Abstract

An O(log n) algorithm for computing the nth convergents of periodic continued fractions is presented. It can be applied to solve the second-order linear recurrences with constant coeffients very efficiently. We also use it to approximate the quadratic numbers in O(log m) time for a result with m-digit precision. Our method can be thought as a generalization of the matrix-vector approach for computing the Fibonacci numbers. It is easy to implement because there are only some matrix multiplications and a division operation involved.

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