# Mathematics

## Mathematical Operators

`Base.:-` — Method

    -(x)

Unary minus operator.

See also: `abs`, `flipsign`.

Examples

    julia> -1
    -1

    julia> -(2)
    -2

    julia> -[1 2; 3 4]
    2×2 Matrix{Int64}:
     -1  -2
     -3  -4

source
`Base.:+` — Function

    +(x, y...)

Addition operator. `x+y+z+...` calls this function with all arguments, i.e. `+(x, y, z, ...)`.

Examples

    julia> 1 + 20 + 4
    25

    julia> +(1, 20, 4)
    25

source

    dt::Date + t::Time -> DateTime

The addition of a `Date` with a `Time` produces a `DateTime`. The hour, minute, second, and millisecond parts of the `Time` are used along with the year, month, and day of the `Date` to create the new `DateTime`. Non-zero microseconds or nanoseconds in the `Time` type will result in an `InexactError` being thrown.

source
`Base.:-` — Method

    -(x, y)

Subtraction operator.

Examples

    julia> 2 - 3
    -1

    julia> -(2, 4.5)
    -2.5

source
`Base.:*` — Method

    *(x, y...)

Multiplication operator. `x*y*z*...` calls this function with all arguments, i.e. `*(x, y, z, ...)`.

Examples

    julia> 2 * 7 * 8
    112

    julia> *(2, 7, 8)
    112

source
`Base.:/` — Function

    /(x, y)

Right division operator: multiplication of `x` by the inverse of `y` on the right. Gives floating-point results for integer arguments.

Examples

    julia> 1/2
    0.5

    julia> 4/2
    2.0

    julia> 4.5/2
    2.25

source
`Base.:\` — Method

    \(x, y)

Left division operator: multiplication of `y` by the inverse of `x` on the left. Gives floating-point results for integer arguments.

Examples

    julia> 3 \ 6
    2.0

    julia> inv(3) * 6
    2.0

    julia> A = [4 3; 2 1]; x = [5, 6];

    julia> A \ x
    2-element Vector{Float64}:
      6.5
     -7.0

    julia> inv(A) * x
    2-element Vector{Float64}:
      6.5
     -7.0

source
`Base.:^` — Method

    ^(x, y)

Exponentiation operator. If `x` is a matrix, computes matrix exponentiation.

If `y` is an `Int` literal (e.g. `2` in `x^2` or `-3` in `x^-3`), the Julia code `x^y` is transformed by the compiler to `Base.literal_pow(^, x, Val(y))`, to enable compile-time specialization on the value of the exponent. (As a default fallback we have `Base.literal_pow(^, x, Val(y)) = ^(x,y)`, where usually `^ == Base.^` unless `^` has been defined in the calling namespace.) If `y` is a negative integer literal, then `Base.literal_pow` transforms the operation to `inv(x)^-y` by default, where `-y` is positive.

Examples

    julia> 3^5
    243

    julia> A = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> A^3
    2×2 Matrix{Int64}:
     37   54
     81  118

source
`Base.fma` — Function

    fma(x, y, z)

Computes `x*y+z` without rounding the intermediate result `x*y`. On some systems this is significantly more expensive than `x*y+z`. `fma` is used to improve accuracy in certain algorithms. See `muladd`.

source
`Base.muladd` — Function

    muladd(x, y, z)

Combined multiply-add: computes `x*y+z`, but allowing the add and multiply to be merged with each other or with surrounding operations for performance. For example, this may be implemented as an `fma` if the hardware supports it efficiently. The result can be different on different machines and can also be different on the same machine due to constant propagation or other optimizations. See `fma`.

Examples

    julia> muladd(3, 2, 1)
    7

    julia> 3 * 2 + 1
    7

source

    muladd(A, y, z)

Combined multiply-add, `A*y .+ z`, for matrix-matrix or matrix-vector multiplication. The result is always the same size as `A*y`, but `z` may be smaller, or a scalar.

Julia 1.6

These methods require Julia 1.6 or later.

Examples

    julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; z=[0, 100];

    julia> muladd(A, B, z)
    2×2 Matrix{Float64}:
       3.0    3.0
     107.0  107.0

source
`Base.inv` — Method

    inv(x)

Return the multiplicative inverse of `x`, such that `x*inv(x)` or `inv(x)*x` yields `one(x)` (the multiplicative identity) up to roundoff errors.

If `x` is a number, this is essentially the same as `one(x)/x`, but for some types `inv(x)` may be slightly more efficient.

Examples

    julia> inv(2)
    0.5

    julia> inv(1 + 2im)
    0.2 - 0.4im

    julia> inv(1 + 2im) * (1 + 2im)
    1.0 + 0.0im

    julia> inv(2//3)
    3//2

Julia 1.2

`inv(::Missing)` requires at least Julia 1.2.

source
`Base.div` — Function

    div(x, y)
    ÷(x, y)

The quotient from Euclidean (integer) division. Generally equivalent to a mathematical operation x/y without a fractional part.

See also: `cld`, `fld`, `rem`, `divrem`.

Examples

    julia> 9 ÷ 4
    2

    julia> -5 ÷ 3
    -1

    julia> 5.0 ÷ 2
    2.0

    julia> div.(-5:5, 3)'
    1×11 adjoint(::Vector{Int64}) with eltype Int64:
     -1  -1  -1  0  0  0  0  0  1  1  1

source
`Base.fld` — Function

    fld(x, y)

Largest integer less than or equal to `x/y`. Equivalent to `div(x, y, RoundDown)`.

See also `div`, `cld`, `fld1`.

Examples

    julia> fld(7.3,5.5)
    1.0

    julia> fld.(-5:5, 3)'
    1×11 adjoint(::Vector{Int64}) with eltype Int64:
     -2  -2  -1  -1  -1  0  0  0  1  1  1

Because `fld(x, y)` implements strictly correct floored rounding based on the true value of floating-point numbers, unintuitive situations can arise. For example:

    julia> fld(6.0,0.1)
    59.0
    julia> 6.0/0.1
    60.0
    julia> 6.0/big(0.1)
    59.99999999999999666933092612453056361837965690217069245739573412231113406246995

What is happening here is that the true value of the floating-point number written as `0.1` is slightly larger than the numerical value 1/10 while `6.0` represents the number 6 precisely. Therefore the true value of `6.0 / 0.1` is slightly less than 60. When doing division, this is rounded to precisely `60.0`, but `fld(6.0, 0.1)` always takes the floor or the true value, so the result is `59.0`.

source
`Base.cld` — Function

    cld(x, y)

Smallest integer larger than or equal to `x/y`. Equivalent to `div(x, y, RoundUp)`.

See also `div`, `fld`.

Examples

    julia> cld(5.5,2.2)
    3.0

    julia> cld.(-5:5, 3)'
    1×11 adjoint(::Vector{Int64}) with eltype Int64:
     -1  -1  -1  0  0  0  1  1  1  2  2

source
`Base.mod` — Function

    mod(x::Integer, r::AbstractUnitRange)

Find `y` in the range `r` such that $x ≡ y (mod n)$, where `n = length(r)`, i.e. `y = mod(x - first(r), n) + first(r)`.

See also `mod1`.

Examples

    julia> mod(0, Base.OneTo(3))
    3

    julia> mod(3, 0:2)
    0

Julia 1.3

This method requires at least Julia 1.3.

source

    mod(x, y)
    rem(x, y, RoundDown)

The reduction of `x` modulo `y`, or equivalently, the remainder of `x` after floored division by `y`, i.e. `x - y*fld(x,y)` if computed without intermediate rounding.

The result will have the same sign as `y`, and magnitude less than `abs(y)` (with some exceptions, see note below).

Note

When used with floating point values, the exact result may not be representable by the type, and so rounding error may occur. In particular, if the exact result is very close to `y`, then it may be rounded to `y`.

See also: `rem`, `div`, `fld`, `mod1`, `invmod`.

    julia> mod(8, 3)
    2

    julia> mod(9, 3)
    0

    julia> mod(8.9, 3)
    2.9000000000000004

    julia> mod(eps(), 3)
    2.220446049250313e-16

    julia> mod(-eps(), 3)
    3.0

    julia> mod.(-5:5, 3)'
    1×11 adjoint(::Vector{Int64}) with eltype Int64:
     1  2  0  1  2  0  1  2  0  1  2

source

    rem(x::Integer, T::Type{<:Integer}) -> T
    mod(x::Integer, T::Type{<:Integer}) -> T
    %(x::Integer, T::Type{<:Integer}) -> T

Find `y::T` such that `x` ≡ `y` (mod n), where n is the number of integers representable in `T`, and `y` is an integer in `[typemin(T),typemax(T)]`. If `T` can represent any integer (e.g. `T == BigInt`), then this operation corresponds to a conversion to `T`.

Examples

    julia> 129 % Int8
    -127

source
`Base.rem` — Function

    rem(x, y)
    %(x, y)

Remainder from Euclidean division, returning a value of the same sign as `x`, and smaller in magnitude than `y`. This value is always exact.

See also: `div`, `mod`, `mod1`, `divrem`.

