`Base.:-`

Method
-(x)

Unary minus operator.

source`Base.:+`

Function
+(x, y...)

Addition operator. `x+y+z+...`

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

.

`Base.:-`

Method
-(x, y)

Subtraction operator.

source`Base.:*`

Method
*(x, y...)

Multiplication operator. `x*y*z*...`

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

.

`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.

`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.

julia> 3 \ 6 2.0 julia> inv(3) * 6 2.0 julia> A = [1 2; 3 4]; x = [5, 6]; julia> A \ x 2-element Array{Float64,1}: -4.0 4.5 julia> inv(A) * x 2-element Array{Float64,1}: -4.0 4.5source

`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.)

julia> 3^5 243 julia> A = [1 2; 3 4] 2×2 Array{Int64,2}: 1 2 3 4 julia> A^3 2×2 Array{Int64,2}: 37 54 81 118source

`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`

.

`Base.muladd`

Function
muladd(x, y, z)

Combined multiply-add, computes `x*y+z`

in an efficient manner. This may on some systems be equivalent to `x*y+z`

, or to `fma(x,y,z)`

. `muladd`

is used to improve performance. See `fma`

.

**Example**

julia> muladd(3, 2, 1) 7 julia> 3 * 2 + 1 7source

`Base.div`

Function
div(x, y) ÷(x, y)

The quotient from Euclidean division. Computes `x/y`

, truncated to an integer.

julia> 9 ÷ 4 2 julia> -5 ÷ 3 -1source

`Base.fld`

Function
fld(x, y)

Largest integer less than or equal to `x/y`

.

julia> fld(7.3,5.5) 1.0source

`Base.cld`

Function
cld(x, y)

Smallest integer larger than or equal to `x/y`

.

julia> cld(5.5,2.2) 3.0source

`Base.mod`

Function
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`

.

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) 3.0source

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`

.

julia> 129 % Int8 -127source

`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.

julia> x = 15; y = 4; julia> x % y 3 julia> x == div(x, y) * y + rem(x, y) truesource

`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.if

`r == RoundToZero`

, then the result is in the interval $[0, 2π]$ if`x`

is positive,. or $[-2π, 0]$ otherwise.if

`r == RoundDown`

, then the result is in the interval $[0, 2π]$.if

`r == RoundUp`

, then the result is in the interval $[-2π, 0]$.

**Example**

julia> rem2pi(7pi/4, RoundNearest) -0.7853981633974485 julia> rem2pi(7pi/4, RoundDown) 5.497787143782138source

`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`

.

**Example**

julia> mod2pi(9*pi/4) 0.7853981633974481source

`Base.divrem`

Function
divrem(x, y)

The quotient and remainder from Euclidean division. Equivalent to `(div(x,y), rem(x,y))`

or `(x÷y, x%y)`

.

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. Equivalent to `(fld(x,y), mod(x,y))`

.

`Base.fld1`

Function
fld1(x, y)

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

See also: `mod1`

.

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) truesource

`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`

.

julia> mod1(4, 2) 2 julia> mod1(4, 3) 1source

`Base.fldmod1`

Function
fldmod1(x, y)

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

.

`Base.://`

Function
//(num, den)

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

result.

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

`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`

. If `T`

is not provided, it defaults to `Int`

.

julia> rationalize(5.6) 28//5 julia> a = rationalize(BigInt, 10.3) 103//10 julia> typeof(numerator(a)) BigIntsource

`Base.numerator`

Function
numerator(x)

Numerator of the rational representation of `x`

.

julia> numerator(2//3) 2 julia> numerator(4) 4source

`Base.denominator`

Function
denominator(x)

Denominator of the rational representation of `x`

.

julia> denominator(2//3) 3 julia> denominator(4) 1source

`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`

.

julia> Int8(3) << 2 12 julia> bits(Int8(3)) "00000011" julia> bits(Int8(12)) "00001100"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 BitArray{1}: true false true false false julia> B << 1 5-element BitArray{1}: false true false false false julia> B << -1 5-element BitArray{1}: false true false true falsesource

`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`

.

julia> Int8(13) >> 2 3 julia> bits(Int8(13)) "00001101" julia> bits(Int8(3)) "00000011" julia> Int8(-14) >> 2 -4 julia> bits(Int8(-14)) "11110010" julia> bits(Int8(-4)) "11111100"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`

.

**Example**

julia> B = BitVector([true, false, true, false, false]) 5-element BitArray{1}: true false true false false julia> B >> 1 5-element BitArray{1}: false true false true false julia> B >> -1 5-element BitArray{1}: false true false false falsesource

`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)`

.

julia> Int8(-14) >>> 2 60 julia> bits(Int8(-14)) "11110010" julia> bits(Int8(60)) "00111100"

`BigInt`

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

.

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

Unsigned right bitshift operator, `B >>> n`

. Equivalent to `B >> n`

. See `>>`

for details and examples.

`Base.colon`

Function
colon(start, [step], stop)

Called by `:`

syntax for constructing ranges.

julia> colon(1, 2, 5) 1:2:5source

:(start, [step], stop)

Range operator. `a:b`

constructs a range from `a`

to `b`

with a step size of 1, and `a:s:b`

is similar but uses a step size of `s`

. These syntaxes call the function `colon`

. The colon is also used in indexing to select whole dimensions.

`Base.range`

Function
range(start, [step], length)

Construct a range by length, given a starting value and optional step (defaults to 1).

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.

`Base.StepRangeLen`

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

A range `r`

where `r[i]`

produces values of type `T`

, parametrized 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.

