IntervalArithmetic.jl

https://juliaintervals.github.io/IntervalArithmetic.jl/stable/
Admin
IntervalArithmetic.jl is a Julia package for validated numerics that performs calculations using...
Tokens:43,146
Snippets:388
Trust Score:9.7
Update:1 month ago
Show doc for...
Tokens:
Raw
### interval method usage examples

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Demonstrates creating intervals with different types and formats.

```julia
julia> using IntervalArithmeticjulia> setdisplay(:full);julia> interval(1//1, π)Interval{Rational{Int64}}(1//1, 85563208//27235615, com, true)julia> interval(Rational{Int32}, 1//1, π)Interval{Rational{Int32}}(1//1, 85563208//27235615, com, true)julia> interval(1, π)Interval{Float64}(1.0, 3.1415926535897936, com, true)julia> interval(BigFloat, 1, π)Interval{BigFloat}(1.0, 3.141592653589793238462643383279502884197169399375105820974944592307816406286233, com, true)
```

--------------------------------

### IntervalArithmetic.Symbols.:.. Example

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/symbols

Example of using the `..` operator with rational and floating-point bounds. Requires `using IntervalArithmetic` and `using IntervalArithmetic.Symbols`, and `setdisplay(:full)` for full display.

```julia
julia> using IntervalArithmetic
julia> using IntervalArithmetic.Symbols
julia> setdisplay(:full);
julia> (1//1)..π
Interval{Rational{Int64}}(1//1, 85563208//27235615, com, true)
julia> 0.1..0.3
Interval{Float64}(0.1, 0.3, com, true)
```

--------------------------------

### fastpown usage examples

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Demonstrates the use of fastpown for interval exponentiation.

```julia
julia> using IntervalArithmeticjulia> setdisplay(:full);julia> fastpown(bareinterval(2, 3), 2)BareInterval{Float64}(4.0, 9.0)julia> fastpown(interval(-4, -2), 2)Interval{Float64}(4.0, 16.0, com, true)julia> fastpown(interval(-1, 1), -3)Interval{Float64}(-Inf, Inf, trv, true)
```

--------------------------------

### Install IntervalArithmetic.jl

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable

Use the Julia Pkg manager to add the IntervalArithmetic package to your environment.

```julia
julia> using Pkg # Julia v1.10 or abovejulia> Pkg.add("IntervalArithmetic")
```

--------------------------------

### IntervalArithmetic.Symbols.:± Example

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/symbols

Example of using the `±` operator with floating-point and rational midpoints. Requires `using IntervalArithmetic` and `using IntervalArithmetic.Symbols`, and `setdisplay(:full)` for full display.

```julia
julia> using IntervalArithmetic
julia> using IntervalArithmetic.Symbols
julia> setdisplay(:full);
julia> 0 ± π
Interval{Float64}(-3.1415926535897936, 3.1415926535897936, com, true)
julia> 0//1 ± π
Interval{Rational{Int64}}(-85563208//27235615, 85563208//27235615, com, true)
```

--------------------------------

### Using IntervalArithmetic.Symbols

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/symbols

Example of using the Symbols submodule to create intervals and perform comparisons with unicode operators. Requires `using IntervalArithmetic, IntervalArithmetic.Symbols`.

```julia
julia> using IntervalArithmetic, IntervalArithmetic.Symbols
julia> a = 0 .. 2
Interval{Float64}(0.0, 2.0, com, true)
julia> b = 1 ± 0.5
Interval{Float64}(0.5, 1.5, com, true)
julia> a ≛ a
true
julia> b ⊑ a
true
julia> b ⋤ a
true
julia> ∅ ⪽ ℝ
true
```

--------------------------------

### Common interval and sqrt with com decoration

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Creates a common interval and computes its square root. Both the input and output are common (com) because the interval is closed, bounded, non-empty, and the square root is continuous over [1/2, 3]. This example demonstrates the Schauder Fixed-Point Theorem: the computation proves the existence of a fixed point of the square root in [1/2, 3].

```julia
julia> x = interval(0.5, 3)
Interval{Float64}(0.5, 3.0, com, true)
julia> sqrt(x)
Interval{Float64}(0.7071067811865475, 1.7320508075688774, com, true)
```

