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q.mli
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(**
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).
*)
(** {1 Types} *)
type t = {
num: Z.t; (** Numerator. *)
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).
*)
(** {1 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]). *)
(** {1 Inspection} *)
val num: t -> Z.t
(** Get the numerator. *)
val den: t -> Z.t
(** Get the denominator. *)
(** {1 Testing} *)
type kind =
| ZERO (** 0 *)
| INF (** infinity, i.e. 1/0 *)
| MINF (** minus infinity, i.e. -1/0 *)
| UNDEF (** undefined, i.e., 0/0 *)
| 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). *)
(** {1 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. *)
(** {1 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
(** Addition. *)
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]. *)
(** {1 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].
*)
(** {1 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].
@since 1.8 *)
val (<): t -> t -> bool
(** Same as [lt].
@since 1.8 *)
val (>): t -> t -> bool
(** Same as [gt].
@since 1.8 *)
val (<=): t -> t -> bool
(** Same as [leq].
@since 1.8 *)
val (>=): t -> t -> bool
(** Same as [geq].
@since 1.8 *)
val (<>): t -> t -> bool
(** [a <> b] is equivalent to [not (equal a b)].
@since 1.8 *)