`Base.:==`

Function
==(x, y)

Generic equality operator, giving a single `Bool`

result. 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.

Follows IEEE semantics for floating-point numbers.

Collections should generally implement `==`

by calling `==`

recursively on all contents.

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.

source`Base.:!=`

Function
!=(x, y) ≠(x,y)

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

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

instead.

julia> 3 != 2 true julia> "foo" ≠ "foo" falsesource

`Base.:!==`

Function
!==(x, y) ≢(x,y)

Equivalent to `!(x === y)`

.

julia> a = [1, 2]; b = [1, 2]; julia> a ≢ b true julia> a ≢ a falsesource

`Base.:<`

Function
<(x, y)

Less-than comparison operator. New numeric types should implement this function for two arguments of the new type. Because of the behavior of floating-point NaN values, `<`

implements a partial order. Types with a canonical partial order should implement `<`

, and types with a canonical total order should implement `isless`

.

julia> 'a' < 'b' true julia> "abc" < "abd" true julia> 5 < 3 falsesource

`Base.:<=`

Function
<=(x, y) ≤(x,y)

Less-than-or-equals comparison operator.

julia> 'a' <= 'b' true julia> 7 ≤ 7 ≤ 9 true julia> "abc" ≤ "abc" true julia> 5 <= 3 falsesource

`Base.:>`

Function
>(x, y)

Greater-than comparison operator. Generally, new types should implement `<`

instead of this function, and rely on the fallback definition `>(x, y) = y < x`

.

julia> 'a' > 'b' false julia> 7 > 3 > 1 true julia> "abc" > "abd" false julia> 5 > 3 truesource

`Base.:>=`

Function
>=(x, y) ≥(x,y)

Greater-than-or-equals comparison operator.

julia> 'a' >= 'b' false julia> 7 ≥ 7 ≥ 3 true julia> "abc" ≥ "abc" true julia> 5 >= 3 truesource

`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`

. For floating-point numbers, uses `<`

but throws an error for unordered arguments.

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

`Base.:~`

Function
~(x)

Bitwise not.

julia> ~4 -5 julia> ~10 -11 julia> ~true falsesource

`Base.:&`

Function
&(x, y)

Bitwise and.

julia> 4 & 10 0 julia> 4 & 12 4source

`Base.:|`

Function
|(x, y)

Bitwise or.

julia> 4 | 10 14 julia> 4 | 1 5source

`Base.xor`

Function
xor(x, y) ⊻(x, y)

Bitwise exclusive or of `x`

and `y`

. 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.

julia> [true; true; false] .⊻ [true; false; false] 3-element BitArray{1}: false true falsesource

`Base.:!`

Function
!(x)

Boolean not.

julia> !true false julia> !false true julia> .![true false true] 1×3 BitArray{2}: false true falsesource

!f::Function

Predicate function negation: when the argument of `!`

is a function, it returns a function which computes the boolean negation of `f`

. Example:

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

`&&`

Keyword
x && y

Short-circuiting boolean AND.

source`||`

Keyword
x || y

Short-circuiting boolean OR.

source`Base.isapprox`

Function
isapprox(x, y; rtol::Real=sqrt(eps), atol::Real=0, nans::Bool=false, norm::Function)

Inexact equality comparison: `true`

if `norm(x-y) <= 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, `rtol`

defaults to `sqrt(eps(typeof(real(x-y))))`

. This corresponds to requiring equality of about half of the significand digits. For other types, `rtol`

defaults to zero.

`x`

and `y`

may also be arrays of numbers, in which case `norm`

defaults to `vecnorm`

but may be changed by passing a `norm::Function`

keyword argument. (For numbers, `norm`

is the same thing as `abs`

.) 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)`

.

julia> 0.1 ≈ (0.1 - 1e-10) true julia> isapprox(10, 11; atol = 2) true julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) truesource

`Base.sin`

Function
sin(x)

Compute sine of `x`

, where `x`

is in radians.

`Base.cos`

Function
cos(x)

Compute cosine of `x`

, where `x`

is in radians.

`Base.tan`

Function
tan(x)

Compute tangent of `x`

, where `x`

is in radians.

`Base.Math.sind`

Function
sind(x)

Compute sine of `x`

, where `x`

is in degrees.

`Base.Math.cosd`

Function
cosd(x)

Compute cosine of `x`

, where `x`

is in degrees.

`Base.Math.tand`

Function
tand(x)

Compute tangent of `x`

, where `x`

is in degrees.

`Base.Math.sinpi`

Function
sinpi(x)

Compute $\sin(\pi x)$ more accurately than `sin(pi*x)`

, especially for large `x`

.

`Base.Math.cospi`

Function
cospi(x)

Compute $\cos(\pi x)$ more accurately than `cos(pi*x)`

, especially for large `x`

.

`Base.sinh`

Function
sinh(x)

Compute hyperbolic sine of `x`

.

`Base.cosh`

Function
cosh(x)

Compute hyperbolic cosine of `x`

.

`Base.tanh`

Function
tanh(x)

Compute hyperbolic tangent of `x`

.

`Base.asin`

Function
asin(x)

Compute the inverse sine of `x`

, where the output is in radians.

`Base.acos`

Function
acos(x)

Compute the inverse cosine of `x`

, where the output is in radians

`Base.atan`

Function
atan(x)

Compute the inverse tangent of `x`

, where the output is in radians.

`Base.Math.atan2`

Function
atan2(y, x)

Compute the inverse tangent of `y/x`

, using the signs of both `x`

and `y`

to determine the quadrant of the return value.

`Base.Math.asind`

Function
asind(x)

Compute the inverse sine of `x`

, where the output is in degrees.