--------------------------------

### Basic linear algebra operations with ArbMatrix and AcbMatrix

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/interfaces/arblib

Convert an interval matrix to ArbMatrix for real operations or AcbMatrix for eigenvalues. The example shows matrix multiplication, inversion, solving a linear system, and computing eigenvalues. Eigenvalues are only supported for AcbMatrix.

```julia
julia> using IntervalArithmetic, Arblib, LinearAlgebra
julia> A = interval.(BigFloat, [1 2; 3 4])
2×2 Matrix{Interval{BigFloat}}:
 Interval{BigFloat}(1.0, 1.0, com, true)  …  Interval{BigFloat}(2.0, 2.0, com, true)
 Interval{BigFloat}(3.0, 3.0, com, true)     Interval{BigFloat}(4.0, 4.0, com, true)

julia> B = ArbMatrix(A)
2×2 Arblib.ArbMatrix:
 1.0  2.0
 3.0  4.0

julia> B * B
2×2 Arblib.ArbMatrix:
  7.0  10.0
 15.0  22.0

julia> inv(B)
2×2 Arblib.ArbMatrix:
 [-2.00{…71 digits…}00 ± 1.44e-76]   [1.00{…71 digits…}00 ± 7.78e-77]
 [1.50{…72 digits…}00 ± 9.07e-77]  [-0.50{…72 digits…}00 ± 3.68e-77]

julia> B \ B
2×2 Arblib.ArbMatrix:
     [1.00{…71 digits…}00 ± 7.20e-77]  [+/- 6.91e-77]
 [+/- 3.24e-77]                            [1.00{…72 digits…}00 ± 5.19e-77]

julia> eigvals(AcbMatrix(B)) # Eigenvalues are only supported for AcbMatrix
2-element Arblib.AcbVector:
 [-0.372281323269014329925305734109464659110132228991396183849938735282950360729 +/- 5.90e-76] + [+/- 2.51e-76]im
 [5.372281323269014329925305734109464659110132228991396183849938735282950360729 +/- 8.90e-76] + [+/- 2.51e-76]im
```

--------------------------------

### Get upper bound with sup

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Returns the supremum of an interval.

```julia
sup(x)
```

--------------------------------

### Elementary functions with Interval{Float64} and Interval{BigFloat}

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/usage

Demonstrates applying elementary functions like sin and cosh to intervals, including switching precision for BigFloat calculations.

```julia
julia> X = interval(1)Interval{Float64}(1.0, 1.0, com, true)julia> sin(X)Interval{Float64}(0.8414709848078965, 0.8414709848078966, com, true)julia> cos(cosh(X))Interval{Float64}(0.027712143770207736, 0.02771214377020796, com, true)julia> setprecision(BigFloat, 53)53julia> Y = big(X)Interval{BigFloat}(1.0, 1.0, com, true)julia> sin(Y)Interval{BigFloat}(0.8414709848078965, 0.84147098480789662, com, true)julia> cos(cosh(Y))Interval{BigFloat}(0.027712143770207736, 0.027712143770207961, com, true)julia> setprecision(BigFloat, 128)128julia> sin(Y)Interval{BigFloat}(0.8414709848078965, 0.84147098480789662, com, true)
```

--------------------------------

### Construct bare and decorated intervals with `bareinterval` and `interval`

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Shows how to construct bare and decorated intervals with `bareinterval` and `interval`. The `setdisplay(:full)` call changes the display format; the output shows the default decoration `com` for `interval`.

```julia
julia> using IntervalArithmetic
julia> setdisplay(:full) # print the interval in full
Display options:
  - format: full
  - decorations: true (ignored)
  - NG flag: true (ignored)
  - significant digits: 6 (ignored)

julia> bareinterval(1, π) # `bareinterval(Float64, 1, π)`
BareInterval{Float64}(1.0, 3.1415926535897936)

julia> interval(1, π) # `interval(Float64, 1, π)`, interval decorated with `com` (common)
Interval{Float64}(1.0, 3.1415926535897936, com, true)
```

--------------------------------

### isguaranteed Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Tests whether the interval is not guaranteed to encompass all possible numerical errors. Examples show usage with different interval types.

```julia
isguaranteed(x::BareInterval)
isguaranteed(x::Interval)
isguaranteed(x::Complex{<:Interval})
```

```julia
julia> using IntervalArithmetic
julia> isguaranteed(bareinterval(1))
true
julia> isguaranteed(interval(1))
true
julia> isguaranteed(convert(Interval{Float64}, 1))
false
julia> isguaranteed(interval(1) + 0)
false
```

