Calkin-Wilf

This package implements the the Calkin-Wilf bijection between integers and rationals, as well as some convenience functions for working with rationals.

The package exports the following functions:

• function fusc(n: number): number Produces elements of Stern's Diatomic Series. This is done with a linear cache, so requests for large indices may end up consuming a lot of memory. The first request for a given index will run in linear time, after whcih requests for any lesser or equal indices will run in constant time. The first 16 elements are hard-coded to initialize the cache, and will always be retrieved in constant time.

• function Q(i: number): [number, number] Using the fusc function, produces the ith rational number in the Calkin-Wilf sequence as a [numerator, denominator] pair. Negative indices are valid, with Q(-i) = -Q(i).

• function F(i: number): [number, number] Using tree traversal, produces the ith rational number in the Calkin-Wilf sequence as a [numerator, denominator] pair. Negative indices are valid, with F(-i) = -F(i).

If you need to generate many sequential elements of the Calkin-Wilk sequence, Q will run in constant time per element, though it may eventually result in the cache consuming a lot of memory. Generating a single rational number at a high index will, however, require linear time. In contrast, F will generate any element of the sequence in logarithmic time in the size of the index. This is sub-optimal for generating long strings o sequential elements, but is asymptotically considerably faster for generating single elements at large indices, and does not cause the fusc cache to consume memory.

• function N(n: number, d: number): number Given a rational number n/d, return its integer position in the Calkin-Wilf sequence. Negative inputs are valid, with N(-n, d) = N(n, -d) = -N(n, d). This function runs in logarithmic time in the size of the returned index.

• function reduce(n: number, d: number): [number, number] Returns an ordered pair representing the rational number n/d in simplest form.

• function left(n: number, d: number): [number, number] Given a positive rational number n/d, returns its left child in the Calkin-Wilf tree.

• function right(n: number, d: number): [number, number] Given a positive rational number n/d, returns its right child in the Calkin-Wilf tree.

• function parent(n: number, d: number): [number, number] Given a positive rational number n/d, returns its parent in the Calkin-Wilf tree.

• function S(n: number, d: number): [number, number] Given a non-negative rational number n/d, returns its successor in the Calkin-Wilf sequence.

• function Sd(n: number, d: number): number Given a non-negative rational number n/d, return the denominator of its successor in the Calkin-Wilf sequence. The numerator of the successor of n/d is trivially equal to d, so this function may be useful to avoid allocating memory for returning the full pair.

• function P(n: number, d: number): [number, number] Given a positive rational number n/d, return its predecessor in the Calkin-Wilf sequence.

• function Pn(n: number, d: number): number Given a positive rational number n/d, returns the numerator of its predecessor in the Calkin-Wilf sequence. The denominator of the predecessor of n/d is trivially equal to n, so this function may be useful to avoid allocating memory for returning the full pair.

• function add(a: number, b: number, c: number, d: number): [number, number] Returns the sum a/b + c/d in simplest terms.

• function sub(a: number, b: number, c: number, d: number): [number, number] Returns the difference a/b - c/d in simplest terms.

• function mul(a: number, b: number, c: number, d: number): [number, number] Returns the product (a/b)(c/d) in simplest terms.

• function div(a: number, b: number, c: number, d: number): [number, number] Returns the quotient (a/b)/(c/d) in simplest terms.

• function cmp(a: number, b: number, c: number, d: number): -1|0|1 Compares the rational numbers a/b and c/d. This is more efficient than performing a full subtraction.