`Base.Math.acosd`

Function
acosd(x)

Compute the inverse cosine of `x`

, where the output is in degrees.

`Base.Math.atand`

Function
atand(x)

Compute the inverse tangent of `x`

, where the output is in degrees.

`Base.Math.sec`

Function
sec(x)

Compute the secant of `x`

, where `x`

is in radians.

`Base.Math.csc`

Function
csc(x)

Compute the cosecant of `x`

, where `x`

is in radians.

`Base.Math.cot`

Function
cot(x)

Compute the cotangent of `x`

, where `x`

is in radians.

`Base.Math.secd`

Function
secd(x)

Compute the secant of `x`

, where `x`

is in degrees.

`Base.Math.cscd`

Function
cscd(x)

Compute the cosecant of `x`

, where `x`

is in degrees.

`Base.Math.cotd`

Function
cotd(x)

Compute the cotangent of `x`

, where `x`

is in degrees.

`Base.Math.asec`

Function
asec(x)

Compute the inverse secant of `x`

, where the output is in radians.

`Base.Math.acsc`

Function
acsc(x)

Compute the inverse cosecant of `x`

, where the output is in radians.

`Base.Math.acot`

Function
acot(x)

Compute the inverse cotangent of `x`

, where the output is in radians.

`Base.Math.asecd`

Function
asecd(x)

Compute the inverse secant of `x`

, where the output is in degrees.

`Base.Math.acscd`

Function
acscd(x)

Compute the inverse cosecant of `x`

, where the output is in degrees.

`Base.Math.acotd`

Function
acotd(x)

Compute the inverse cotangent of `x`

, where the output is in degrees.

`Base.Math.sech`

Function
sech(x)

Compute the hyperbolic secant of `x`

`Base.Math.csch`

Function
csch(x)

Compute the hyperbolic cosecant of `x`

.

`Base.Math.coth`

Function
coth(x)

Compute the hyperbolic cotangent of `x`

.

`Base.asinh`

Function
asinh(x)

Compute the inverse hyperbolic sine of `x`

.

`Base.acosh`

Function
acosh(x)

Compute the inverse hyperbolic cosine of `x`

.

`Base.atanh`

Function
atanh(x)

Compute the inverse hyperbolic tangent of `x`

.

`Base.Math.asech`

Function
asech(x)

Compute the inverse hyperbolic secant of `x`

.

`Base.Math.acsch`

Function
acsch(x)

Compute the inverse hyperbolic cosecant of `x`

.

`Base.Math.acoth`

Function
acoth(x)

Compute the inverse hyperbolic cotangent of `x`

.

`Base.Math.sinc`

Function
sinc(x)

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

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)`

.

`Base.Math.deg2rad`

Function
deg2rad(x)

Convert `x`

from degrees to radians.

julia> deg2rad(90) 1.5707963267948966source

`Base.Math.rad2deg`

Function
rad2deg(x)

Convert `x`

from radians to degrees.

julia> rad2deg(pi) 180.0source

`Base.Math.hypot`

Function
hypot(x, y)

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

**Examples**

julia> a = 10^10; julia> hypot(a, a) 1.4142135623730951e10 julia> √(a^2 + a^2) # a^2 overflows ERROR: DomainError: sqrt will only return a complex result if called with a complex argument. Try sqrt(complex(x)). Stacktrace: [1] sqrt(::Int64) at ./math.jl:434source

hypot(x...)

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

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.

There is an experimental variant in the `Base.Math.JuliaLibm`

module, which is typically faster and more accurate.

`Base.log`

Method
log(b,x)

Compute the base `b`

logarithm of `x`

. Throws `DomainError`

for negative `Real`

arguments.

julia> log(4,8) 1.5 julia> log(4,2) 0.5

Note

`Base.log2`

Function
log2(x)

Compute the logarithm of `x`

to base 2. Throws `DomainError`

for negative `Real`

arguments.

**Example**

julia> log2(4) 2.0 julia> log2(10) 3.321928094887362source

`Base.log10`

Function
log10(x)

Compute the logarithm of `x`

to base 10. Throws `DomainError`

for negative `Real`

arguments.

**Example**

julia> log10(100) 2.0 julia> log10(2) 0.3010299956639812source

`Base.log1p`

Function
log1p(x)

Accurate natural logarithm of `1+x`

. Throws `DomainError`

for `Real`

arguments less than -1.

There is an experimental variant in the `Base.Math.JuliaLibm`

module, which is typically faster and more accurate.

**Examples**

julia> log1p(-0.5) -0.6931471805599453 julia> log1p(0) 0.0source

`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 $x \times 2^{exp}$.

`Base.exp`

Function
exp(x)

Compute the natural base exponential of `x`

, in other words $e^x$.

`Base.exp2`

Function
exp2(x)

Compute $2^x$.

julia> exp2(5) 32.0source

`Base.exp10`

Function
exp10(x)

Compute $10^x$.

julia> exp10(2) 100.0 julia> exp10(0.2) 1.5848931924611136source

`Base.Math.ldexp`

Function
ldexp(x, n)

Compute $x \times 2^n$.

**Example**

julia> ldexp(5., 2) 20.0source

`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.

**Example**

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

`Base.expm1`

Function
expm1(x)

Accurately compute $e^x-1$.

source`Base.round`

Method
round([T,] x, [digits, [base]], [r::RoundingMode])

Rounds `x`

to an integer value according to the provided `RoundingMode`

, returning a value of the same type as `x`

. When not specifying a rounding mode the global mode will be used (see `rounding`

), which by default is round to the nearest integer (`RoundNearest`

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

julia> round(1.7) 2.0 julia> round(1.5) 2.0 julia> round(2.5) 2.0

The optional `RoundingMode`

argument will change how the number gets rounded.