--------------------------------

### Configure power mode with IntervalArithmetic.configure

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/configuration

Shows how to switch the power implementation between :slow and :fast using IntervalArithmetic.configure, and the effect on the radius of x^3. The :slow mode gives a tighter result (7.105e-15) than the default :fast mode (1.421e-14).

```julia
julia> using IntervalArithmetic
julia> x = interval(π)
Interval{Float64}(3.141592653589793, 3.1415926535897936, com, true)
julia> IntervalArithmetic.configure(; power = :slow)
Configuration options:
  - bound type: Float64
  - flavor: set_based
  - interval rounding: correct
  - power mode: slow
  - matrix multiplication mode: fast
  - number of threads for `:fast` matrix multiplication mode: 2
julia> radius(x^3)
7.105427357601002e-15
julia> IntervalArithmetic.configure(; power = :fast) # default
Configuration options:
  - bound type: Float64
  - flavor: set_based
  - interval rounding: correct
  - power mode: fast
  - matrix multiplication mode: fast
  - number of threads for `:fast` matrix multiplication mode: 2
julia> radius(x^3)
1.4210854715202004e-14
```

--------------------------------

### ExactReal construction

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Demonstrates the construction of an ExactReal value using the exact function.

```julia
julia> exact(0.1)ExactReal{Float64}(0.1000000000000000055511151231257827021181583404541015625)
```

--------------------------------

### fastpown Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Fast power implementation for integer exponents.

```julia
fastpown(x, n)
```

--------------------------------

### Using FLINT special functions via low-level wrapper

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/interfaces/arblib

Shows how to use FLINT special functions not in SpecialFunctions.jl, such as dirichlet_lerch_phi, through the low-level wrapper. Requires using IntervalArithmetic and Arblib.

```julia
julia> using IntervalArithmetic, Arblib
julia> z = interval(BigFloat, 2 + 3im)
Interval{BigFloat}(2.0, 2.0, com, true) + im*Interval{BigFloat}(3.0, 3.0, com, true)
julia> s = interval(BigFloat, 2)
Interval{BigFloat}(2.0, 2.0, com, true)
julia> a = interval(BigFloat, 4)
Interval{BigFloat}(4.0, 4.0, com, true)
julia> interval(Arblib.dirichlet_lerch_phi!(Acb(), Acb(z), Acb(s), Acb(a)))
Interval{BigFloat}(-0.0002662647805245549465036569063000041468826679484039490012317600272567739198984341, -0.0002662647805245549465036569063000041468826679484039490012317600272567739198722642, com, true) + im*Interval{BigFloat}(0.03373697955708857151314153116051267066240962948450506837346469744178310287937996, 0.03373697955708857151314153116051267066240962948450506837346469744178310287944149, com, true)
```

--------------------------------

### Use ExactReal for precise interval arithmetic

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/guarantee

Shows how to use exact() and the @exact macro to ensure literal numbers are treated as exact values within interval computations.

```julia
julia> interval(1)Interval{Float64}(1.0, 1.0, com, true)julia> interval(1) + exact(1)Interval{Float64}(2.0, 2.0, com, true)julia> @exact 2interval(1) + 1 + exp(im * interval(1))Interval{Float64}(3.5403023058681393, 3.5403023058681398, com, true) + im*Interval{Float64}(0.8414709848078965, 0.8414709848078966, com, true)julia> @exact foo(x) = 2x + 1 + exp(im * x)foo (generic function with 1 method)julia> foo(1.)3.5403023058681398 + 0.8414709848078965imjulia> foo(interval(1))Interval{Float64}(3.5403023058681393, 3.5403023058681398, com, true) + im*Interval{Float64}(0.8414709848078965, 0.8414709848078966, com, true)
```

--------------------------------

### Using SpecialFunctions.jl with Arblib

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/interfaces/arblib

Shows how to apply SpecialFunctions.jl functions (besselj0, gamma, zeta) to an Arb interval and convert back to an Interval. Requires using IntervalArithmetic, Arblib, and SpecialFunctions.

```julia
julia> using IntervalArithmetic, Arblib, SpecialFunctions
julia> x = interval(BigFloat, 2)
Interval{BigFloat}(2.0, 2.0, com, true)
julia> interval(besselj0(Arb(x))) # Convert to Arb and then back
Interval{BigFloat}(0.2238907791412356680518274546499486258251544822186076031283497060108539577680058, 0.2238907791412356680518274546499486258251544822186076031283497060108539577680145, com, true)
julia> interval(gamma(Arb(x)))
Interval{BigFloat}(1.0, 1.0, com, true)
julia> interval(zeta(Arb(x)))
Interval{BigFloat}(1.644934066848226436472415166646025189218949901206798437735558229370007470403185, 1.644934066848226436472415166646025189218949901206798437735558229370007470403219, com, true)
```

