zarith
Legend:
Library
Module
Module type
Parameter
Class
Class type

Rationals.

This modules builds arbitrary precision rationals on top of arbitrary integers from module Z.

This file is part of the Zarith library http://forge.ocamlcore.org/projects/zarith . It is distributed under LGPL 2 licensing, with static linking exception. See the LICENSE file included in the distribution.

Copyright (c) 2010-2011 Antoine Miné, Abstraction project. Abstraction is part of the LIENS (Laboratoire d'Informatique de l'ENS), a joint laboratory by: CNRS (Centre national de la recherche scientifique, France), ENS (École normale supérieure, Paris, France), INRIA Rocquencourt (Institut national de recherche en informatique, France).

## Types

`type t = {`
1. `num : Z.t;`
(*

Numerator.

*)
2. `den : Z.t;`
(*

Denominator, >= 0

*)
`}`

A rational is represented as a pair numerator/denominator, reduced to have a non-negative denominator and no common factor. This form is canonical (enabling polymorphic equality and hashing). The representation allows three special numbers: `inf` (1/0), `-inf` (-1/0) and `undef` (0/0).

## Construction

`val make : Z.t -> Z.t -> t`

`make num den` constructs a new rational equal to `num`/`den`. It takes care of putting the rational in canonical form.

`val zero : t`
`val one : t`
`val minus_one : t`

0, 1, -1.

`val inf : t`

1/0.

`val minus_inf : t`

-1/0.

`val undef : t`

0/0.

`val of_bigint : Z.t -> t`
`val of_int : int -> t`
`val of_int32 : int32 -> t`
`val of_int64 : int64 -> t`
`val of_nativeint : nativeint -> t`

Conversions from various integer types.

`val of_ints : int -> int -> t`

Conversion from an `int` numerator and an `int` denominator.

`val of_float : float -> t`

Conversion from a `float`. The conversion is exact, and maps NaN to `undef`.

`val of_string : string -> t`

Converts a string to a rational. Plain integers, `/` separated integer ratios (with optional sign), decimal point and scientific notations are understood. Additionally, the special `inf`, `-inf`, and `undef` are recognized (they can also be typeset respectively as `1/0`, `-1/0`, `0/0`).

## Inspection

`val num : t -> Z.t`

Get the numerator.

`val den : t -> Z.t`

Get the denominator.

## Testing

`type kind = `
1. `| ZERO`
(*

0

*)
2. `| INF`
(*

infinity, i.e. 1/0

*)
3. `| MINF`
(*

minus infinity, i.e. -1/0

*)
4. `| UNDEF`
(*

undefined, i.e., 0/0

*)
5. `| NZERO`
(*

well-defined, non-infinity, non-zero number

*)

Rationals can be categorized into different kinds, depending mainly on whether the numerator and/or denominator is null.

`val classify : t -> kind`

Determines the kind of a rational.

`val is_real : t -> bool`

Whether the argument is non-infinity and non-undefined.

`val sign : t -> int`

Returns 1 if the argument is positive (including inf), -1 if it is negative (including -inf), and 0 if it is null or undefined.

`val compare : t -> t -> int`

`compare x y` compares `x` to `y` and returns 1 if `x` is strictly greater that `y`, -1 if it is strictly smaller, and 0 if they are equal. This is a total ordering. Infinities are ordered in the natural way, while undefined is considered the smallest of all: undef = undef < -inf <= -inf < x < inf <= inf. This is consistent with OCaml's handling of floating-point infinities and NaN.

OCaml's polymorphic comparison will NOT return a result consistent with the ordering of rationals.

`val equal : t -> t -> bool`

Equality testing. Unlike `compare`, this follows IEEE semantics: `undef` <> `undef`.

`val min : t -> t -> t`

Returns the smallest of its arguments.

`val max : t -> t -> t`

Returns the largest of its arguments.

`val leq : t -> t -> bool`

Less than or equal. `leq undef undef` resturns false.

`val geq : t -> t -> bool`

Greater than or equal. `leq undef undef` resturns false.

`val lt : t -> t -> bool`

Less than (not equal).

`val gt : t -> t -> bool`

Greater than (not equal).

## Conversions

`val to_bigint : t -> Z.t`
`val to_int : t -> int`
`val to_int32 : t -> int32`
`val to_int64 : t -> int64`
`val to_nativeint : t -> nativeint`

Convert to integer by truncation. Raises a `Divide_by_zero` if the argument is an infinity or undefined. Raises a `Z.Overflow` if the result does not fit in the destination type.

`val to_string : t -> string`

Converts to human-readable, base-10, `/`-separated rational.

`val to_float : t -> float`

Converts to a floating-point number, using the current floating-point rounding mode. With the default rounding mode, the result is the floating-point number closest to the given rational; ties break to even mantissa.

## Arithmetic operations

In all operations, the result is `undef` if one argument is `undef`. Other operations can return `undef`: such as `inf`-`inf`, `inf`*0, 0/0.

`val neg : t -> t`

Negation.

`val abs : t -> t`

Absolute value.

`val add : t -> t -> t`

`val sub : t -> t -> t`

Subtraction. We have `sub x y` = `add x (neg y)`.

`val mul : t -> t -> t`

Multiplication.

`val inv : t -> t`

Inverse. Note that `inv 0` is defined, and equals `inf`.

`val div : t -> t -> t`

Division. We have `div x y` = `mul x (inv y)`, and `inv x` = `div one x`.

`val mul_2exp : t -> int -> t`

`mul_2exp x n` multiplies `x` by 2 to the power of `n`.

`val div_2exp : t -> int -> t`

`div_2exp x n` divides `x` by 2 to the power of `n`.

## Printing

`val print : t -> unit`

Prints the argument on the standard output.

`val output : out_channel -> t -> unit`

Prints the argument on the specified channel. Also intended to be used as `%a` format printer in `Printf.printf`.

`val sprint : unit -> t -> string`

To be used as `%a` format printer in `Printf.sprintf`.

`val bprint : Buffer.t -> t -> unit`

To be used as `%a` format printer in `Printf.bprintf`.

`val pp_print : Format.formatter -> t -> unit`

Prints the argument on the specified formatter. Also intended to be used as `%a` format printer in `Format.printf`.

## Prefix and infix operators

Classic prefix and infix `int` operators are redefined on `t`.

`val (~-) : t -> t`

Negation `neg`.

`val (~+) : t -> t`

Identity.

`val (+) : t -> t -> t`

Addition `add`.

`val (-) : t -> t -> t`

Subtraction `sub`.

`val (*) : t -> t -> t`

Multiplication `mul`.

`val (/) : t -> t -> t`

Division `div`.

`val (lsl) : t -> int -> t`

Multiplication by a power of two `mul_2exp`.

`val (asr) : t -> int -> t`

Division by a power of two `shift_right`.

`val (~\$) : int -> t`

Conversion from `int`.

`val (//) : int -> int -> t`

Creates a rational from two `int`s.

`val (~\$\$) : Z.t -> t`

Conversion from `Z.t`.

`val (///) : Z.t -> Z.t -> t`

Creates a rational from two `Z.t`.

`val (=) : t -> t -> bool`

Same as `equal`.

`val (<) : t -> t -> bool`

Same as `lt`.

`val (>) : t -> t -> bool`

Same as `gt`.

`val (<=) : t -> t -> bool`

Same as `leq`.

`val (>=) : t -> t -> bool`

Same as `geq`.

`val (<>) : t -> t -> bool`

`a <> b` is equivalent to `not (equal a b)`.