`round(T, x, [r::RoundingMode])`

converts the result to type `T`

, throwing an `InexactError`

if the value is not representable.

`round(x, digits)`

rounds to the specified number of digits after the decimal place (or before if negative). `round(x, digits, base)`

rounds using a base other than 10.

julia> round(pi, 2) 3.14 julia> round(pi, 3, 2) 3.125

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.

julia> x = 1.15 1.15 julia> @sprintf "%.20f" x "1.14999999999999991118" julia> x < 115//100 true julia> round(x, 1) 1.2

`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)`RoundFromZero`

(`BigFloat`

only)

`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).

`Base.Rounding.RoundNearestTiesUp`

Constant
RoundNearestTiesUp

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

behaviour).

`Base.Rounding.RoundToZero`

Constant
RoundToZero

`round`

using this rounding mode is an alias for `trunc`

.

`Base.Rounding.RoundUp`

Constant
RoundUp

`round`

using this rounding mode is an alias for `ceil`

.

`Base.Rounding.RoundDown`

Constant
RoundDown

`round`

using this rounding mode is an alias for `floor`

.

`Base.round`

Method
round(z, RoundingModeReal, RoundingModeImaginary)

Returns 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.

`Base.ceil`

Function
ceil([T,] x, [digits, [base]])

`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.

`digits`

and `base`

work as for `round`

.

`Base.floor`

Function
floor([T,] x, [digits, [base]])

`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.

`digits`

and `base`

work as for `round`

.

`Base.trunc`

Function
trunc([T,] x, [digits, [base]])

`trunc(x)`

returns the nearest integral value of the same type as `x`

whose absolute value is less than or equal to `x`

.

`trunc(T, x)`

converts the result to type `T`

, throwing an `InexactError`

if the value is not representable.

`digits`

and `base`

work as for `round`

.

`Base.unsafe_trunc`

Function
unsafe_trunc(T, x)

`unsafe_trunc(T, x)`

returns the nearest integral value of type `T`

whose absolute value is less than or equal to `x`

. If the value is not representable by `T`

, an arbitrary value will be returned.

`Base.signif`

Function
signif(x, digits, [base])

Rounds (in the sense of `round`

) `x`

so that there are `digits`

significant digits, under a base `base`

representation, default 10. E.g., `signif(123.456, 2)`

is `120.0`

, and `signif(357.913, 4, 2)`

is `352.0`

.

`Base.min`

Function
min(x, y, ...)

Return the minimum of the arguments. See also the `minimum`

function to take the minimum element from a collection.

julia> min(2, 5, 1) 1source

`Base.max`

Function
max(x, y, ...)

Return the maximum of the arguments. See also the `maximum`

function to take the maximum element from a collection.

julia> max(2, 5, 1) 5source

`Base.minmax`

Function
minmax(x, y)

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

. See also: `extrema`

that returns `(minimum(x), maximum(x))`

.

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

`Base.Math.clamp`

Function
clamp(x, lo, hi)

Return `x`

if `lo <= x <= hi`

. If `x < lo`

, return `lo`

. If `x > hi`

, return `hi`

. Arguments are promoted to a common type.

julia> clamp.([pi, 1.0, big(10.)], 2., 9.) 3-element Array{BigFloat,1}: 3.141592653589793238462643383279502884197169399375105820974944592307816406286198 2.000000000000000000000000000000000000000000000000000000000000000000000000000000 9.000000000000000000000000000000000000000000000000000000000000000000000000000000source

`Base.Math.clamp!`

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

Restrict values in `array`

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

.

`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.

julia> abs(-3) 3 julia> abs(1 + im) 1.4142135623730951 julia> abs(typemin(Int64)) -9223372036854775808source

`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.

`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.

`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.

`Base.abs2`

Function
abs2(x)

Squared absolute value of `x`

.

`Base.copysign`

Function
copysign(x, y) -> z

Return `z`

which has the magnitude of `x`

and the same sign as `y`

.

julia> copysign(1, -2) -1 julia> copysign(-1, 2) 1source

`Base.sign`

Function
sign(x)

Return zero if `x==0`

and $x/|x|$ otherwise (i.e., ±1 for real `x`

).

`Base.signbit`

Function
signbit(x)

Returns `true`

if the value of the sign of `x`

is negative, otherwise `false`

.

julia> signbit(-4) true julia> signbit(5) false julia> signbit(5.5) false julia> signbit(-4.1) truesource

`Base.flipsign`

Function
flipsign(x, y)

Return `x`

with its sign flipped if `y`

is negative. For example `abs(x) = flipsign(x,x)`

.

`Base.sqrt`

Function
sqrt(x)

Return $\sqrt{x}$. Throws `DomainError`

for negative `Real`

arguments. Use complex negative arguments instead. The prefix operator `√`

is equivalent to `sqrt`

.

`Base.isqrt`

Function
isqrt(n::Integer)

Integer square root: the largest integer `m`

such that `m*m <= n`

.

julia> isqrt(5) 2source

`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`

.

julia> cbrt(big(27)) 3.000000000000000000000000000000000000000000000000000000000000000000000000000000source

`Base.real`

Method
real(z)

Return the real part of the complex number `z`

.

julia> real(1 + 3im) 1source

`Base.imag`

Function
imag(z)

Return the imaginary part of the complex number `z`

.

julia> imag(1 + 3im) 3source

`Base.reim`

Function
reim(z)

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

.

