sml-interval

Zero-dependency Standard ML library for interval arithmetic with conservative outward rounding.

API

signature INTERVAL =
sig
  type ivl = { lo : real, hi : real }
  exception Divide

  val make      : real * real -> ivl
  val singleton : real -> ivl
  val width     : ivl -> real
  val mid       : ivl -> real
  val contains  : ivl -> real -> bool

  val hull      : ivl -> ivl -> ivl
  val intersect : ivl -> ivl -> ivl option

  val add    : ivl -> ivl -> ivl
  val sub    : ivl -> ivl -> ivl
  val mul    : ivl -> ivl -> ivl
  val divide : ivl -> ivl -> ivl   (* raises Divide if rhs contains 0 *)
  val neg    : ivl -> ivl
  val abs_   : ivl -> ivl
  val sqr    : ivl -> ivl
  val sqrt_  : ivl -> ivl

  val exp_   : ivl -> ivl
  val ln_    : ivl -> ivl
  val sin_   : ivl -> ivl
  val cos_   : ivl -> ivl

  val intervalNewton : {f: ivl -> ivl, fPrime: ivl -> ivl, x0: ivl, tol: real}
    -> ivl list
end

Worked example

(* Bound sqrt(2) using interval Newton *)
fun f  x = Interval.sub (Interval.sqr x) (Interval.singleton 2.0)
fun fp x = Interval.mul (Interval.singleton 2.0) x
val roots = Interval.intervalNewton
  {f=f, fPrime=fp, x0=Interval.make(1.0, 2.0), tol=1e~10}
(* roots contains an interval [a,b] with |a-b| < 1e-10 enclosing sqrt(2) *)

(* Range of sin on [0, pi] *)
val range = Interval.sin_ (Interval.make (0.0, Math.pi))
(* range = [-eps, 1+eps] where eps is a tiny outward-rounding bound *)

Example

make example builds and runs examples/demo.sml, which walks through basic interval arithmetic, set operations, elementary functions, and the interval-Newton method bounding sqrt(2) (output is byte-identical under MLton and Poly/ML):

Basic arithmetic:
  a = [1.000000, 2.000000], b = [3.000000, 4.000000]
  a + b = [4.000000, 6.000000]
  a * b = [3.000000, 8.000000]
  width a = 1.000000, mid a = 1.500000

Set operations:
  hull([1,3],[2,5])      = [1.000000, 5.000000]
  intersect([1,4],[3,6]) = [3.000000, 4.000000]

Elementary functions:
  sqrt([4,9]) = [2.000000, 3.000000]
  exp([0,1])  = [1.000000, 2.718282]

Interval Newton method for sqrt(2):
  roots found: 1
    [1.414214, 1.414214] width=0.000000

Scope and limitations

• Outward rounding uses a small relative nudge (≈1e-14) rather than true directed rounding, so enclosures are conservative but not guaranteed tight.
• sin_ and cos_ check for critical points within the interval; for wide intervals (>2π) they return [-1, 1].
• The interval Newton method is a simple fixed-point iteration; convergence is not guaranteed for all inputs.
• Does not support extended intervals, interval matrices, or multi-dimensional arithmetic.

Build and test

Requires MLton and Poly/ML in PATH.

make all-tests