Examples

    julia> x = 15; y = 4;

    julia> x % y
    3

    julia> x == div(x, y) * y + rem(x, y)
    true

    julia> rem.(-5:5, 3)'
    1×11 adjoint(::Vector{Int64}) with eltype Int64:
     -2  -1  0  -2  -1  0  1  2  0  1  2

source
`Base.Math.rem2pi` — Function

    rem2pi(x, r::RoundingMode)

Compute the remainder of `x` after integer division by `2π`, with the quotient rounded according to the rounding mode `r`. In other words, the quantity

    x - 2π*round(x/(2π),r)

without any intermediate rounding. This internally uses a high precision approximation of 2π, and so will give a more accurate result than `rem(x,2π,r)`

- if `r == RoundNearest`, then the result is in the interval $[-π, π]$. This will generally be the most accurate result. See also `RoundNearest`.

- if `r == RoundToZero`, then the result is in the interval $[0, 2π]$ if `x` is positive,. or $[-2π, 0]$ otherwise. See also `RoundToZero`.

- if `r == RoundDown`, then the result is in the interval $[0, 2π]$. See also `RoundDown`.

- if `r == RoundUp`, then the result is in the interval $[-2π, 0]$. See also `RoundUp`.

Examples

    julia> rem2pi(7pi/4, RoundNearest)
    -0.7853981633974485

    julia> rem2pi(7pi/4, RoundDown)
    5.497787143782138

source
`Base.Math.mod2pi` — Function

    mod2pi(x)

Modulus after division by `2π`, returning in the range $[0,2π)$.

This function computes a floating point representation of the modulus after division by numerically exact `2π`, and is therefore not exactly the same as `mod(x,2π)`, which would compute the modulus of `x` relative to division by the floating-point number `2π`.

Note

Depending on the format of the input value, the closest representable value to 2π may be less than 2π. For example, the expression `mod2pi(2π)` will not return `0`, because the intermediate value of `2*π` is a `Float64` and `2*Float64(π) < 2*big(π)`. See `rem2pi` for more refined control of this behavior.

Examples

    julia> mod2pi(9*pi/4)
    0.7853981633974481

source
`Base.divrem` — Function

    divrem(x, y, r::RoundingMode=RoundToZero)

The quotient and remainder from Euclidean division. Equivalent to `(div(x,y,r), rem(x,y,r))`. Equivalently, with the default value of `r`, this call is equivalent to `(x÷y, x%y)`.

See also: `fldmod`, `cld`.

Examples

    julia> divrem(3,7)
    (0, 3)

    julia> divrem(7,3)
    (2, 1)

source
`Base.fldmod` — Function

    fldmod(x, y)

The floored quotient and modulus after division. A convenience wrapper for `divrem(x, y, RoundDown)`. Equivalent to `(fld(x,y), mod(x,y))`.

See also: `fld`, `cld`, `fldmod1`.

source
`Base.fld1` — Function

    fld1(x, y)

Flooring division, returning a value consistent with `mod1(x,y)`

See also `mod1`, `fldmod1`.

Examples

    julia> x = 15; y = 4;

    julia> fld1(x, y)
    4

    julia> x == fld(x, y) * y + mod(x, y)
    true

    julia> x == (fld1(x, y) - 1) * y + mod1(x, y)
    true

source
`Base.mod1` — Function

    mod1(x, y)

Modulus after flooring division, returning a value `r` such that `mod(r, y) == mod(x, y)` in the range $(0, y]$ for positive `y` and in the range $[y,0)$ for negative `y`.

See also `fld1`, `fldmod1`.

Examples

    julia> mod1(4, 2)
    2

    julia> mod1(4, 3)
    1

source
`Base.fldmod1` — Function

    fldmod1(x, y)

Return `(fld1(x,y), mod1(x,y))`.

See also `fld1`, `mod1`.

source
`Base.://` — Function

    //(num, den)

Divide two integers or rational numbers, giving a `Rational` result.

Examples

    julia> 3 // 5
    3//5

    julia> (3 // 5) // (2 // 1)
    3//10

source
`Base.rationalize` — Function

    rationalize([T<:Integer=Int,] x; tol::Real=eps(x))

Approximate floating point number `x` as a `Rational` number with components of the given integer type. The result will differ from `x` by no more than `tol`.

Examples

    julia> rationalize(5.6)
    28//5

    julia> a = rationalize(BigInt, 10.3)
    103//10

    julia> typeof(numerator(a))
    BigInt

source
`Base.numerator` — Function

    numerator(x)

Numerator of the rational representation of `x`.

Examples

    julia> numerator(2//3)
    2

    julia> numerator(4)
    4

source
`Base.denominator` — Function

    denominator(x)

Denominator of the rational representation of `x`.

Examples

    julia> denominator(2//3)
    3

    julia> denominator(4)
    1

source
`Base.:<<` — Function

    <<(x, n)

Left bit shift operator, `x << n`. For `n >= 0`, the result is `x` shifted left by `n` bits, filling with `0`s. This is equivalent to `x * 2^n`. For `n < 0`, this is equivalent to `x >> -n`.

Examples

    julia> Int8(3) << 2
    12

    julia> bitstring(Int8(3))
    "00000011"

    julia> bitstring(Int8(12))
    "00001100"

See also `>>`, `>>>`, `exp2`, `ldexp`.

source

    <<(B::BitVector, n) -> BitVector

Left bit shift operator, `B << n`. For `n >= 0`, the result is `B` with elements shifted `n` positions backwards, filling with `false` values. If `n < 0`, elements are shifted forwards. Equivalent to `B >> -n`.

Examples

    julia> B = BitVector([true, false, true, false, false])
    5-element BitVector:
     1
     0
     1
     0
     0

    julia> B << 1
    5-element BitVector:
     0
     1
     0
     0
     0

    julia> B << -1
    5-element BitVector:
     0
     1
     0
     1
     0

source
`Base.:>>` — Function

    >>(x, n)

Right bit shift operator, `x >> n`. For `n >= 0`, the result is `x` shifted right by `n` bits, where `n >= 0`, filling with `0`s if `x >= 0`, `1`s if `x < 0`, preserving the sign of `x`. This is equivalent to `fld(x, 2^n)`. For `n < 0`, this is equivalent to `x << -n`.

Examples

    julia> Int8(13) >> 2
    3

    julia> bitstring(Int8(13))
    "00001101"

    julia> bitstring(Int8(3))
    "00000011"

    julia> Int8(-14) >> 2
    -4

    julia> bitstring(Int8(-14))
    "11110010"

    julia> bitstring(Int8(-4))
    "11111100"

See also `>>>`, `<<`.

source

    >>(B::BitVector, n) -> BitVector

Right bit shift operator, `B >> n`. For `n >= 0`, the result is `B` with elements shifted `n` positions forward, filling with `false` values. If `n < 0`, elements are shifted backwards. Equivalent to `B << -n`.

Examples

    julia> B = BitVector([true, false, true, false, false])
    5-element BitVector:
     1
     0
     1
     0
     0

    julia> B >> 1
    5-element BitVector:
     0
     1
     0
     1
     0

    julia> B >> -1
    5-element BitVector:
     0
     1
     0
     0
     0

source
`Base.:>>>` — Function

    >>>(x, n)

Unsigned right bit shift operator, `x >>> n`. For `n >= 0`, the result is `x` shifted right by `n` bits, where `n >= 0`, filling with `0`s. For `n < 0`, this is equivalent to `x << -n`.

For `Unsigned` integer types, this is equivalent to `>>`. For `Signed` integer types, this is equivalent to `signed(unsigned(x) >> n)`.

Examples

    julia> Int8(-14) >>> 2
    60

    julia> bitstring(Int8(-14))
    "11110010"

    julia> bitstring(Int8(60))
    "00111100"

`BigInt`s are treated as if having infinite size, so no filling is required and this is equivalent to `>>`.

See also `>>`, `<<`.

source

    >>>(B::BitVector, n) -> BitVector

Unsigned right bitshift operator, `B >>> n`. Equivalent to `B >> n`. See `>>` for details and examples.

source
`Base.bitrotate` — Function

    bitrotate(x::Base.BitInteger, k::Integer)

`bitrotate(x, k)` implements bitwise rotation. It returns the value of `x` with its bits rotated left `k` times. A negative value of `k` will rotate to the right instead.

Julia 1.5

This function requires Julia 1.5 or later.

See also: `<<`, `circshift`, `BitArray`.

    julia> bitrotate(UInt8(114), 2)
    0xc9

    julia> bitstring(bitrotate(0b01110010, 2))
    "11001001"

    julia> bitstring(bitrotate(0b01110010, -2))
    "10011100"

    julia> bitstring(bitrotate(0b01110010, 8))
    "01110010"

source
`Base.::` — Function

    (:)(start, [step], stop)

Range operator. `a:b` constructs a range from `a` to `b` with a step size of 1 (a `UnitRange`) , and `a:s:b` is similar but uses a step size of `s` (a `StepRange`).

`:` is also used in indexing to select whole dimensions and for `Symbol` literals, as in e.g. `:hello`.

source

    (:)(start::CartesianIndex, [step::CartesianIndex], stop::CartesianIndex)

Construct `CartesianIndices` from two `CartesianIndex` and an optional step.

Julia 1.1

This method requires at least Julia 1.1.

Julia 1.6

The step range method start:step:stop requires at least Julia 1.6.