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

`Base.conj`

Function
conj(z)

Compute the complex conjugate of a complex number `z`

.

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

conj(v::RowVector)

Returns a `ConjArray`

lazy view of the input, where each element is conjugated.

**Example**

julia> v = [1+im, 1-im].' 1×2 RowVector{Complex{Int64},Array{Complex{Int64},1}}: 1+1im 1-1im julia> conj(v) 1×2 RowVector{Complex{Int64},ConjArray{Complex{Int64},1,Array{Complex{Int64},1}}}: 1-1im 1+1imsource

`Base.angle`

Function
angle(z)

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

.

`Base.cis`

Function
cis(z)

Return $\exp(iz)$.

source`Base.binomial`

Function
binomial(n,k)

Number of ways to choose `k`

out of `n`

items.

`Base.factorial`

Function
factorial(n)

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. If `n`

is not an `Integer`

, `factorial(n)`

is equivalent to `gamma(n+1)`

.

julia> factorial(6) 720 julia> factorial(21) ERROR: OverflowError() [...] julia> factorial(21.0) 5.109094217170944e19 julia> factorial(big(21)) 51090942171709440000source

`Base.gcd`

Function
gcd(x,y)

Greatest common (positive) divisor (or zero if `x`

and `y`

are both zero).

julia> gcd(6,9) 3 julia> gcd(6,-9) 3source

`Base.lcm`

Function
lcm(x,y)

Least common (non-negative) multiple.

julia> lcm(2,3) 6 julia> lcm(-2,3) 6source

`Base.gcdx`

Function
gcdx(x,y)

Computes the greatest common (positive) divisor of `x`

and `y`

and their Bézout coefficients, i.e. the integer coefficients `u`

and `v`

that satisfy $ux+vy = d = gcd(x,y)$. $gcdx(x,y)$ returns $(d,u,v)$.

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.

`Base.ispow2`

Function
ispow2(n::Integer) -> Bool

Test whether `n`

is a power of two.

julia> ispow2(4) true julia> ispow2(5) falsesource

`Base.nextpow2`

Function
nextpow2(n::Integer)

The smallest power of two not less than `n`

. Returns 0 for `n==0`

, and returns `-nextpow2(-n)`

for negative arguments.

julia> nextpow2(16) 16 julia> nextpow2(17) 32source

`Base.prevpow2`

Function
prevpow2(n::Integer)

The largest power of two not greater than `n`

. Returns 0 for `n==0`

, and returns `-prevpow2(-n)`

for negative arguments.

julia> prevpow2(5) 4source

`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.

`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.

`Base.nextprod`

Function
nextprod([k_1, k_2,...], n)

Next integer greater than or equal to `n`

that can be written as $\prod k_i^{p_i}$ for integers $p_1$, $p_2$, etc.

julia> nextprod([2, 3], 105) 108 julia> 2^2 * 3^3 108source

`Base.invmod`

Function
invmod(x,m)

Take the inverse of `x`

modulo `m`

: `y`

such that $x y = 1 \pmod m$, with $div(x,y) = 0$. This is undefined for $m = 0$, or if $gcd(x,m) \neq 1$.

julia> invmod(2,5) 3 julia> invmod(2,3) 2 julia> invmod(5,6) 5source

`Base.powermod`

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

Compute $x^p \pmod m$.

source`Base.Math.gamma`

Function
gamma(x)

Compute the gamma function of `x`

.

`Base.Math.lgamma`

Function
lgamma(x)

Compute the logarithm of the absolute value of `gamma`

for `Real`

`x`

, while for `Complex`

`x`

compute the principal branch cut of the logarithm of `gamma(x)`

(defined for negative `real(x)`

by analytic continuation from positive `real(x)`

).

`Base.Math.lfact`

Function
lfact(x)

Compute the logarithmic factorial of a nonnegative integer `x`

. Equivalent to `lgamma`

of `x + 1`

, but `lgamma`

extends this function to non-integer `x`

.

`Base.Math.beta`

Function
beta(x, y)

Euler integral of the first kind $\operatorname{B}(x,y) = \Gamma(x)\Gamma(y)/\Gamma(x+y)$.

source`Base.Math.lbeta`

Function
lbeta(x, y)

Natural logarithm of the absolute value of the `beta`

function $\log(|\operatorname{B}(x,y)|)$.

`Base.ndigits`

Function
ndigits(n::Integer, b::Integer=10)

Compute the number of digits in integer `n`

written in base `b`

.

`Base.widemul`

Function
widemul(x, y)

Multiply `x`

and `y`

, giving the result as a larger type.

julia> widemul(Float32(3.), 4.) 1.200000000000000000000000000000000000000000000000000000000000000000000000000000e+01source

`Base.Math.@evalpoly`

Macro
@evalpoly(z, c...)

Evaluate the polynomial $\sum_k c[k] z^{k-1}$ 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.

julia> @evalpoly(3, 1, 0, 1) 10 julia> @evalpoly(2, 1, 0, 1) 5 julia> @evalpoly(2, 1, 1, 1) 7source

`Base.mean`

Function
mean(f::Function, v)

Apply the function `f`

to each element of `v`

and take the mean.

julia> mean(√, [1, 2, 3]) 1.3820881233139908 julia> mean([√1, √2, √3]) 1.3820881233139908source

mean(v[, region])

Compute the mean of whole array `v`

, or optionally along the dimensions in `region`

.

Note

Julia does not ignore `NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended.

`Base.mean!`

Function
mean!(r, v)

Compute the mean of `v`

over the singleton dimensions of `r`

, and write results to `r`

.