--------------------------------

### Interval comparisons

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/usage

Shows how comparisons between intervals work, noting that ambiguous comparisons throw an InconclusiveBooleanOperation error.

```julia
julia> interval(1) < interval(2)truejulia> interval(1, 5) < interval(7, 9)truejulia> interval(1, 5) < interval(4.99, 9)ERROR: InconclusiveBooleanOperation: The operation `Interval{Float64}(1.0, 5.0, com, true) < Interval{Float64}(4.99, 9.0, com, true)` cannot be determined unambiguously. See the documentation for more information. See also `strictprecedes`.julia> interval(1.23) == interval(1.23)truejulia> interval(1.23) == interval(4.99, 9)falsejulia> interval(1.23) == interval(1.2, 1.3)ERROR: InconclusiveBooleanOperation: The operation `Interval{Float64}(1.23, 1.23, com, true) == Interval{Float64}(1.2, 1.3, com, true)` cannot be determined unambiguously. See the documentation for more information. See also `isequal_interval`.
```

--------------------------------

### Using @exact for exact interval arithmetic

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Demonstrates the use of the @exact macro to ensure interval arithmetic operations maintain exact representations where possible.

```julia
julia> using IntervalArithmeticjulia> setdisplay(:infsup);julia> f(x) = 1.2 * x + 0.1f (generic function with 1 method)julia> f(interval(1, 2))[1.29999, 2.5]_com_NGjulia> @exact g(x) = 1.2 * x + 0.1g (generic function with 1 method)julia> g(interval(1, 2))[1.29999, 2.5]_comjulia> g(1.4)1.78
```

--------------------------------

### Compare Float64 0.1 with rational 1//10

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Demonstrates that the floating-point value 0.1 is slightly greater than the rational 1//10 because 1/10 cannot be represented exactly in binary floating-point arithmetic at any precision.

```julia
julia> x > 1//10true
```

--------------------------------

### exact Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Wraps a number into an ExactReal.

```julia
exact(x)
```

--------------------------------

### Step-by-step evaluation of asin ∘ sqrt with trv

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Shows the individual steps of the composed function f = asin ∘ sqrt. sqrt(x) has trv because x contains negative values; sqrt(y) has com; asin(sqrt(y)) has trv because sqrt(y) contains values strictly greater than 1. This demonstrates where loose evaluation occurred in the composed function.

```julia
julia> sqrt(x) # `f(x)` has the decoration is `trv` since `x` contains negative values
Interval{Float64}(0.0, 1.7320508075688774, trv, true)
julia> sqrt(y) # the decoration is `com`
Interval{Float64}(0.0, 1.7320508075688774, com, true)
julia> asin(sqrt(y)) # `f(x)` has the decoration is `trv` since `sqrt(y)` contains values stricly greater than `1`
Interval{Float64}(0.0, 1.5707963267948968, trv, true)
```

--------------------------------

### Construct exact interval with I"0.1" string macro

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Uses the string macro I"..." to parse the decimal string and construct an interval that encloses the true real number 1/10. This is the recommended way to obtain an interval containing the exact real value.

```julia
julia> I"0.1"Interval{Float64}(0.09999999999999999, 0.1, com, true)
```

--------------------------------

### ..(a, b)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/symbols

Creates an interval [a, b] using bounds. This is semantically equivalent to interval(a, b).

```APIDOC
## ..(a, b)

### Description
Create the interval [a, b] according to the IEEE Standard 1788-2015. This is semantically equivalent to `interval(a, b)`.

### Method
`..`

### Parameters
- `a` - Lower bound
- `b` - Upper bound

### Request Example
julia
(1//1)..π
0.1..0.3


### Response
Returns an `Interval` object.

### Response Example
julia
Interval{Rational{Int64}}(1//1, 85563208//27235615, com, true)
Interval{Float64}(0.1, 0.3, com, true)

```

--------------------------------

### Converting between Interval and Arblib types

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/interfaces/arblib

Demonstrates conversion between Interval and Arb/Acb types. Shows default BigFloat behavior and explicit type specification.

```julia
julia> using IntervalArithmetic, Arblibjulia> x = interval(π)Interval{Float64}(3.141592653589793, 3.1415926535897936, com, true)julia> Arb(x)[3.141592653589793 +/- 5.61e-16]julia> y = Arb(π)[3.1415926535897932384626433832795028841971693993751058209749445923078164062862 +/- 1.93e-77]julia> interval(y) # Defaults to BigFloatInterval{BigFloat}(3.141592653589793238462643383279502884197169399375105820974944592307816406286163, 3.141592653589793238462643383279502884197169399375105820974944592307816406286233, com, true)julia> interval(Float64, y) # Other type can be explicitly specifiedInterval{Float64}(3.141592653589793, 3.1415926535897936, com, true)julia> z = complex(interval(2), interval(1 // 3))Interval{Float64}(2.0, 2.0, com, true) + im*Interval{Float64}(0.3333333333333333, 0.33333333333333337, com, true)julia> Acb(z)2.0 + [0.3333333333333333 +/- 7.04e-17]imjulia> w = Acb(2, 1 // 3)2.0 + [0.33333333333333333333333333333333333333333333333333333333333333333333333333333 +/- 4.78e-78]imjulia> interval(w)Interval{BigFloat}(2.0, 2.0, com, true) + im*Interval{BigFloat}(0.3333333333333333333333333333333333333333333333333333333333333333333333333333261, 0.3333333333333333333333333333333333333333333333333333333333333333333333333333348, com, true)julia> interval(Float64, w)Interval{Float64}(2.0, 2.0, com, true) + im*Interval{Float64}(0.3333333333333333, 0.33333333333333337, com, true)
```