Examples

    julia> I = CartesianIndex(2,1);

    julia> J = CartesianIndex(3,3);

    julia> I:J
    2×3 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
     CartesianIndex(2, 1)  CartesianIndex(2, 2)  CartesianIndex(2, 3)
     CartesianIndex(3, 1)  CartesianIndex(3, 2)  CartesianIndex(3, 3)

    julia> I:CartesianIndex(1, 2):J
    2×2 CartesianIndices{2, Tuple{StepRange{Int64, Int64}, StepRange{Int64, Int64}}}:
     CartesianIndex(2, 1)  CartesianIndex(2, 3)
     CartesianIndex(3, 1)  CartesianIndex(3, 3)

source
`Base.range` — Function

    range(start, stop, length)
    range(start, stop; length, step)
    range(start; length, stop, step)
    range(;start, length, stop, step)

Construct a specialized array with evenly spaced elements and optimized storage (an `AbstractRange`) from the arguments. Mathematically a range is uniquely determined by any three of `start`, `step`, `stop` and `length`. Valid invocations of range are:

- Call `range` with any three of `start`, `step`, `stop`, `length`.
- Call `range` with two of `start`, `stop`, `length`. In this case `step` will be assumed

to be one. If both arguments are Integers, a `UnitRange` will be returned.

Examples

    julia> range(1, length=100)
    1:100

    julia> range(1, stop=100)
    1:100

    julia> range(1, step=5, length=100)
    1:5:496

    julia> range(1, step=5, stop=100)
    1:5:96

    julia> range(1, 10, length=101)
    1.0:0.09:10.0

    julia> range(1, 100, step=5)
    1:5:96

    julia> range(stop=10, length=5)
    6:10

    julia> range(stop=10, step=1, length=5)
    6:1:10

    julia> range(start=1, step=1, stop=10)
    1:1:10

If `length` is not specified and `stop - start` is not an integer multiple of `step`, a range that ends before `stop` will be produced.

    julia> range(1, 3.5, step=2)
    1.0:2.0:3.0

Special care is taken to ensure intermediate values are computed rationally. To avoid this induced overhead, see the `LinRange` constructor.

Julia 1.1

`stop` as a positional argument requires at least Julia 1.1.

Julia 1.7

The versions without keyword arguments and `start` as a keyword argument require at least Julia 1.7.

source
`Base.OneTo` — Type

    Base.OneTo(n)

Define an `AbstractUnitRange` that behaves like `1:n`, with the added distinction that the lower limit is guaranteed (by the type system) to be 1.

source
`Base.StepRangeLen` — Type

    StepRangeLen{T,R,S}(ref::R, step::S, len, [offset=1]) where {T,R,S}
    StepRangeLen(       ref::R, step::S, len, [offset=1]) where {  R,S}

A range `r` where `r[i]` produces values of type `T` (in the second form, `T` is deduced automatically), parameterized by a `ref`erence value, a `step`, and the `len`gth. By default `ref` is the starting value `r[1]`, but alternatively you can supply it as the value of `r[offset]` for some other index `1 <= offset <= len`. In conjunction with `TwicePrecision` this can be used to implement ranges that are free of roundoff error.

source
`Base.:==` — Function

    ==(x, y)

Generic equality operator. Falls back to `===`. Should be implemented for all types with a notion of equality, based on the abstract value that an instance represents. For example, all numeric types are compared by numeric value, ignoring type. Strings are compared as sequences of characters, ignoring encoding. For collections, `==` is generally called recursively on all contents, though other properties (like the shape for arrays) may also be taken into account.

This operator follows IEEE semantics for floating-point numbers: `0.0 == -0.0` and `NaN != NaN`.

The result is of type `Bool`, except when one of the operands is `missing`, in which case `missing` is returned (three-valued logic). For collections, `missing` is returned if at least one of the operands contains a `missing` value and all non-missing values are equal. Use `isequal` or `===` to always get a `Bool` result.

Implementation

New numeric types should implement this function for two arguments of the new type, and handle comparison to other types via promotion rules where possible.

`isequal` falls back to `==`, so new methods of `==` will be used by the `Dict` type to compare keys. If your type will be used as a dictionary key, it should therefore also implement `hash`.

If some type defines `==`, `isequal`, and `isless` then it should also implement `<` to ensure consistency of comparisons.

source
`Base.:!=` — Function

    !=(x, y)
    ≠(x,y)

Not-equals comparison operator. Always gives the opposite answer as `==`.

Implementation

New types should generally not implement this, and rely on the fallback definition `!=(x,y) = !(x==y)` instead.

Examples

    julia> 3 != 2
    true

    julia> "foo" ≠ "foo"
    false

source

    !=(x)

Create a function that compares its argument to `x` using `!=`, i.e. a function equivalent to `y -> y != x`. The returned function is of type `Base.Fix2{typeof(!=)}`, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
`Base.:!==` — Function

    !==(x, y)
    ≢(x,y)

Always gives the opposite answer as `===`.

Examples

    julia> a = [1, 2]; b = [1, 2];

    julia> a ≢ b
    true

    julia> a ≢ a
    false

source
`Base.:<` — Function

    <(x, y)

Less-than comparison operator. Falls back to `isless`. Because of the behavior of floating-point NaN values, this operator implements a partial order.

Implementation

New numeric types with a canonical partial order should implement this function for two arguments of the new type. Types with a canonical total order should implement `isless` instead.

Examples

    julia> 'a' < 'b'
    true

    julia> "abc" < "abd"
    true

    julia> 5 < 3
    false

source

    <(x)

Create a function that compares its argument to `x` using `<`, i.e. a function equivalent to `y -> y < x`. The returned function is of type `Base.Fix2{typeof(<)}`, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
`Base.:<=` — Function

    <=(x, y)
    ≤(x,y)

Less-than-or-equals comparison operator. Falls back to `(x < y) | (x == y)`.

Examples

    julia> 'a' <= 'b'
    true

    julia> 7 ≤ 7 ≤ 9
    true

    julia> "abc" ≤ "abc"
    true

    julia> 5 <= 3
    false

source

    <=(x)

Create a function that compares its argument to `x` using `<=`, i.e. a function equivalent to `y -> y <= x`. The returned function is of type `Base.Fix2{typeof(<=)}`, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
`Base.:>` — Function

    >(x, y)

Greater-than comparison operator. Falls back to `y < x`.

Implementation

Generally, new types should implement `<` instead of this function, and rely on the fallback definition `>(x, y) = y < x`.

Examples

    julia> 'a' > 'b'
    false

    julia> 7 > 3 > 1
    true

    julia> "abc" > "abd"
    false

    julia> 5 > 3
    true

source

    >(x)

Create a function that compares its argument to `x` using `>`, i.e. a function equivalent to `y -> y > x`. The returned function is of type `Base.Fix2{typeof(>)}`, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
`Base.:>=` — Function

    >=(x, y)
    ≥(x,y)

Greater-than-or-equals comparison operator. Falls back to `y <= x`.

Examples

    julia> 'a' >= 'b'
    false

    julia> 7 ≥ 7 ≥ 3
    true

    julia> "abc" ≥ "abc"
    true

    julia> 5 >= 3
    true

source

    >=(x)

Create a function that compares its argument to `x` using `>=`, i.e. a function equivalent to `y -> y >= x`. The returned function is of type `Base.Fix2{typeof(>=)}`, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
`Base.cmp` — Function

    cmp(x,y)

Return -1, 0, or 1 depending on whether `x` is less than, equal to, or greater than `y`, respectively. Uses the total order implemented by `isless`.

Examples

    julia> cmp(1, 2)
    -1

    julia> cmp(2, 1)
    1

    julia> cmp(2+im, 3-im)
    ERROR: MethodError: no method matching isless(::Complex{Int64}, ::Complex{Int64})
    [...]

source

    cmp(<, x, y)

Return -1, 0, or 1 depending on whether `x` is less than, equal to, or greater than `y`, respectively. The first argument specifies a less-than comparison function to use.

source

    cmp(a::AbstractString, b::AbstractString) -> Int

Compare two strings. Return `0` if both strings have the same length and the character at each index is the same in both strings. Return `-1` if `a` is a prefix of `b`, or if `a` comes before `b` in alphabetical order. Return `1` if `b` is a prefix of `a`, or if `b` comes before `a` in alphabetical order (technically, lexicographical order by Unicode code points).

Examples

    julia> cmp("abc", "abc")
    0

    julia> cmp("ab", "abc")
    -1

    julia> cmp("abc", "ab")
    1

    julia> cmp("ab", "ac")
    -1

    julia> cmp("ac", "ab")
    1

    julia> cmp("α", "a")
    1

    julia> cmp("b", "β")
    -1

source
`Base.:~` — Function

    ~(x)

Bitwise not.

See also: `!`, `&`, `|`.

Examples

    julia> ~4
    -5

    julia> ~10
    -11

    julia> ~true
    false

source
`Base.:&` — Function

    x & y

Bitwise and. Implements three-valued logic, returning `missing` if one operand is `missing` and the other is `true`. Add parentheses for function application form: `(&)(x, y)`.

See also: `|`, `xor`, `&&`.

Examples

    julia> 4 & 10
    0

    julia> 4 & 12
    4

    julia> true & missing
    missing

    julia> false & missing
    false

source
`Base.:|` — Function

    x | y

Bitwise or. Implements three-valued logic, returning `missing` if one operand is `missing` and the other is `false`.