`Base.std`

Function
std(v[, region]; corrected::Bool=true, mean=nothing)

Compute the sample standard deviation of a vector or array `v`

, optionally along dimensions in `region`

. The algorithm returns an estimator of the generative distribution's standard deviation under the assumption that each entry of `v`

is an IID drawn from that generative distribution. This computation is equivalent to calculating `sqrt(sum((v - mean(v)).^2) / (length(v) - 1))`

. A pre-computed `mean`

may be provided. If `corrected`

is `true`

, then the sum is scaled with `n-1`

, whereas the sum is scaled with `n`

if `corrected`

is `false`

where `n = length(x)`

.

Note

Julia does not ignore `NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended.

`Base.stdm`

Function
stdm(v, m::Number; corrected::Bool=true)

Compute the sample standard deviation of a vector `v`

with known mean `m`

. If `corrected`

is `true`

, then the sum is scaled with `n-1`

, whereas the sum is scaled with `n`

if `corrected`

is `false`

where `n = length(x)`

.

Note

Julia does not ignore `NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended.

`Base.var`

Function
var(v[, region]; corrected::Bool=true, mean=nothing)

Compute the sample variance of a vector or array `v`

, optionally along dimensions in `region`

. The algorithm will return an estimator of the generative distribution's variance under the assumption that each entry of `v`

is an IID drawn from that generative distribution. This computation is equivalent to calculating `sum(abs2, v - mean(v)) / (length(v) - 1)`

. If `corrected`

is `true`

, then the sum is scaled with `n-1`

, whereas the sum is scaled with `n`

if `corrected`

is `false`

where `n = length(x)`

. The mean `mean`

over the region may be provided.

Note

`NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended.

`Base.varm`

Function
varm(v, m[, region]; corrected::Bool=true)

Compute the sample variance of a collection `v`

with known mean(s) `m`

, optionally over `region`

. `m`

may contain means for each dimension of `v`

. If `corrected`

is `true`

, then the sum is scaled with `n-1`

, whereas the sum is scaled with `n`

if `corrected`

is `false`

where `n = length(x)`

.

Note

`NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended.

`Base.middle`

Function
middle(x)

Compute the middle of a scalar value, which is equivalent to `x`

itself, but of the type of `middle(x, x)`

for consistency.

middle(x, y)

Compute the middle of two reals `x`

and `y`

, which is equivalent in both value and type to computing their mean (`(x + y) / 2`

).

middle(range)

Compute the middle of a range, which consists of computing the mean of its extrema. Since a range is sorted, the mean is performed with the first and last element.

julia> middle(1:10) 5.5source

middle(a)

Compute the middle of an array `a`

, which consists of finding its extrema and then computing their mean.

julia> a = [1,2,3.6,10.9] 4-element Array{Float64,1}: 1.0 2.0 3.6 10.9 julia> middle(a) 5.95source

`Base.median`

Function
median(v[, region])

Compute the median of an entire array `v`

, or, optionally, along the dimensions in `region`

. For an even number of elements no exact median element exists, so the result is equivalent to calculating mean of two median elements.

Note

`NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended.

`Base.median!`

Function
median!(v)

Like `median`

, but may overwrite the input vector.

`Base.quantile`

Function
quantile(v, p; sorted=false)

Compute the quantile(s) of a vector `v`

at a specified probability or vector `p`

. The keyword argument `sorted`

indicates whether `v`

can be assumed to be sorted.

The `p`

should be on the interval [0,1], and `v`

should not have any `NaN`

values.

Quantiles are computed via linear interpolation between the points `((k-1)/(n-1), v[k])`

, for `k = 1:n`

where `n = length(v)`

. This corresponds to Definition 7 of Hyndman and Fan (1996), and is the same as the R default.

Note

Julia does not ignore `NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended. `quantile`

will throw an `ArgumentError`

in the presence of `NaN`

values in the data array.

Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages",

*The American Statistician*, Vol. 50, No. 4, pp. 361-365

`Base.quantile!`

Function
quantile!([q, ] v, p; sorted=false)

Compute the quantile(s) of a vector `v`

at the probabilities `p`

, with optional output into array `q`

(if not provided, a new output array is created). The keyword argument `sorted`

indicates whether `v`

can be assumed to be sorted; if `false`

(the default), then the elements of `v`

may be partially sorted.

The elements of `p`

should be on the interval [0,1], and `v`

should not have any `NaN`

values.

Quantiles are computed via linear interpolation between the points `((k-1)/(n-1), v[k])`

, for `k = 1:n`

where `n = length(v)`

. This corresponds to Definition 7 of Hyndman and Fan (1996), and is the same as the R default.

Note

Julia does not ignore `NaN`

values in the computation. For applications requiring the handling of missing data, the `DataArrays.jl`

package is recommended. `quantile!`

will throw an `ArgumentError`

in the presence of `NaN`

values in the data array.

Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages",

*The American Statistician*, Vol. 50, No. 4, pp. 361-365

`Base.cov`

Function
cov(x[, corrected=true])

Compute the variance of the vector `x`

. If `corrected`

is `true`

(the default) then the sum is scaled with `n-1`

, whereas the sum is scaled with `n`

if `corrected`

is `false`

where `n = length(x)`

.

cov(X[, vardim=1, corrected=true])

Compute the covariance matrix of the matrix `X`

along the dimension `vardim`

. If `corrected`

is `true`

(the default) then the sum is scaled with `n-1`

, whereas the sum is scaled with `n`

if `corrected`

is `false`

where `n = size(X, vardim)`

.

cov(x, y[, corrected=true])

Compute the covariance between the vectors `x`

and `y`

. If `corrected`

is `true`

(the default), computes $\frac{1}{n-1}\sum_{i=1}^n (x_i-\bar x) (y_i-\bar y)^*$ where $*$ denotes the complex conjugate and `n = length(x) = length(y)`

. If `corrected`

is `false`

, computes $rac{1}{n}sum_{i=1}^n (x_i-\bar x) (y_i-\bar y)^*$.

cov(X, Y[, vardim=1, corrected=true])

Compute the covariance between the vectors or matrices `X`

and `Y`

along the dimension `vardim`

. If `corrected`

is `true`

(the default) then the sum is scaled with `n-1`

, whereas the sum is scaled with `n`

if `corrected`

is `false`

where `n = size(X, vardim) = size(Y, vardim)`

.

