package interval_base

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Usual arithmetic operators

interval_base
- Interval
  - DIRECTED
  - High
  - I
    
    U
  - Low
  - T

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Interval operations. Locally open this module — using e.g. I.(...) — to redefine classical arithmetic operators for interval arithmetic.

include T with type number = float and type t = t

type number = float

Numbers type on which intervals are defined.

type t = t

The type of intervals.

val zero : t

Neutral element for addition.

val one : t

Neutral element for multiplication.

val pi : t

π with bounds properly rounded.

val two_pi : t

2π with bounds properly rounded.

val half_pi : t

π/2 with bounds properly rounded.

val e : t

e (Euler's constant) with bounds properly rounded.

val entire : t

The entire set of numbers.

since 1.5

val v : number -> number -> t

v a b returns the interval [a, b]. BEWARE that, unless you take care, if you use v a b with literal values for a and/or b, the resulting interval may not contain these values because the compiler will round them to binary numbers before passing them to v.

raises Invalid_argument
if the interval [a, b] is equal to [-∞,-∞] or [+∞,+∞] or one of the bounds is NaN.

val low : t -> number

low t returns the lower bound of the interval.

val high : t -> number

high t returns the higher bound of the interval.

val of_int : int -> t

Returns the interval containing the conversion of an integer to the number type.

val to_string : ?fmt:(number -> 'b, 'a, 'b) format -> t -> string

to_string i return a string representation of the interval i.

parameter fmt
is the format used to print the two bounds of i. Default: "%g" for float numbers.

val pr : out_channel -> t -> unit

Print the interval to the channel. To be used with Printf format "%a".

val pp : Format.formatter -> t -> unit

Print the interval to the formatter. To be used with Format format "%a".

val fmt : (number -> 'b, 'a, 'b) format -> (t -> 'c, 'd, 'e, 'c) format4

fmt number_fmt returns a format to print intervals where each component is printed with number_fmt.

Example: Printf.printf ("%s = " ^^ fmt "%.10f" ^^ "\n") name i.

Boolean functions

val compare_f : t -> number -> int

compare_f a x returns

1 if high(a) < x,
0 if low(a) ≤ x ≤ high(a), i.e., if x ∈ a, and
-1 if x < low(a).

val is_bounded : t -> bool

is_bounded x says whether the interval is bounded, i.e., -∞ < low(x) and high(x) < ∞.

since 1.5

val is_entire : t -> bool

is_entire x says whether x is the entire interval.

since 1.5

val equal : t -> t -> bool

equal a b says whether the two intervals are the same.

since 1.5

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

Synonym for equal.

since 1.5

val subset : t -> t -> bool

subset x y returns true iff x ⊆ y.

since 1.5

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

x <= y says whether x is weakly less than y i.e., ∀ξ ∈ x, ∃η ∈ y, ξ ≤ η and ∀η ∈ y, ∃ξ ∈ x, ξ ≤ η.

since 1.5

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

x >= y says whether x is weakly greater than y i.e., ∀ξ ∈ x, ∃η ∈ y, ξ ≥ η and ∀η ∈ y, ∃ξ ∈ x, ξ ≥ η.

since 1.5

val precedes : t -> t -> bool

precedes x y returns true iff x is to the left but may touch y.

since 1.5

val interior : t -> t -> bool

interior x y returns true if x is interior to y in the topological sense. For example interior entire entire is true.

since 1.5

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

x < y says whether x is strictly weakly less than y i.e., ∀ξ ∈ x, ∃η ∈ y, ξ < η and ∀η ∈ y, ∃ξ ∈ x, ξ < η.

since 1.5

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

x > y iff y < x.

since 1.5

val strict_precedes : t -> t -> bool

strict_precedes x y returns true iff x is to the left and does not touch y.

since 1.5

val disjoint : t -> t -> bool

disjoint x y returns true iff x ∩ y = ∅.

since 1.5

Operations

val size : t -> t

size a returns an interval containing the true length of the interval high a - low a.

val size_high : t -> number

size_high a returns the length of the interval high a - low a rounded up.

val size_low : t -> number

size_low a returns the length of the interval high a - low a rounded down.

val sgn : t -> t

sgn a returns the sign of each bound, e.g., for floats [float (compare (low a) 0.), float (compare (high a) 0.)].

val truncate : t -> t

truncate a returns the integer interval containing a, that is [floor(low a), ceil(high a)].

val abs : t -> t

abs a returns the absolute value of the interval, that is

a if low a ≥ 0.,
~- a if high a ≤ 0., and
[0, max (- low a) (high a)] otherwise.

val hull : t -> t -> t

hull a b returns the smallest interval containing a and b, that is [min (low a) (low b), max (high a) (high b)].

val inter_exn : t -> t -> t

inter_exn x y returns the intersection of x and y.

raises Domain_error
if the intersection is empty.

since 1.5

val inter : t -> t -> t option

inter_exn x y returns Some z where z is the intersection of x and y if it is not empty and None if the intersection is empty.

since 1.5

val max : t -> t -> t

max a b returns the "maximum" of the intervals a and b, that is [max (low a) (low b), max (high a) (high b)].

val min : t -> t -> t

min a b returns the "minimum" of the intervals a and b, that is [min (low a) (low b), min (high a) (high b)].

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

a + b returns [low a +. low b, high a +. high b] properly rounded.

val (+.) : t -> number -> t

a +. x returns [low a +. x, high a +. x] properly rounded.

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

x +: a returns [a +. low a, x +. high a] properly rounded.

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

a - b returns [low a -. high b, high a -. low b] properly rounded.

val (-.) : t -> number -> t

a -. x returns [low a -. x, high a -. x] properly rounded.

val (-:) : number -> t -> t

x -: a returns [x -. high a, x -. low a] properly rounded.

val (~-) : t -> t

~- a is the unary negation, it returns [-high a, -low a].

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

a * b multiplies a by b according to interval arithmetic and returns the proper result. If a=zero or b=zero then zero is returned.

val (*.) : number -> t -> t

x *. a multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero is returned.

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

a *. x multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero is returned.

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

a / b divides the first interval by the second according to interval arithmetic and returns the proper result. Raise Interval.Division_by_zero if b=zero.

val (/.) : t -> number -> t

a /. x divides a by x according to interval arithmetic and returns the proper result. Raise Interval.Division_by_zero if x=0.0.

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

x /: a divides x by a according to interval arithmetic and returns the result. Raise Interval.Division_by_zero if a=zero.

val inv : t -> t

inv a returns 1. /: a but is more efficient. Raise Interval.Division_by_zero if a=zero.

type 'a one_or_two =

| One of 'a
| Two of 'a * 'a

val invx : t -> t one_or_two

invx a is the extended division. When 0 ∉ a, the result is One(inv a). If 0 ∈ a, then the two natural intervals (properly rounded) Two([-∞, 1/(low a)], [1/(high a), +∞]) are returned. Raise Interval.Division_by_zero if a=zero.

val cancelminus : t -> t -> t

cancelminus x y returns the tightest interval z such that x ⊆ z + y. If no such z exists, it returns entire.

since 1.5

val cancelplus : t -> t -> t

cancelplus x y returns the tightest interval z such that x ⊆ z - y. If no such z exists, it returns entire.

since 1.5

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

a**n returns interval a raised to nth power according to interval arithmetic. If n=0 then one is returned.

raises Domain_error
if n < 0 and a=zero.

Usual arithmetic operators

module U : sig ... end

Module undoing the redeclaration of usual infix operators +, +., etc. in case it is needed locally, while this module is open.