See also: `&`, `xor`, `||`.

Examples

    julia> 4 | 10
    14

    julia> 4 | 1
    5

    julia> true | missing
    true

    julia> false | missing
    missing

source
`Base.xor` — Function

    xor(x, y)
    ⊻(x, y)

Bitwise exclusive or of `x` and `y`. Implements three-valued logic, returning `missing` if one of the arguments is `missing`.

The infix operation `a ⊻ b` is a synonym for `xor(a,b)`, and `⊻` can be typed by tab-completing `\xor` or `\veebar` in the Julia REPL.

Examples

    julia> xor(true, false)
    true

    julia> xor(true, true)
    false

    julia> xor(true, missing)
    missing

    julia> false ⊻ false
    false

    julia> [true; true; false] .⊻ [true; false; false]
    3-element BitVector:
     0
     1
     0

source
`Base.nand` — Function

    nand(x, y)
    ⊼(x, y)

Bitwise nand (not and) of `x` and `y`. Implements three-valued logic, returning `missing` if one of the arguments is `missing`.

The infix operation `a ⊼ b` is a synonym for `nand(a,b)`, and `⊼` can be typed by tab-completing `\nand` or `\barwedge` in the Julia REPL.

Examples

    julia> nand(true, false)
    true

    julia> nand(true, true)
    false

    julia> nand(true, missing)
    missing

    julia> false ⊼ false
    true

    julia> [true; true; false] .⊼ [true; false; false]
    3-element BitVector:
     0
     1
     1

source
`Base.nor` — Function

    nor(x, y)
    ⊽(x, y)

Bitwise nor (not or) of `x` and `y`. Implements three-valued logic, returning `missing` if one of the arguments is `missing`.

The infix operation `a ⊽ b` is a synonym for `nor(a,b)`, and `⊽` can be typed by tab-completing `\nor` or `\veebar` in the Julia REPL.

Examples

    julia> nor(true, false)
    false

    julia> nor(true, true)
    false

    julia> nor(true, missing)
    false

    julia> false ⊽ false
    true

    julia> [true; true; false] .⊽ [true; false; false]
    3-element BitVector:
     0
     0
     1

source
`Base.:!` — Function

    !(x)

Boolean not. Implements three-valued logic, returning `missing` if `x` is `missing`.

See also `~` for bitwise not.

Examples

    julia> !true
    false

    julia> !false
    true

    julia> !missing
    missing

    julia> .![true false true]
    1×3 BitMatrix:
     0  1  0

source

    !f::Function

Predicate function negation: when the argument of `!` is a function, it returns a function which computes the boolean negation of `f`.

See also `∘`.

Examples

    julia> str = "∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"
    "∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"

    julia> filter(isletter, str)
    "εδxyδfxfyε"

    julia> filter(!isletter, str)
    "∀  > 0, ∃  > 0: |-| <  ⇒ |()-()| < "

source
`&&` — Keyword

    x && y

Short-circuiting boolean AND.

See also `&`, the ternary operator `? :`, and the manual section on control flow.

Examples

    julia> x = 3;

    julia> x > 1 && x < 10 && x isa Int
    true

    julia> x < 0 && error("expected positive x")
    false

source
`||` — Keyword

    x || y

Short-circuiting boolean OR.

See also: `|`, `xor`, `&&`.

Examples

    julia> pi < 3 || ℯ < 3
    true

    julia> false || true || println("neither is true!")
    true

source

## Mathematical Functions

`Base.isapprox` — Function

    isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])

Inexact equality comparison. Two numbers compare equal if their relative distance or their absolute distance is within tolerance bounds: `isapprox` returns `true` if `norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`. The default `atol` is zero and the default `rtol` depends on the types of `x` and `y`. The keyword argument `nans` determines whether or not NaN values are considered equal (defaults to false).

For real or complex floating-point values, if an `atol > 0` is not specified, `rtol` defaults to the square root of `eps` of the type of `x` or `y`, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significand digits. Otherwise, e.g. for integer arguments or if an `atol > 0` is supplied, `rtol` defaults to zero.

The `norm` keyword defaults to `abs` for numeric `(x,y)` and to `LinearAlgebra.norm` for arrays (where an alternative `norm` choice is sometimes useful). When `x` and `y` are arrays, if `norm(x-y)` is not finite (i.e. `±Inf` or `NaN`), the comparison falls back to checking whether all elements of `x` and `y` are approximately equal component-wise.

The binary operator `≈` is equivalent to `isapprox` with the default arguments, and `x ≉ y` is equivalent to `!isapprox(x,y)`.

Note that `x ≈ 0` (i.e., comparing to zero with the default tolerances) is equivalent to `x == 0` since the default `atol` is `0`. In such cases, you should either supply an appropriate `atol` (or use `norm(x) ≤ atol`) or rearrange your code (e.g. use `x ≈ y` rather than `x - y ≈ 0`). It is not possible to pick a nonzero `atol` automatically because it depends on the overall scaling (the "units") of your problem: for example, in `x - y ≈ 0`, `atol=1e-9` is an absurdly small tolerance if `x` is the radius of the Earth in meters, but an absurdly large tolerance if `x` is the radius of a Hydrogen atom in meters.

Julia 1.6

Passing the `norm` keyword argument when comparing numeric (non-array) arguments requires Julia 1.6 or later.

Examples

    julia> isapprox(0.1, 0.15; atol=0.05)
    true

    julia> isapprox(0.1, 0.15; rtol=0.34)
    true

    julia> isapprox(0.1, 0.15; rtol=0.33)
    false

    julia> 0.1 + 1e-10 ≈ 0.1
    true

    julia> 1e-10 ≈ 0
    false

    julia> isapprox(1e-10, 0, atol=1e-8)
    true

    julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`
    true

source

    isapprox(x; kwargs...) / ≈(x; kwargs...)

Create a function that compares its argument to `x` using `≈`, i.e. a function equivalent to `y -> y ≈ x`.

The keyword arguments supported here are the same as those in the 2-argument `isapprox`.

Julia 1.5

This method requires Julia 1.5 or later.

source
`Base.sin` — Method

    sin(x)

Compute sine of `x`, where `x` is in radians.

See also [`sind`], [`sinpi`], [`sincos`], [`cis`].

source
`Base.cos` — Method

    cos(x)

Compute cosine of `x`, where `x` is in radians.

See also [`cosd`], [`cospi`], [`sincos`], [`cis`].

source
`Base.Math.sincos` — Method

    sincos(x)

Simultaneously compute the sine and cosine of `x`, where `x` is in radians, returning a tuple `(sine, cosine)`.

See also `cis`, `sincospi`, `sincosd`.

source
`Base.tan` — Method

    tan(x)

Compute tangent of `x`, where `x` is in radians.

source
`Base.Math.sind` — Function

    sind(x)

Compute sine of `x`, where `x` is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.cosd` — Function

    cosd(x)

Compute cosine of `x`, where `x` is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.tand` — Function

    tand(x)

Compute tangent of `x`, where `x` is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.sincosd` — Function

    sincosd(x)

Simultaneously compute the sine and cosine of `x`, where `x` is in degrees.

Julia 1.3

This function requires at least Julia 1.3.

source
`Base.Math.sinpi` — Function

    sinpi(x)

Compute $\sin(\pi x)$ more accurately than `sin(pi*x)`, especially for large `x`.

See also `sind`, `cospi`, `sincospi`.

source
`Base.Math.cospi` — Function

    cospi(x)

Compute $\cos(\pi x)$ more accurately than `cos(pi*x)`, especially for large `x`.

source
`Base.Math.sincospi` — Function

    sincospi(x)

Simultaneously compute `sinpi(x)` and `cospi(x)` (the sine and cosine of `π*x`, where `x` is in radians), returning a tuple `(sine, cosine)`.

Julia 1.6

This function requires Julia 1.6 or later.

See also: `cispi`, `sincosd`, `sinpi`.

source
`Base.sinh` — Method

    sinh(x)

Compute hyperbolic sine of `x`.

source
`Base.cosh` — Method

    cosh(x)

Compute hyperbolic cosine of `x`.

source
`Base.tanh` — Method

    tanh(x)

Compute hyperbolic tangent of `x`.

source
`Base.asin` — Method

    asin(x)

Compute the inverse sine of `x`, where the output is in radians.

source
`Base.acos` — Method

    acos(x)

Compute the inverse cosine of `x`, where the output is in radians

source
`Base.atan` — Method

    atan(y)
    atan(y, x)

Compute the inverse tangent of `y` or `y/x`, respectively.

For one argument, this is the angle in radians between the positive x-axis and the point (1, y), returning a value in the interval $[-\pi/2, \pi/2]$.