--------------------------------

### interval([T,] a, b, d = com; format = :infsup)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Create an interval according to the IEEE Standard 1788-2015.

```APIDOC
## interval([T,] a, b, d = com; format = :infsup)

### Description
Create the interval [a,b] according to the IEEE Standard 1788-2015. The validity of the interval is checked by `is_valid_interval`: if `true` then an `Interval{T}` is constructed, otherwise an NaI (Not an Interval) is returned.

### Method
Function

### Endpoint
`interval([T,] a, b, d = com; format = :infsup)`

### Parameters
- **T** (Type) - Optional - The type of the interval endpoints.
- **a** (Real) - Required - Lower bound.
- **b** (Real) - Required - Upper bound.
- **d** (Decoration) - Optional - Decoration, default `com`.
- **format** (Symbol) - Optional - Format, default `:infsup`.

### Request Example
```julia
interval(1//1, π)
interval(Rational{Int32}, 1//1, π)
interval(1, π)
interval(BigFloat, 1, π)
```

### Response
#### Success Response
Returns an `Interval{T}`.

#### Response Example
```julia
Interval{Rational{Int64}}(1//1, 85563208//27235615, com, true)
Interval{Rational{Int32}}(1//1, 85563208//27235615, com, true)
Interval{Float64}(1.0, 3.1415926535897936, com, true)
Interval{BigFloat}(1.0, 3.141592653589793238462643383279502884197169399375105820974944592307816406286233, com, true)
```

### Notes
See also: `±`, `..`, `@I_str`.
```

--------------------------------

### Configure interval display modes with setdisplay

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/usage

Use setdisplay to globally control how intervals are represented, including options for infsup, full, and midpoint formats, as well as significant digits and decorations.

```julia
julia> using IntervalArithmeticjulia> a = interval(1.1, pi) # default displayInterval{Float64}(1.1, 3.1415926535897936, com, true)julia> setdisplay(; sigdigits = 10)Display options:  - format: full  - decorations: true (ignored)  - NG flag: true (ignored)  - significant digits: 10 (ignored)julia> aInterval{Float64}(1.1, 3.1415926535897936, com, true)julia> setdisplay(:full)Display options:  - format: full  - decorations: true (ignored)  - NG flag: true (ignored)  - significant digits: 10 (ignored)julia> aInterval{Float64}(1.1, 3.1415926535897936, com, true)julia> setdisplay(:midpoint)Display options:  - format: midpoint  - decorations: true  - NG flag: true  - significant digits: 10julia> a(2.120796327 ± 1.020796327)_comjulia> setdisplay(; sigdigits = 4)Display options:  - format: midpoint  - decorations: true  - NG flag: true  - significant digits: 4julia> a(2.121 ± 1.021)_comjulia> setdisplay(:infsup)Display options:  - format: infsup  - decorations: true  - NG flag: true  - significant digits: 4julia> a[1.1, 3.142]_com
```

--------------------------------

### interval method definition

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Creates an interval [a,b] according to IEEE Standard 1788-2015.

```julia
interval([T,] a, b, d = com; format = :infsup)
```

--------------------------------

### isweakless Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Tests whether inf(x) ≤ inf(y) and sup(x) ≤ sup(y).

```julia
isweakless(x, y)
```

--------------------------------

### Evaluate function over interval with IntervalArithmetic.jl

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/intro

Shows how to define a function and evaluate it over an interval using IntervalArithmetic.jl. The output shows the resulting interval and its decoration.

```julia
julia> using IntervalArithmetic
julia> f(x) = x^2 - 2
f (generic function with 1 method)
julia> x = interval(3, 4)
Interval{Float64}(3.0, 4.0, com, true)
julia> f(x)
Interval{Float64}(7.0, 14.0, com, false)
```