`Base.cor`

Function
cor(x)

Return the number one.

sourcecor(X[, vardim=1])

Compute the Pearson correlation matrix of the matrix `X`

along the dimension `vardim`

.

cor(x, y)

Compute the Pearson correlation between the vectors `x`

and `y`

.

cor(X, Y[, vardim=1])

Compute the Pearson correlation between the vectors or matrices `X`

and `Y`

along the dimension `vardim`

.

Fast Fourier transform (FFT) functions in Julia are implemented by calling functions from FFTW.

`Base.DFT.fft`

Function
fft(A [, dims])

Performs a multidimensional FFT of the array `A`

. The optional `dims`

argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. Most efficient if the size of `A`

along the transformed dimensions is a product of small primes; see `nextprod()`

. See also `plan_fft()`

for even greater efficiency.

A one-dimensional FFT computes the one-dimensional discrete Fourier transform (DFT) as defined by

\[\operatorname{DFT}(A)[k] = \sum_{n=1}^{\operatorname{length}(A)} \exp\left(-i\frac{2\pi (n-1)(k-1)}{\operatorname{length}(A)} \right) A[n].\]A multidimensional FFT simply performs this operation along each transformed dimension of `A`

.

Note

Julia starts FFTW up with 1 thread by default. Higher performance is usually possible by increasing number of threads. Use

`FFTW.set_num_threads(Sys.CPU_CORES)`

to use as many threads as cores on your system.This performs a multidimensional FFT by default. FFT libraries in other languages such as Python and Octave perform a one-dimensional FFT along the first non-singleton dimension of the array. This is worth noting while performing comparisons. For more details, refer to the Noteworthy Differences from other Languages section of the manual.

`Base.DFT.fft!`

Function
fft!(A [, dims])

Same as `fft`

, but operates in-place on `A`

, which must be an array of complex floating-point numbers.

`Base.DFT.ifft`

Function
ifft(A [, dims])

Multidimensional inverse FFT.

A one-dimensional inverse FFT computes

\[\operatorname{IDFT}(A)[k] = \frac{1}{\operatorname{length}(A)} \sum_{n=1}^{\operatorname{length}(A)} \exp\left(+i\frac{2\pi (n-1)(k-1)} {\operatorname{length}(A)} \right) A[n].\]A multidimensional inverse FFT simply performs this operation along each transformed dimension of `A`

.

`Base.DFT.ifft!`

Function
ifft!(A [, dims])

Same as `ifft`

, but operates in-place on `A`

.

`Base.DFT.bfft`

Function
bfft(A [, dims])

Similar to `ifft`

, but computes an unnormalized inverse (backward) transform, which must be divided by the product of the sizes of the transformed dimensions in order to obtain the inverse. (This is slightly more efficient than `ifft`

because it omits a scaling step, which in some applications can be combined with other computational steps elsewhere.)

`Base.DFT.bfft!`

Function
bfft!(A [, dims])

Same as `bfft`

, but operates in-place on `A`

.

`Base.DFT.plan_fft`

Function
plan_fft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Pre-plan an optimized FFT along given dimensions (`dims`

) of arrays matching the shape and type of `A`

. (The first two arguments have the same meaning as for `fft`

.) Returns an object `P`

which represents the linear operator computed by the FFT, and which contains all of the information needed to compute `fft(A, dims)`

quickly.

To apply `P`

to an array `A`

, use `P * A`

; in general, the syntax for applying plans is much like that of matrices. (A plan can only be applied to arrays of the same size as the `A`

for which the plan was created.) You can also apply a plan with a preallocated output array `Â`

by calling `A_mul_B!(Â, plan, A)`

. (For `A_mul_B!`

, however, the input array `A`

must be a complex floating-point array like the output `Â`

.) You can compute the inverse-transform plan by `inv(P)`

and apply the inverse plan with `P \ Â`

(the inverse plan is cached and reused for subsequent calls to `inv`

or `\`

), and apply the inverse plan to a pre-allocated output array `A`

with `A_ldiv_B!(A, P, Â)`

.

The `flags`

argument is a bitwise-or of FFTW planner flags, defaulting to `FFTW.ESTIMATE`

. e.g. passing `FFTW.MEASURE`

or `FFTW.PATIENT`

will instead spend several seconds (or more) benchmarking different possible FFT algorithms and picking the fastest one; see the FFTW manual for more information on planner flags. The optional `timelimit`

argument specifies a rough upper bound on the allowed planning time, in seconds. Passing `FFTW.MEASURE`

or `FFTW.PATIENT`

may cause the input array `A`

to be overwritten with zeros during plan creation.

`plan_fft!`

is the same as `plan_fft`

but creates a plan that operates in-place on its argument (which must be an array of complex floating-point numbers). `plan_ifft`

and so on are similar but produce plans that perform the equivalent of the inverse transforms `ifft`

and so on.