For two arguments, this is the angle in radians between the positive x-axis and the point (x, y), returning a value in the interval $[-\pi, \pi]$. This corresponds to a standard `atan2` function. Note that by convention `atan(0.0,x)` is defined as $\pi$ and `atan(-0.0,x)` is defined as $-\pi$ when `x < 0`.

source
`Base.Math.asind` — Function

    asind(x)

Compute the inverse sine of `x`, where the output is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.acosd` — Function

    acosd(x)

Compute the inverse cosine of `x`, where the output is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.atand` — Function

    atand(y)
    atand(y,x)

Compute the inverse tangent of `y` or `y/x`, respectively, where the output is in degrees.

source
`Base.Math.sec` — Method

    sec(x)

Compute the secant of `x`, where `x` is in radians.

source
`Base.Math.csc` — Method

    csc(x)

Compute the cosecant of `x`, where `x` is in radians.

source
`Base.Math.cot` — Method

    cot(x)

Compute the cotangent of `x`, where `x` is in radians.

source
`Base.Math.secd` — Function

    secd(x)

Compute the secant of `x`, where `x` is in degrees.

source
`Base.Math.cscd` — Function

    cscd(x)

Compute the cosecant of `x`, where `x` is in degrees.

source
`Base.Math.cotd` — Function

    cotd(x)

Compute the cotangent of `x`, where `x` is in degrees.

source
`Base.Math.asec` — Method

    asec(x)

Compute the inverse secant of `x`, where the output is in radians.

source
`Base.Math.acsc` — Method

    acsc(x)

Compute the inverse cosecant of `x`, where the output is in radians.

source
`Base.Math.acot` — Method

    acot(x)

Compute the inverse cotangent of `x`, where the output is in radians.

source
`Base.Math.asecd` — Function

    asecd(x)

Compute the inverse secant of `x`, where the output is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.acscd` — Function

    acscd(x)

Compute the inverse cosecant of `x`, where the output is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.acotd` — Function

    acotd(x)

Compute the inverse cotangent of `x`, where the output is in degrees. If `x` is a matrix, `x` needs to be a square matrix.

source
`Base.Math.sech` — Method

    sech(x)

Compute the hyperbolic secant of `x`.

source
`Base.Math.csch` — Method

    csch(x)

Compute the hyperbolic cosecant of `x`.

source
`Base.Math.coth` — Method

    coth(x)

Compute the hyperbolic cotangent of `x`.

source
`Base.asinh` — Method

    asinh(x)

Compute the inverse hyperbolic sine of `x`.

source
`Base.acosh` — Method

    acosh(x)

Compute the inverse hyperbolic cosine of `x`.

source
`Base.atanh` — Method

    atanh(x)

Compute the inverse hyperbolic tangent of `x`.

source
`Base.Math.asech` — Method

    asech(x)

Compute the inverse hyperbolic secant of `x`.

source
`Base.Math.acsch` — Method

    acsch(x)

Compute the inverse hyperbolic cosecant of `x`.

source
`Base.Math.acoth` — Method

    acoth(x)

Compute the inverse hyperbolic cotangent of `x`.

source
`Base.Math.sinc` — Function

    sinc(x)

Compute $\sin(\pi x) / (\pi x)$ if $x \neq 0$, and $1$ if $x = 0$.

See also `cosc`, its derivative.

source
`Base.Math.cosc` — Function

    cosc(x)

Compute $\cos(\pi x) / x - \sin(\pi x) / (\pi x^2)$ if $x \neq 0$, and $0$ if $x = 0$. This is the derivative of `sinc(x)`.

source
`Base.Math.deg2rad` — Function

    deg2rad(x)

Convert `x` from degrees to radians.

See also: `rad2deg`, `sind`.

Examples

    julia> deg2rad(90)
    1.5707963267948966

source
`Base.Math.rad2deg` — Function

    rad2deg(x)

Convert `x` from radians to degrees.

Examples

    julia> rad2deg(pi)
    180.0

source
`Base.Math.hypot` — Function

    hypot(x, y)

Compute the hypotenuse $\sqrt{|x|^2+|y|^2}$ avoiding overflow and underflow.

This code is an implementation of the algorithm described in: An Improved Algorithm for `hypot(a,b)` by Carlos F. Borges The article is available online at ArXiv at the link https://arxiv.org/abs/1904.09481

    hypot(x...)

Compute the hypotenuse $\sqrt{\sum |x_i|^2}$ avoiding overflow and underflow.

Examples

    julia> a = Int64(10)^10;

    julia> hypot(a, a)
    1.4142135623730951e10

    julia> √(a^2 + a^2) # a^2 overflows
    ERROR: DomainError with -2.914184810805068e18:
    sqrt will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
    Stacktrace:
    [...]

    julia> hypot(3, 4im)
    5.0

    julia> hypot(-5.7)
    5.7

    julia> hypot(3, 4im, 12.0)
    13.0

source
`Base.log` — Method

    log(x)

Compute the natural logarithm of `x`. Throws `DomainError` for negative `Real` arguments. Use complex negative arguments to obtain complex results.

See also [`log1p`], [`log2`], [`log10`].

Examples

    julia> log(2)
    0.6931471805599453

    julia> log(-3)
    ERROR: DomainError with -3.0:
    log will only return a complex result if called with a complex argument. Try log(Complex(x)).
    Stacktrace:
     [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
    [...]

source
`Base.log` — Method

    log(b,x)

Compute the base `b` logarithm of `x`. Throws `DomainError` for negative `Real` arguments.

Examples

    julia> log(4,8)
    1.5

    julia> log(4,2)
    0.5

    julia> log(-2, 3)
    ERROR: DomainError with -2.0:
    log will only return a complex result if called with a complex argument. Try log(Complex(x)).
    Stacktrace:
     [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
    [...]

    julia> log(2, -3)
    ERROR: DomainError with -3.0:
    log will only return a complex result if called with a complex argument. Try log(Complex(x)).
    Stacktrace:
     [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
    [...]

Note

If `b` is a power of 2 or 10, `log2` or `log10` should be used, as these will typically be faster and more accurate. For example,

    julia> log(100,1000000)
    2.9999999999999996

    julia> log10(1000000)/2
    3.0

source
`Base.log2` — Function

    log2(x)

Compute the logarithm of `x` to base 2. Throws `DomainError` for negative `Real` arguments.

See also: `exp2`, `ldexp`, `ispow2`.

Examples

    julia> log2(4)
    2.0

    julia> log2(10)
    3.321928094887362

    julia> log2(-2)
    ERROR: DomainError with -2.0:
    log2 will only return a complex result if called with a complex argument. Try log2(Complex(x)).
    Stacktrace:
     [1] throw_complex_domainerror(f::Symbol, x::Float64) at ./math.jl:31
    [...]

source
`Base.log10` — Function

    log10(x)

Compute the logarithm of `x` to base 10. Throws `DomainError` for negative `Real` arguments.

Examples

    julia> log10(100)
    2.0

    julia> log10(2)
    0.3010299956639812

    julia> log10(-2)
    ERROR: DomainError with -2.0:
    log10 will only return a complex result if called with a complex argument. Try log10(Complex(x)).
    Stacktrace:
     [1] throw_complex_domainerror(f::Symbol, x::Float64) at ./math.jl:31
    [...]

source
`Base.log1p` — Function

    log1p(x)

Accurate natural logarithm of `1+x`. Throws `DomainError` for `Real` arguments less than -1.

Examples

    julia> log1p(-0.5)
    -0.6931471805599453

    julia> log1p(0)
    0.0

    julia> log1p(-2)
    ERROR: DomainError with -2.0:
    log1p will only return a complex result if called with a complex argument. Try log1p(Complex(x)).
    Stacktrace:
     [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
    [...]

source
`Base.Math.frexp` — Function

    frexp(val)

Return `(x,exp)` such that `x` has a magnitude in the interval $[1/2, 1)$ or 0, and `val` is equal to $x \times 2^{exp}$.

Examples

    julia> frexp(12.8)
    (0.8, 4)

source
`Base.exp` — Method

    exp(x)

Compute the natural base exponential of `x`, in other words $ℯ^x$.

See also `exp2`, `exp10` and `cis`.

Examples

    julia> exp(1.0)
    2.718281828459045

    julia> exp(im * pi) == cis(pi)
    true

source
`Base.exp2` — Function

    exp2(x)

Compute the base 2 exponential of `x`, in other words $2^x$.

See also `ldexp`, `<<`.

Examples

    julia> exp2(5)
    32.0

    julia> 2^5
    32

    julia> exp2(63) > typemax(Int)
    true

source
`Base.exp10` — Function

    exp10(x)

Compute the base 10 exponential of `x`, in other words $10^x$.

Examples

    julia> exp10(2)
    100.0

    julia> 10^2
    100

source
`Base.Math.ldexp` — Function

    ldexp(x, n)

Compute $x \times 2^n$.

Examples

    julia> ldexp(5., 2)
    20.0

source
`Base.Math.modf` — Function

    modf(x)

Return a tuple `(fpart, ipart)` of the fractional and integral parts of a number. Both parts have the same sign as the argument.

Examples

    julia> modf(3.5)
    (0.5, 3.0)

    julia> modf(-3.5)
    (-0.5, -3.0)

source
`Base.expm1` — Function

    expm1(x)

Accurately compute $e^x-1$. It avoids the loss of precision involved in the direct evaluation of exp(x)-1 for small values of x.

Examples

    julia> expm1(1e-16)
    1.0e-16

    julia> exp(1e-16) - 1
    0.0

source
`Base.round` — Method

    round([T,] x, [r::RoundingMode])
    round(x, [r::RoundingMode]; digits::Integer=0, base = 10)
    round(x, [r::RoundingMode]; sigdigits::Integer, base = 10)

Rounds the number `x`.

Without keyword arguments, `x` is rounded to an integer value, returning a value of type `T`, or of the same type of `x` if no `T` is provided. An `InexactError` will be thrown if the value is not representable by `T`, similar to `convert`.