--------------------------------

### Using @interval macro for expression wrapping

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Shows how to use the @interval macro to walk through an expression and wrap function arguments with the atomic constructor, optionally specifying the bound type.

```julia
julia> using IntervalArithmeticjulia> setdisplay(:full);julia> @interval sin(1) # Float64 is the default bound typeInterval{Float64}(0.8414709848078965, 0.8414709848078966, com, true)julia> @interval Float32 sin(1)Interval{Float32}(0.84147096f0, 0.841471f0, com, true)julia> @interval sin(1) exp(1)Interval{Float64}(0.8414709848078965, 2.7182818284590455, com, true)julia> @interval Float64 sin(1) exp(1)Interval{Float64}(0.8414709848078965, 2.7182818284590455, com, true)
```

--------------------------------

### mince! Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

In-place version of the mince function.

```julia
mince!(v, x, n)
```

--------------------------------

### Compare `sqrt` on bare and decorated intervals

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Compares `sqrt` on a bare interval and a decorated interval. The bare interval silently discards the negative part, while the decorated interval returns a `trv` decoration, indicating the result is trivial.

```julia
julia> sqrt(bareinterval(-1, 1)) # `sqrt(bareinterval(0, 1))`
BareInterval{Float64}(0.0, 1.0)

julia> sqrt(interval(-1, 1)) # interval decorated with `trv` (trivial)
Interval{Float64}(0.0, 1.0, trv, true)
```

--------------------------------

### precedes Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Tests whether every element of x is less than or equal to every element of y.

```julia
precedes(x, y)
```

--------------------------------

### Unbounded interval and sqrt with dac decoration

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Creates an unbounded interval and computes its square root. Both intervals are unbounded, so the maximum possible decoration is dac (defined and continuous).

```julia
julia> x = interval(3, Inf)
Interval{Float64}(3.0, Inf, dac, true)
julia> sqrt(x)
Interval{Float64}(1.7320508075688772, Inf, dac, true)
```

--------------------------------

### fastpown(x, n)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

A faster implementation of `pown(x, n)` that may return a larger interval but obeys the same domain convention.

```APIDOC
## fastpown(x, n)

### Description
A faster implementation of `pown(x, n)`, at the cost of maybe returning a larger interval. In particular, it obeys the same domain convention so negative values of `x` are allowed.

### Method
Function

### Endpoint
`fastpown(x, n)`

### Parameters
- **x** (interval) - Required - The interval to be raised to the power.
- **n** (integer) - Required - The exponent.

### Request Example
```julia
fastpown(bareinterval(2, 3), 2)
fastpown(interval(-4, -2), 2)
fastpown(interval(-1, 1), -3)
```

### Response
#### Success Response
Returns an interval (possibly larger than the exact result).

#### Response Example
```julia
BareInterval{Float64}(4.0, 9.0)
Interval{Float64}(4.0, 16.0, com, true)
Interval{Float64}(-Inf, Inf, trv, true)
```

### See Also
`pown`, `pow`, `fastpow`
```

--------------------------------

### configure function signature

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/configuration

Signature of the configure function. Each keyword defaults to the value currently in use, so only the given options are modified.

```julia
configure(; numtype=Float64, flavor=:set_based, rounding=:correct, power=:fast, matmul=:fast, nthreads=IntervalArithmetic.default_threads())
```

--------------------------------

### decoration Method for Interval

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Returns the decoration of an interval x.

```julia
decoration(x::Interval)
```

--------------------------------

### mince Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Splits an interval x into n intervals of the same diameter. Unbounded intervals are rejected.

```julia
mince(x, n)
```

--------------------------------

### Compute π with forward sum using IntervalArithmetic.jl

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/examples/pi

Computes rigorous bounds on π by splitting the Basel sum into a finite sum and a tail bound, then returns the midpoint and radius of the resulting interval. The output shows the midpoint is correct to about 10 decimal places.

```julia
using IntervalArithmetic
function forward_sum(N)
    S_N = interval(0)
    for i ∈ 1:N
        S_N += interval(1) / interval(i) ^ interval(2)
    end
    T_N = interval(1) / interval(N, N+1)
    S = S_N + T_N
    return sqrt(interval(6) * S)
end
pi_interval = forward_sum(10^6)
midradius(pi_interval)
```

```text
(3.14159265358984, 1.0649436887888442e-10)
```

--------------------------------

### isstrictless Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Tests whether inf(x) < inf(y) and sup(x) < sup(y), with adjustments for infinite values.

```julia
isstrictless(x, y)
```

--------------------------------

### Evaluate function over semi-infinite interval with IntervalArithmetic.jl

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/intro

Shows how to evaluate a function over a semi-infinite interval using IntervalArithmetic.jl. The output shows the resulting interval and its decoration.

```julia
julia> f(interval(3, Inf))
Interval{Float64}(7.0, Inf, dac, false)
```

