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.(Int8[-128 -127 -126 0 126 127])  # overflow at typemin(Int8)
1×6 Matrix{Int8}:
 -128  127  126  0  126  127

julia> maximum(abs, [1, -2, 3, -4])
4

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

log(x)

Compute the natural logarithm of x.

Throws DomainError for negative Real arguments. Use complex arguments to obtain complex results. Has a branch cut along the negative real axis, for which -0.0im is taken to be below the axis.

See also ℯ, log1p, log2, log10.

Examples

julia> log(2)
0.6931471805599453

julia> log(-3)
ERROR: DomainError with -3.0:
log was called with a negative real argument but 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(-3 + 0im)
1.0986122886681098 + 3.141592653589793im

julia> log(-3 - 0.0im)
1.0986122886681098 - 3.141592653589793im

julia> log.(exp.(-1:1))
3-element Vector{Float64}:
 -1.0
  0.0
  1.0

sqrt(x)

Return $\sqrt{x}$.

Throws DomainError for negative Real arguments. Use complex negative arguments instead. Note that sqrt has a branch cut along the negative real axis.

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> sqrt(-81 - 0.0im)  # -0.0im is below the branch cut
0.0 - 9.0im

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

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 truncated value is not representable a T.

Keywords digits, sigdigits and base work as for round.

To support trunc for a new type, define Base.round(x::NewType, ::RoundingMode{:ToZero}).

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

min(x, y, ...)

Return the minimum of the arguments, with respect to isless. If any of the arguments is missing, return missing. See also the minimum function to take the minimum element from a collection.

Examples

julia> min(2, 5, 1)
1

julia> min(4, missing, 6)
missing

max(x, y, ...)

Return the maximum of the arguments, with respect to isless. If any of the arguments is missing, return missing. See also the maximum function to take the maximum element from a collection.

Examples

julia> max(2, 5, 1)
5

julia> max(5, missing, 6)
missing

sum(A::AbstractArray; dims)

Sum elements of an array over the given dimensions.

Examples

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

julia> sum(A, dims=1)
1×2 Matrix{Int64}:
 4  6

julia> sum(A, dims=2)
2×1 Matrix{Int64}:
 3
 7

sort(A; dims::Integer, alg::Algorithm=defalg(A), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort a multidimensional array A along the given dimension. See sort! for a description of possible keyword arguments.

To sort slices of an array, refer to sortslices.

Examples

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

julia> sort(A, dims = 1)
2×2 Matrix{Int64}:
 1  2
 4  3

julia> sort(A, dims = 2)
2×2 Matrix{Int64}:
 3  4
 1  2

sin(x)

Compute sine of x, where x is in radians.

See also sind, sinpi, sincos, cis, asin.

Examples

julia> round.(sin.(range(0, 2pi, length=9)'), digits=3)
1×9 Matrix{Float64}:
 0.0  0.707  1.0  0.707  0.0  -0.707  -1.0  -0.707  -0.0

julia> sind(45)
0.7071067811865476

julia> sinpi(1/4)
0.7071067811865475

julia> round.(sincos(pi/6), digits=3)
(0.5, 0.866)

julia> round(cis(pi/6), digits=3)
0.866 + 0.5im

julia> round(exp(im*pi/6), digits=3)
0.866 + 0.5im

cos(x)

Compute cosine of x, where x is in radians.

See also cosd, cospi, sincos, cis.

tan(x)

Compute tangent of x, where x is in radians.

asin(x)

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

See also asind for output in degrees.

Examples

julia> asin.((0, 1/2, 1))
(0.0, 0.5235987755982989, 1.5707963267948966)

julia> asind.((0, 1/2, 1))
(0.0, 30.000000000000004, 90.0)

acos(x)

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

atan(y)
atan(y, x)

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

For one real 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.

See also atand for degrees.

Examples

julia> rad2deg(atan(-1/√3))
-30.000000000000004

julia> rad2deg(atan(-1, √3))
-30.000000000000004

julia> rad2deg(atan(1, -√3))
150.0

asinh(x)

Compute the inverse hyperbolic sine of x.

acosh(x)

Compute the inverse hyperbolic cosine of x.

atanh(x)

Compute the inverse hyperbolic tangent of x.

mean(A::AbstractArray; dims)

Compute the mean of an array over the given dimensions.

Examples

julia> using Statistics

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

julia> mean(A, dims=1)
1×2 Matrix{Float64}:
 2.0  3.0

julia> mean(A, dims=2)
2×1 Matrix{Float64}:
 1.5
 3.5

cloglog(x)

Compute the complementary log-log, log(-log(1 - x)).

cexpexp(x)

Compute the complementary double exponential, 1 - exp(-exp(x)).

logit(x)

The logit or log-odds transformation, defined as

\[\operatorname{logit}(x) = \log\left(\frac{x}{1-x}\right)\]

for $0 < x < 1$.

Its inverse is the logistic function.

logistic(x)

The logistic sigmoid function mapping a real number to a value in the interval $[0,1]$,

\[\sigma(x) = \frac{1}{e^{-x} + 1} = \frac{e^x}{1+e^x}.\]

Its inverse is the logit function.

equals(x, y)

Returns 1 if $x$ is equal to $y$, 0 otherwise.

inprod(a, b)

Inner product of $a$ and $b$.

inverse(m::AbstractMatrix)

Inverse of matrix $\mathbf{m}$.

logdet(::AbstractMatrix)

Logarithm of the determinant of matrix $\mathbf{v}$.

logfact(x)

Logarithm of the factorial of $x$.

loggam(x)

Logarithm of the gamma function of $x$.

icloglog(x)

Inverse complementary log-log function of $x$. Alias for cexpexp(x).

mexp(x::AbstractMatrix)

Matrix exponential of $\mathbf{x}$.

phi(x)

Cumulative distribution function (CDF) of the standard normal distribution evaluated at $x$.

pow(a, b)

Return $a$ raised to the power of $b$.

rank(v::AbstractVector, i::Integer)

Return the rank of the $i$-th element of $\mathbf{v}$.

ranked(v::AbstractVector, i::Integer)

Return the $i$-th element of $\mathbf{v}$ sorted in ascending order.

sd(v::AbstractVector)

Return the standard deviation of the input vector $\mathbf{v}$.

softplus(x)

Return the softplus function of x, defined as $\log(1 + \exp(x))$.

_step(x)

Return 1 if $x$ is greater than 0, and 0 otherwise.

arcsin(x)

See asin.

arcsinh(x)

See asinh.

arccos(x)

See acos.

arccosh(x)

See acosh.

arctan(x)

See atan.

arctanh(x)

See atanh.