every competing rational

a / b

with

0 < b \leq q

. Every simple continued-fraction convergent has this property.

With

M (a) = (a 1 10)

M (a_{0}) \dots M (a_{n}) = (p_{n} q_{n} p_{n - 1} q_{n - 1}) .

p_{n} / q_{n} = [a_{0}, \dots, a_{n}]

, then appending a positive tail

x

gives

[a_{0}, \dots, a_{n}, x] = \frac{p _{n} x + p _{n - 1}}{q _{n} x + q _{n - 1}} .

Because each matrix

M (a) = (a 1 10)

is symmetric, reversing the partial quotients transposes their product. If the convergents of

[a_{0}, \dots, a_{n}]

are

p_{j} / q_{j}

, then the last two convergents of

[a_{n}, \dots, a_{0}]

are

\frac{p _{n}}{p _{n - 1}} and \frac{q _{n}}{q _{n - 1}} .

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