--------------------------------

### Interval set operations

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/usage

Demonstrates that standard set operations are disallowed for intervals and must be replaced by their *_interval equivalents.

```julia
julia> issubset(interval(1, 2), interval(2))ERROR: ArgumentError: `issubset` is purposely not supported for intervals. See instead `issubset_interval`julia> issubset_interval(interval(1, 2), interval(2))falsejulia> intersect(interval(1, 2), interval(2))ERROR: ArgumentError: `intersect` is purposely not supported for intervals. See instead `intersect_interval`julia> intersect_interval(interval(1, 2), interval(2))Interval{Float64}(2.0, 2.0, trv, true)
```

--------------------------------

### mig Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Calculates the mignitude of x, implementing the IEEE Standard 1788-2015 Table 9.2.

```julia
mig(x)
```

--------------------------------

### isbounded method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Tests whether x is empty or has finite bounds.

```julia
isbounded(x)
```

--------------------------------

### mince!(v, x, n)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

In-place version of `mince`. Splits an interval `x` into `n` intervals of the same diameter and stores the result in `v`.

```APIDOC
## `mince!(v, x, n)`

### Description
In-place version of `mince`. Splits an interval `x` into `n` intervals of the same diameter and stores the result in `v`.

### Method
Function call

### Endpoint
`mince!(v, x, n)`

### Parameters
- **v** (vector) - Required - The vector to store the resulting intervals.
- **x** (interval) - Required - The interval to split.
- **n** (integer) - Required - The number of intervals to split into.

### Return Value
Returns the modified vector `v`.

```

--------------------------------

### extended_div Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Performs two-output division, corresponding to mulRevToPair(y, x) of the IEEE Standard 1788-2015.

```julia
extended_div(x, y)
```

--------------------------------

### isbounded(x)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Test whether an interval is empty or has finite bounds.

```APIDOC
## isbounded(x)

### Description
Test whether `x` is empty or has finite bounds.

### Method
Function

### Endpoint
`isbounded(x)`

### Parameters
- **x** (interval) - Required - The interval to test.

### Response
#### Success Response
Returns a boolean indicating whether the interval is bounded.
```

--------------------------------

### Perform basic interval arithmetic

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/usage

Basic arithmetic operations like addition are defined such that the result is the smallest interval containing all possible results of the operation on the input intervals.

```julia
julia> using IntervalArithmeticjulia> setdisplay(:full)Display options:  - format: full  - decorations: true (ignored)  - NG flag: true (ignored)  - significant digits: 4 (ignored)julia> X = interval(0, 1)Interval{Float64}(0.0, 1.0, com, true)julia> Y = interval(1, 2)Interval{Float64}(1.0, 2.0, com, true)julia> X + YInterval{Float64}(1.0, 3.0, com, true)
```

```julia
julia> X - XInterval{Float64}(-1.0, 1.0, com, true)
```

--------------------------------

### iscommon(x)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Test whether an interval is not empty and bounded.

```APIDOC
## iscommon(x)

### Description
Test whether `x` is not empty and bounded. Implements the `isCommonInterval` function of the IEEE Standard 1788-2015.

### Method
Function

### Endpoint
`iscommon(x)`

### Parameters
- **x** (interval) - Required - The interval to test.

### Response
#### Success Response
Returns a boolean indicating whether the interval is common.

### Notes
This does not take into consideration the decoration of the interval.
```

--------------------------------

### iscommon method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Tests whether x is not empty and bounded.

```julia
iscommon(x)
```

--------------------------------

### ⪯(x, y)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/symbols

Unicode alias of precedes. Precedes relation.

```APIDOC
## ⪯(x, y)

### Description
Unicode alias of `precedes`. Precedes relation.

### Method
`⪯`

### Parameters
- `x` - First interval
- `y` - Second interval

### Response
Returns a boolean indicating whether `x` precedes `y`.
```

--------------------------------

### Check interval guarantee status

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/guarantee

Demonstrates how implicit conversion results in an 'NG' label, while explicit construction via interval() is marked as guaranteed.

```julia
julia> using IntervalArithmeticjulia> setdisplay(; sigdigits = 6)Display options:  - format: full  - decorations: true (ignored)  - NG flag: true (ignored)  - significant digits: 6 (ignored)julia> convert(Interval{Float64}, 1.) # considered "not guaranteed" as this call can be done implicitlyInterval{Float64}(1.0, 1.0, com, false)julia> interval(1) # considered "guaranteed" as the user explicitly constructed the intervalInterval{Float64}(1.0, 1.0, com, true)
```