If the `digits` keyword argument is provided, it rounds to the specified number of digits after the decimal place (or before if negative), in base `base`.

If the `sigdigits` keyword argument is provided, it rounds to the specified number of significant digits, in base `base`.

The `RoundingMode` `r` controls the direction of the rounding; the default is `RoundNearest`, which rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer. Note that `round` may give incorrect results if the global rounding mode is changed (see `rounding`).

Examples

    julia> round(1.7)
    2.0

    julia> round(Int, 1.7)
    2

    julia> round(1.5)
    2.0

    julia> round(2.5)
    2.0

    julia> round(pi; digits=2)
    3.14

    julia> round(pi; digits=3, base=2)
    3.125

    julia> round(123.456; sigdigits=2)
    120.0

    julia> round(357.913; sigdigits=4, base=2)
    352.0

Note

Rounding to specified digits in bases other than 2 can be inexact when operating on binary floating point numbers. For example, the `Float64` value represented by `1.15` is actually less than 1.15, yet will be rounded to 1.2.

# Examples

    julia> x = 1.15
    1.15

    julia> @sprintf "%.20f" x
    "1.14999999999999991118"

    julia> x < 115//100
    true

    julia> round(x, digits=1)
    1.2

Extensions

To extend `round` to new numeric types, it is typically sufficient to define `Base.round(x::NewType, r::RoundingMode)`.

source
`Base.Rounding.RoundingMode` — Type

    RoundingMode

A type used for controlling the rounding mode of floating point operations (via `rounding`/`setrounding` functions), or as optional arguments for rounding to the nearest integer (via the `round` function).

Currently supported rounding modes are:

- `RoundNearest` (default)
- `RoundNearestTiesAway`
- `RoundNearestTiesUp`
- `RoundToZero`
- `RoundFromZero` (`BigFloat` only)
- `RoundUp`
- `RoundDown`

source
`Base.Rounding.RoundNearest` — Constant

    RoundNearest

The default rounding mode. Rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer.

source
`Base.Rounding.RoundNearestTiesAway` — Constant

    RoundNearestTiesAway

Rounds to nearest integer, with ties rounded away from zero (C/C++ `round` behaviour).

source
`Base.Rounding.RoundNearestTiesUp` — Constant

    RoundNearestTiesUp

Rounds to nearest integer, with ties rounded toward positive infinity (Java/JavaScript `round` behaviour).

source
`Base.Rounding.RoundToZero` — Constant

    RoundToZero

`round` using this rounding mode is an alias for `trunc`.

source
`Base.Rounding.RoundFromZero` — Constant

    RoundFromZero

Rounds away from zero. This rounding mode may only be used with `T == BigFloat` inputs to `round`.

Examples

    julia> BigFloat("1.0000000000000001", 5, RoundFromZero)
    1.06

source
`Base.Rounding.RoundUp` — Constant

    RoundUp

`round` using this rounding mode is an alias for `ceil`.

source
`Base.Rounding.RoundDown` — Constant

    RoundDown

`round` using this rounding mode is an alias for `floor`.

source
`Base.round` — Method

    round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]])
    round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; digits=, base=10)
    round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; sigdigits=, base=10)

Return the nearest integral value of the same type as the complex-valued `z` to `z`, breaking ties using the specified `RoundingMode`s. The first `RoundingMode` is used for rounding the real components while the second is used for rounding the imaginary components.

Example

    julia> round(3.14 + 4.5im)
    3.0 + 4.0im

source
`Base.ceil` — Function

    ceil([T,] x)
    ceil(x; digits::Integer= [, base = 10])
    ceil(x; sigdigits::Integer= [, base = 10])

`ceil(x)` returns the nearest integral value of the same type as `x` that is greater than or equal to `x`.

`ceil(T, x)` converts the result to type `T`, throwing an `InexactError` if the value is not representable.

Keywords `digits`, `sigdigits` and `base` work as for `round`.

source
`Base.floor` — Function

    floor([T,] x)
    floor(x; digits::Integer= [, base = 10])
    floor(x; sigdigits::Integer= [, base = 10])

`floor(x)` returns the nearest integral value of the same type as `x` that is less than or equal to `x`.

`floor(T, x)` converts the result to type `T`, throwing an `InexactError` if the value is not representable.

Keywords `digits`, `sigdigits` and `base` work as for `round`.

source
`Base.trunc` — Function

    trunc([T,] x)
    trunc(x; digits::Integer= [, base = 10])
    trunc(x; sigdigits::Integer= [, base = 10])

`trunc(x)` returns the nearest integral value of the same type as `x` whose absolute value is less than or equal to the absolute value of `x`.

`trunc(T, x)` converts the result to type `T`, throwing an `InexactError` if the value is not representable.

Keywords `digits`, `sigdigits` and `base` work as for `round`.

See also: `%`, `floor`, `unsigned`, `unsafe_trunc`.

Examples

    julia> trunc(2.22)
    2.0

    julia> trunc(-2.22, digits=1)
    -2.2

    julia> trunc(Int, -2.22)
    -2

source
`Base.unsafe_trunc` — Function

    unsafe_trunc(T, x)

Return the nearest integral value of type `T` whose absolute value is less than or equal to the absolute value of `x`. If the value is not representable by `T`, an arbitrary value will be returned. See also `trunc`.

Examples

    julia> unsafe_trunc(Int, -2.2)
    -2

    julia> unsafe_trunc(Int, NaN)
    -9223372036854775808

source
`Base.min` — Function

    min(x, y, ...)

Return the minimum of the arguments (with respect to `isless`). See also the `minimum` function to take the minimum element from a collection.

Examples

    julia> min(2, 5, 1)
    1

source
`Base.max` — Function

    max(x, y, ...)

Return the maximum of the arguments (with respect to `isless`). See also the `maximum` function to take the maximum element from a collection.

Examples

    julia> max(2, 5, 1)
    5

source
`Base.minmax` — Function

    minmax(x, y)

Return `(min(x,y), max(x,y))`.

See also `extrema` that returns `(minimum(x), maximum(x))`.

Examples

    julia> minmax('c','b')
    ('b', 'c')

source
`Base.Math.clamp` — Function

    clamp(x, lo, hi)

Return `x` if `lo <= x <= hi`. If `x > hi`, return `hi`. If `x < lo`, return `lo`. Arguments are promoted to a common type.

See also `clamp!`, `min`, `max`.

Examples

    julia> clamp.([pi, 1.0, big(10)], 2.0, 9.0)
    3-element Vector{BigFloat}:
     3.141592653589793238462643383279502884197169399375105820974944592307816406286198
     2.0
     9.0

    julia> clamp.([11, 8, 5], 10, 6)  # an example where lo > hi
    3-element Vector{Int64}:
      6
      6
     10

source

    clamp(x, T)::T

Clamp `x` between `typemin(T)` and `typemax(T)` and convert the result to type `T`.

See also `trunc`.

Examples

    julia> clamp(200, Int8)
    127

    julia> clamp(-200, Int8)
    -128

    julia> trunc(Int, 4pi^2)
    39

source

    clamp(x::Integer, r::AbstractUnitRange)

Clamp `x` to lie within range `r`.

Julia 1.6

This method requires at least Julia 1.6.

source
`Base.Math.clamp!` — Function

    clamp!(array::AbstractArray, lo, hi)

Restrict values in `array` to the specified range, in-place. See also `clamp`.

Examples

    julia> row = collect(-4:4)';

    julia> clamp!(row, 0, Inf)
    1×9 adjoint(::Vector{Int64}) with eltype Int64:
     0  0  0  0  0  1  2  3  4

    julia> clamp.((-4:4)', 0, Inf)
    1×9 Matrix{Float64}:
     0.0  0.0  0.0  0.0  0.0  1.0  2.0  3.0  4.0

source
`Base.abs` — Function

    abs(x)

The absolute value of `x`.

When `abs` is applied to signed integers, overflow may occur, resulting in the return of a negative value. This overflow occurs only when `abs` is applied to the minimum representable value of a signed integer. That is, when `x == typemin(typeof(x))`, `abs(x) == x < 0`, not `-x` as might be expected.

See also: `abs2`, `unsigned`, `sign`.

Examples

    julia> abs(-3)
    3

    julia> abs(1 + im)
    1.4142135623730951

    julia> abs(typemin(Int64))
    -9223372036854775808

source
`Base.Checked.checked_abs` — Function

    Base.checked_abs(x)

Calculates `abs(x)`, checking for overflow errors where applicable. For example, standard two's complement signed integers (e.g. `Int`) cannot represent `abs(typemin(Int))`, thus leading to an overflow.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_neg` — Function

    Base.checked_neg(x)

Calculates `-x`, checking for overflow errors where applicable. For example, standard two's complement signed integers (e.g. `Int`) cannot represent `-typemin(Int)`, thus leading to an overflow.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_add` — Function

    Base.checked_add(x, y)

Calculates `x+y`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_sub` — Function

    Base.checked_sub(x, y)

Calculates `x-y`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_mul` — Function

    Base.checked_mul(x, y)

Calculates `x*y`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_div` — Function

    Base.checked_div(x, y)

Calculates `div(x,y)`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_rem` — Function

    Base.checked_rem(x, y)

Calculates `x%y`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_fld` — Function

    Base.checked_fld(x, y)

Calculates `fld(x,y)`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_mod` — Function

    Base.checked_mod(x, y)

Calculates `mod(x,y)`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_cld` — Function

    Base.checked_cld(x, y)

Calculates `cld(x,y)`, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.add_with_overflow` — Function

    Base.add_with_overflow(x, y) -> (r, f)

Calculates `r = x+y`, with the flag `f` indicating whether overflow has occurred.

source
`Base.Checked.sub_with_overflow` — Function

    Base.sub_with_overflow(x, y) -> (r, f)

Calculates `r = x-y`, with the flag `f` indicating whether overflow has occurred.

source
`Base.Checked.mul_with_overflow` — Function

    Base.mul_with_overflow(x, y) -> (r, f)

Calculates `r = x*y`, with the flag `f` indicating whether overflow has occurred.

source
`Base.abs2` — Function

    abs2(x)

Squared absolute value of `x`.

Examples

    julia> abs2(-3)
    9

source
`Base.copysign` — Function

    copysign(x, y) -> z

Return `z` which has the magnitude of `x` and the same sign as `y`.

Examples

    julia> copysign(1, -2)
    -1

    julia> copysign(-1, 2)
    1

source
`Base.sign` — Function

    sign(x)

Return zero if `x==0` and $x/|x|$ otherwise (i.e., ±1 for real `x`).

See also `signbit`, `zero`, `copysign`, `flipsign`.

Examples

    julia> sign(-4.0)
    -1.0

    julia> sign(99)
    1

    julia> sign(-0.0)
    -0.0

    julia> sign(0 + im)
    0.0 + 1.0im

source
`Base.signbit` — Function

    signbit(x)

Returns `true` if the value of the sign of `x` is negative, otherwise `false`.

See also `sign` and `copysign`.

Examples

    julia> signbit(-4)
    true

    julia> signbit(5)
    false

    julia> signbit(5.5)
    false

    julia> signbit(-4.1)
    true

source
`Base.flipsign` — Function

    flipsign(x, y)

Return `x` with its sign flipped if `y` is negative. For example `abs(x) = flipsign(x,x)`.

Examples

    julia> flipsign(5, 3)
    5

    julia> flipsign(5, -3)
    -5

source
`Base.sqrt` — Method

    sqrt(x)

Return $\sqrt{x}$. Throws `DomainError` for negative `Real` arguments. Use complex negative arguments instead. The prefix operator `√` is equivalent to `sqrt`.

See also: `hypot`.

Examples

    julia> sqrt(big(81))
    9.0

    julia> sqrt(big(-81))
    ERROR: DomainError with -81.0:
    NaN result for non-NaN input.
    Stacktrace:
     [1] sqrt(::BigFloat) at ./mpfr.jl:501
    [...]

    julia> sqrt(big(complex(-81)))
    0.0 + 9.0im

    julia> .√(1:4)
    4-element Vector{Float64}:
     1.0
     1.4142135623730951
     1.7320508075688772
     2.0

source

    sqrt(A::AbstractMatrix)

If `A` has no negative real eigenvalues, compute the principal matrix square root of `A`, that is the unique matrix $X$ with eigenvalues having positive real part such that $X^2 = A$. Otherwise, a nonprincipal square root is returned.

If `A` is real-symmetric or Hermitian, its eigendecomposition (`eigen`) is used to compute the square root. For such matrices, eigenvalues λ that appear to be slightly negative due to roundoff errors are treated as if they were zero More precisely, matrices with all eigenvalues `≥ -rtol*(max |λ|)` are treated as semidefinite (yielding a Hermitian square root), with negative eigenvalues taken to be zero. `rtol` is a keyword argument to `sqrt` (in the Hermitian/real-symmetric case only) that defaults to machine precision scaled by `size(A,1)`.

Otherwise, the square root is determined by means of the Björck-Hammarling method [BH83], which computes the complex Schur form (`schur`) and then the complex square root of the triangular factor. If a real square root exists, then an extension of this method [H87] that computes the real Schur form and then the real square root of the quasi-triangular factor is instead used.

Examples

    julia> A = [4 0; 0 4]
    2×2 Matrix{Int64}:
     4  0
     0  4

    julia> sqrt(A)
    2×2 Matrix{Float64}:
     2.0  0.0
     0.0  2.0

source
`Base.isqrt` — Function

    isqrt(n::Integer)

Integer square root: the largest integer `m` such that `m*m <= n`.

    julia> isqrt(5)
    2

source
`Base.Math.cbrt` — Function

    cbrt(x::Real)

Return the cube root of `x`, i.e. $x^{1/3}$. Negative values are accepted (returning the negative real root when $x < 0$).

The prefix operator `∛` is equivalent to `cbrt`.

Examples

    julia> cbrt(big(27))
    3.0

    julia> cbrt(big(-27))
    -3.0

source
`Base.real` — Method

    real(z)

Return the real part of the complex number `z`.

See also: `imag`, `reim`, `complex`, `isreal`, `Real`.

Examples

    julia> real(1 + 3im)
    1

source
`Base.imag` — Function

    imag(z)

Return the imaginary part of the complex number `z`.

See also: `conj`, `reim`, `adjoint`, `angle`.

Examples

    julia> imag(1 + 3im)
    3

source
`Base.reim` — Function

    reim(z)

Return both the real and imaginary parts of the complex number `z`.

Examples

    julia> reim(1 + 3im)
    (1, 3)

source
`Base.conj` — Function

    conj(z)

Compute the complex conjugate of a complex number `z`.

See also: `angle`, `adjoint`.

Examples

    julia> conj(1 + 3im)
    1 - 3im

source
`Base.angle` — Function

    angle(z)

Compute the phase angle in radians of a complex number `z`.

See also: `atan`, `cis`.

Examples

    julia> rad2deg(angle(1 + im))
    45.0

    julia> rad2deg(angle(1 - im))
    -45.0

    julia> rad2deg(angle(-1 - im))
    -135.0

source
`Base.cis` — Function

    cis(z)

Return $\exp(iz)$.

See also `cispi`, `angle`.

Examples

    julia> cis(π) ≈ -1
    true

source

    cis(A::AbstractMatrix)

Compute $\exp(i A)$ for a square matrix $A$.

Julia 1.7

Support for using `cis` with matrices was added in Julia 1.7.

Examples

    julia> cis([π 0; 0 π]) ≈ -I
    true

source
`Base.cispi` — Function

    cispi(z)

Compute $\exp(i\pi x)$ more accurately than `cis(pi*x)`, especially for large `x`.

Examples

    julia> cispi(1)
    -1.0 + 0.0im

    julia> cispi(0.25 + 1im)
    0.030556854645952924 + 0.030556854645952924im

Julia 1.6

This function requires Julia 1.6 or later.

source
`Base.binomial` — Function

    binomial(n::Integer, k::Integer)

The binomial coefficient $\binom{n}{k}$, being the coefficient of the $k$th term in the polynomial expansion of $(1+x)^n$.

If $n$ is non-negative, then it is the number of ways to choose `k` out of `n` items:

\[\binom{n}{k} = \frac{n!}{k! (n-k)!}\]

where $n!$ is the `factorial` function.

If $n$ is negative, then it is defined in terms of the identity

\[\binom{n}{k} = (-1)^k \binom{k-n-1}{k}\]

See also `factorial`.

Examples

    julia> binomial(5, 3)
    10

    julia> factorial(5) ÷ (factorial(5-3) * factorial(3))
    10

    julia> binomial(-5, 3)
    -35

External links

- Binomial coefficient on Wikipedia.

source
`Base.factorial` — Function

    factorial(n::Integer)

Factorial of `n`. If `n` is an `Integer`, the factorial is computed as an integer (promoted to at least 64 bits). Note that this may overflow if `n` is not small, but you can use `factorial(big(n))` to compute the result exactly in arbitrary precision.

See also `binomial`.

Examples

    julia> factorial(6)
    720

    julia> factorial(21)
    ERROR: OverflowError: 21 is too large to look up in the table; consider using `factorial(big(21))` instead
    Stacktrace:
    [...]

    julia> factorial(big(21))
    51090942171709440000

External links

- Factorial on Wikipedia.

source
`Base.gcd` — Function

    gcd(x, y...)

Greatest common (positive) divisor (or zero if all arguments are zero). The arguments may be integer and rational numbers.

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

    julia> gcd(6,9)
    3

    julia> gcd(6,-9)
    3

    julia> gcd(6,0)
    6

    julia> gcd(0,0)
    0

    julia> gcd(1//3,2//3)
    1//3

    julia> gcd(1//3,-2//3)
    1//3

    julia> gcd(1//3,2)
    1//3

    julia> gcd(0, 0, 10, 15)
    5

source
`Base.lcm` — Function

    lcm(x, y...)

Least common (positive) multiple (or zero if any argument is zero). The arguments may be integer and rational numbers.

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

    julia> lcm(2,3)
    6

    julia> lcm(-2,3)
    6

    julia> lcm(0,3)
    0

    julia> lcm(0,0)
    0

    julia> lcm(1//3,2//3)
    2//3

    julia> lcm(1//3,-2//3)
    2//3

    julia> lcm(1//3,2)
    2//1

    julia> lcm(1,3,5,7)
    105

source
`Base.gcdx` — Function

    gcdx(a, b)

Computes the greatest common (positive) divisor of `a` and `b` and their Bézout coefficients, i.e. the integer coefficients `u` and `v` that satisfy $ua+vb = d = gcd(a, b)$. $gcdx(a, b)$ returns $(d, u, v)$.