--------------------------------

### Calculate interval powers with pown and pow

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/usage

The pown and pow functions handle powers with different domain constraints. Arguments outside the domain of definition result in empty intervals with trv decoration.

```julia
julia> x = interval(-4, -2)Interval{Float64}(-4.0, -2.0, com, true)julia> pown(x, 2)Interval{Float64}(4.0, 16.0, com, true)julia> pow(x, interval(2))∅_trv
```

```julia
julia> x ^ interval(2)Interval{Float64}(4.0, 16.0, com, true)julia> x ^ interval(2.5)∅_trv
```

--------------------------------

### emptyinterval Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Creates an empty interval, which is an exception to the rule that the lower bound is less than or equal to the upper bound.

```julia
emptyinterval(T=[default_numtype()])
```

--------------------------------

### inf method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Returns the lower bound, or infimum, of x.

```julia
inf(x)
```

--------------------------------

### bareinterval construction

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Creates bare intervals using different types, noting that floating point literals are rounded to the nearest value.

```julia
julia> using IntervalArithmeticjulia> setdisplay(:full);julia> bareinterval(1//1, π)BareInterval{Rational{Int64}}(1//1, 85563208//27235615)julia> bareinterval(Rational{Int32}, 1//1, π)BareInterval{Rational{Int32}}(1//1, 85563208//27235615)julia> bareinterval(1, π)BareInterval{Float64}(1.0, 3.1415926535897936)julia> bareinterval(BigFloat, 1, π)BareInterval{BigFloat}(1.0, 3.141592653589793238462643383279502884197169399375105820974944592307816406286233)
```

--------------------------------

### Use infix operators .. and ± from IntervalArithmetic.Symbols

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Requires explicitly importing the IntervalArithmetic.Symbols submodule. The infix operator .. creates an interval in infsup format, while ± creates one in midpoint format; both are aliases for interval.

```julia
julia> using IntervalArithmetic.Symbolsjulia> 0.1 .. 0.2 # interval(0.1, 0.2; format = :infsup)Interval{Float64}(0.1, 0.2, com, true)julia> 0.1 ± 0.2 # interval(0.1, 0.2; format = :midpoint)Interval{Float64}(-0.1, 0.30000000000000004, com, true)
```

--------------------------------

### Assign floating-point literal 0.1 to x

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/construction

Shows that assigning the literal 0.1 to x stores a Float64 approximation, not the exact real number 1/10. This motivates using the string constructor for exact intervals.

```julia
julia> x = 0.10.1
```

--------------------------------

### mag Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Calculates the magnitude of an interval.

```julia
mag(x)
```

--------------------------------

### decoration Method for BareInterval

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Returns the default decoration for a BareInterval based on its value.

```julia
decoration(x::BareInterval)
```

--------------------------------

### midradius Method

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Calculates the midpoint and radius of an interval x, as required by IEEE Standard 1788-2015 Section 10.5.9.

```julia
midradius(x)
```

--------------------------------

### ExactReal usage with intervals

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Shows how ExactReal prevents the 'NG' (not guaranteed) flag when performing arithmetic with intervals.

```julia
julia> using IntervalArithmeticjulia> setdisplay(:full);julia> 0.5 * interval(1)Interval{Float64}(0.5, 0.5, com, false)julia> exact(0.5) * interval(1)Interval{Float64}(0.5, 0.5, com, true)julia> setdisplay(:infsup);julia> [1, interval(2)]2-element Vector{Interval{Float64}}: [1.0, 1.0]_com_NG [2.0, 2.0]_comjulia> [exact(1), interval(2)]2-element Vector{Interval{Float64}}: [1.0, 1.0]_com [2.0, 2.0]_com
```

--------------------------------

### mince(x, n)

Source: https://juliaintervals.github.io/IntervalArithmetic.jl/stable/manual/api

Splits an interval `x` into `n` intervals of the same diameter. An unbounded `x` has no such splitting and is rejected; use `bisect` instead. Also splits the `i`-th component of a vector `x` in `n[i]` intervals of the same diameter; `n` can be a tuple of integers, or a single integer in which case the same `n` is used for all components.

```APIDOC
## `mince(x, n)`

### Description
Splits an interval `x` into `n` intervals of the same diameter. An unbounded `x` has no such splitting and is rejected; use `bisect` instead. Also splits the `i`-th component of a vector `x` in `n[i]` intervals of the same diameter; `n` can be a tuple of integers, or a single integer in which case the same `n` is used for all components.

### Method
Function call

### Endpoint
`mince(x, n)`

### Parameters
- **x** (interval or vector) - Required - The interval or vector of intervals to split.
- **n** (integer or tuple) - Required - The number of intervals to split into, or a tuple of integers for each component.

### Return Value
Returns a vector of intervals resulting from the split.

```
Tokens:
Raw