The arguments may be integer and rational numbers.

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

    julia> gcdx(12, 42)
    (6, -3, 1)

    julia> gcdx(240, 46)
    (2, -9, 47)

Note

Bézout coefficients are not uniquely defined. `gcdx` returns the minimal Bézout coefficients that are computed by the extended Euclidean algorithm. (Ref: D. Knuth, TAoCP, 2/e, p. 325, Algorithm X.) For signed integers, these coefficients `u` and `v` are minimal in the sense that $|u| < |y/d|$ and $|v| < |x/d|$. Furthermore, the signs of `u` and `v` are chosen so that `d` is positive. For unsigned integers, the coefficients `u` and `v` might be near their `typemax`, and the identity then holds only via the unsigned integers' modulo arithmetic.

source
`Base.ispow2` — Function

    ispow2(n::Number) -> Bool

Test whether `n` is an integer power of two.

See also `count_ones`, `prevpow`, `nextpow`.

Examples

    julia> ispow2(4)
    true

    julia> ispow2(5)
    false

    julia> ispow2(4.5)
    false

    julia> ispow2(0.25)
    true

    julia> ispow2(1//8)
    true

Julia 1.6

Support for non-`Integer` arguments was added in Julia 1.6.

source
`Base.nextpow` — Function

    nextpow(a, x)

The smallest `a^n` not less than `x`, where `n` is a non-negative integer. `a` must be greater than 1, and `x` must be greater than 0.

See also `prevpow`.

Examples

    julia> nextpow(2, 7)
    8

    julia> nextpow(2, 9)
    16

    julia> nextpow(5, 20)
    25

    julia> nextpow(4, 16)
    16

source
`Base.prevpow` — Function

    prevpow(a, x)

The largest `a^n` not greater than `x`, where `n` is a non-negative integer. `a` must be greater than 1, and `x` must not be less than 1.

See also `nextpow`, `isqrt`.

Examples

    julia> prevpow(2, 7)
    4

    julia> prevpow(2, 9)
    8

    julia> prevpow(5, 20)
    5

    julia> prevpow(4, 16)
    16

source
`Base.nextprod` — Function

    nextprod(factors::Union{Tuple,AbstractVector}, n)

Next integer greater than or equal to `n` that can be written as $\prod k_i^{p_i}$ for integers $p_1$, $p_2$, etcetera, for factors $k_i$ in `factors`.

Examples

    julia> nextprod((2, 3), 105)
    108

    julia> 2^2 * 3^3
    108

Julia 1.6

The method that accepts a tuple requires Julia 1.6 or later.

source
`Base.invmod` — Function

    invmod(n, m)

Take the inverse of `n` modulo `m`: `y` such that $n y = 1 \pmod m$, and $div(y,m) = 0$. This will throw an error if $m = 0$, or if $gcd(n,m) \neq 1$.

Examples

    julia> invmod(2,5)
    3

    julia> invmod(2,3)
    2

    julia> invmod(5,6)
    5

source
`Base.powermod` — Function

    powermod(x::Integer, p::Integer, m)

Compute $x^p \pmod m$.

Examples

    julia> powermod(2, 6, 5)
    4

    julia> mod(2^6, 5)
    4

    julia> powermod(5, 2, 20)
    5

    julia> powermod(5, 2, 19)
    6

    julia> powermod(5, 3, 19)
    11

source
`Base.ndigits` — Function

    ndigits(n::Integer; base::Integer=10, pad::Integer=1)

Compute the number of digits in integer `n` written in base `base` (`base` must not be in `[-1, 0, 1]`), optionally padded with zeros to a specified size (the result will never be less than `pad`).

See also `digits`, `count_ones`.

Examples

    julia> ndigits(12345)
    5

    julia> ndigits(1022, base=16)
    3

    julia> string(1022, base=16)
    "3fe"

    julia> ndigits(123, pad=5)
    5

    julia> ndigits(-123)
    3

source
`Base.add_sum` — Function

    Base.add_sum(x, y)

The reduction operator used in `sum`. The main difference from `+` is that small integers are promoted to `Int`/`UInt`.

source
`Base.widemul` — Function

    widemul(x, y)

Multiply `x` and `y`, giving the result as a larger type.

See also `promote`, `Base.add_sum`.

Examples

    julia> widemul(Float32(3.0), 4.0) isa BigFloat
    true

    julia> typemax(Int8) * typemax(Int8)
    1

    julia> widemul(typemax(Int8), typemax(Int8))  # == 127^2
    16129

source
`Base.Math.evalpoly` — Function

    evalpoly(x, p)

Evaluate the polynomial $\sum_k x^{k-1} p[k]$ for the coefficients `p[1]`, `p[2]`, ...; that is, the coefficients are given in ascending order by power of `x`. Loops are unrolled at compile time if the number of coefficients is statically known, i.e. when `p` is a `Tuple`. This function generates efficient code using Horner's method if `x` is real, or using a Goertzel-like [DK62] algorithm if `x` is complex.

Julia 1.4

This function requires Julia 1.4 or later.

Example

    julia> evalpoly(2, (1, 2, 3))
    17

source
`Base.Math.@evalpoly` — Macro

    @evalpoly(z, c...)

Evaluate the polynomial $\sum_k z^{k-1} c[k]$ for the coefficients `c[1]`, `c[2]`, ...; that is, the coefficients are given in ascending order by power of `z`. This macro expands to efficient inline code that uses either Horner's method or, for complex `z`, a more efficient Goertzel-like algorithm.

See also `evalpoly`.

Examples

    julia> @evalpoly(3, 1, 0, 1)
    10

    julia> @evalpoly(2, 1, 0, 1)
    5

    julia> @evalpoly(2, 1, 1, 1)
    7

source
`Base.FastMath.@fastmath` — Macro

    @fastmath expr

Execute a transformed version of the expression, which calls functions that may violate strict IEEE semantics. This allows the fastest possible operation, but results are undefined – be careful when doing this, as it may change numerical results.

This sets the LLVM Fast-Math flags, and corresponds to the `-ffast-math` option in clang. See the notes on performance annotations for more details.

Examples

    julia> @fastmath 1+2
    3

    julia> @fastmath(sin(3))
    0.1411200080598672

source

## Customizable binary operators

Some unicode characters can be used to define new binary operators that support infix notation. For example `⊗(x,y) = kron(x,y)` defines the `⊗` (otimes) function to be the Kronecker product, and one can call it as binary operator using infix syntax: `C = A ⊗ B` as well as with the usual prefix syntax `C = ⊗(A,B)`.

Other characters that support such extensions include \odot `⊙` and \oplus `⊕`

The complete list is in the parser code: https://github.com/JuliaLang/julia/blob/master/src/julia-parser.scm

Those that are parsed like `*` (in terms of precedence) include `* / ÷ % & ⋅ ∘ × |\\| ∩ ∧ ⊗ ⊘ ⊙ ⊚ ⊛ ⊠ ⊡ ⊓ ∗ ∙ ∤ ⅋ ≀ ⊼ ⋄ ⋆ ⋇ ⋉ ⋊ ⋋ ⋌ ⋏ ⋒ ⟑ ⦸ ⦼ ⦾ ⦿ ⧶ ⧷ ⨇ ⨰ ⨱ ⨲ ⨳ ⨴ ⨵ ⨶ ⨷ ⨸ ⨻ ⨼ ⨽ ⩀ ⩃ ⩄ ⩋ ⩍ ⩎ ⩑ ⩓ ⩕ ⩘ ⩚ ⩜ ⩞ ⩟ ⩠ ⫛ ⊍ ▷ ⨝ ⟕ ⟖ ⟗` and those that are parsed like `+` include `+ - |\|| ⊕ ⊖ ⊞ ⊟ |++| ∪ ∨ ⊔ ± ∓ ∔ ∸ ≏ ⊎ ⊻ ⊽ ⋎ ⋓ ⧺ ⧻ ⨈ ⨢ ⨣ ⨤ ⨥ ⨦ ⨧ ⨨ ⨩ ⨪ ⨫ ⨬ ⨭ ⨮ ⨹ ⨺ ⩁ ⩂ ⩅ ⩊ ⩌ ⩏ ⩐ ⩒ ⩔ ⩖ ⩗ ⩛ ⩝ ⩡ ⩢ ⩣` There are many others that are related to arrows, comparisons, and powers.

- BH83Åke Björck and Sven Hammarling, "A Schur method for the square root of a matrix", Linear Algebra and its Applications, 52-53, 1983, 127-140. doi:10.1016/0024-3795(83)80010-X
- H87Nicholas J. Higham, "Computing real square roots of a real matrix", Linear Algebra and its Applications, 88-89, 1987, 405-430. doi:10.1016/0024-3795(87)90118-2
- DK62Donald Knuth, Art of Computer Programming, Volume 2: Seminumerical Algorithms, Sec. 4.6.4.

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This document was generated with Documenter.jl on Wednesday 7 July 2021. Using Julia version 1.7.0-beta3.0.