`Base.DFT.plan_ifft`

Function
plan_ifft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Same as `plan_fft`

, but produces a plan that performs inverse transforms `ifft`

.

`Base.DFT.plan_bfft`

Function
plan_bfft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Same as `plan_fft`

, but produces a plan that performs an unnormalized backwards transform `bfft`

.

`Base.DFT.plan_fft!`

Function
plan_fft!(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Same as `plan_fft`

, but operates in-place on `A`

.

`Base.DFT.plan_ifft!`

Function
plan_ifft!(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Same as `plan_ifft`

, but operates in-place on `A`

.

`Base.DFT.plan_bfft!`

Function
plan_bfft!(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Same as `plan_bfft`

, but operates in-place on `A`

.

`Base.DFT.rfft`

Function
rfft(A [, dims])

Multidimensional FFT of a real array `A`

, exploiting the fact that the transform has conjugate symmetry in order to save roughly half the computational time and storage costs compared with `fft`

. If `A`

has size `(n_1, ..., n_d)`

, the result has size `(div(n_1,2)+1, ..., n_d)`

.

The optional `dims`

argument specifies an iterable subset of one or more dimensions of `A`

to transform, similar to `fft`

. Instead of (roughly) halving the first dimension of `A`

in the result, the `dims[1]`

dimension is (roughly) halved in the same way.

`Base.DFT.irfft`

Function
irfft(A, d [, dims])

Inverse of `rfft`

: for a complex array `A`

, gives the corresponding real array whose FFT yields `A`

in the first half. As for `rfft`

, `dims`

is an optional subset of dimensions to transform, defaulting to `1:ndims(A)`

.

`d`

is the length of the transformed real array along the `dims[1]`

dimension, which must satisfy `div(d,2)+1 == size(A,dims[1])`

. (This parameter cannot be inferred from `size(A)`

since both `2*size(A,dims[1])-2`

as well as `2*size(A,dims[1])-1`

are valid sizes for the transformed real array.)

`Base.DFT.brfft`

Function
brfft(A, d [, dims])

Similar to `irfft`

but computes an unnormalized inverse transform (similar to `bfft`

), which must be divided by the product of the sizes of the transformed dimensions (of the real output array) in order to obtain the inverse transform.

`Base.DFT.plan_rfft`

Function
plan_rfft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Pre-plan an optimized real-input FFT, similar to `plan_fft`

except for `rfft`

instead of `fft`

. The first two arguments, and the size of the transformed result, are the same as for `rfft`

.

`Base.DFT.plan_brfft`

Function
plan_brfft(A, d [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Pre-plan an optimized real-input unnormalized transform, similar to `plan_rfft`

except for `brfft`

instead of `rfft`

. The first two arguments and the size of the transformed result, are the same as for `brfft`

.

`Base.DFT.plan_irfft`

Function
plan_irfft(A, d [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)

Pre-plan an optimized inverse real-input FFT, similar to `plan_rfft`

except for `irfft`

and `brfft`

, respectively. The first three arguments have the same meaning as for `irfft`

.

`Base.DFT.FFTW.dct`

Function
dct(A [, dims])

Performs a multidimensional type-II discrete cosine transform (DCT) of the array `A`

, using the unitary normalization of the DCT. The optional `dims`

argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. Most efficient if the size of `A`

along the transformed dimensions is a product of small primes; see `nextprod`

. See also `plan_dct`

for even greater efficiency.

`Base.DFT.FFTW.dct!`

Function
dct!(A [, dims])

Same as `dct!`

, except that it operates in-place on `A`

, which must be an array of real or complex floating-point values.

`Base.DFT.FFTW.idct`

Function
idct(A [, dims])

Computes the multidimensional inverse discrete cosine transform (DCT) of the array `A`

(technically, a type-III DCT with the unitary normalization). The optional `dims`

argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. Most efficient if the size of `A`

along the transformed dimensions is a product of small primes; see `nextprod`

. See also `plan_idct`

for even greater efficiency.

`Base.DFT.FFTW.idct!`

Function
idct!(A [, dims])

Same as `idct!`

, but operates in-place on `A`

.

`Base.DFT.FFTW.plan_dct`

Function
plan_dct(A [, dims [, flags [, timelimit]]])

Pre-plan an optimized discrete cosine transform (DCT), similar to `plan_fft`

except producing a function that computes `dct`

. The first two arguments have the same meaning as for `dct`

.

`Base.DFT.FFTW.plan_dct!`

Function
plan_dct!(A [, dims [, flags [, timelimit]]])

Same as `plan_dct`

, but operates in-place on `A`

.

`Base.DFT.FFTW.plan_idct`

Function
plan_idct(A [, dims [, flags [, timelimit]]])

Pre-plan an optimized inverse discrete cosine transform (DCT), similar to `plan_fft`

except producing a function that computes `idct`

. The first two arguments have the same meaning as for `idct`

.

`Base.DFT.FFTW.plan_idct!`

Function
plan_idct!(A [, dims [, flags [, timelimit]]])

Same as `plan_idct`

, but operates in-place on `A`

.

`Base.DFT.fftshift`

Method
fftshift(x)

Swap the first and second halves of each dimension of `x`

.

`Base.DFT.fftshift`

Method
fftshift(x,dim)

Swap the first and second halves of the given dimension or iterable of dimensions of array `x`

.

`Base.DFT.ifftshift`

Function
ifftshift(x, [dim])

Undoes the effect of `fftshift`

.

`Base.DSP.filt`

Function
filt(b, a, x, [si])

Apply filter described by vectors `a`

and `b`

to vector `x`

, with an optional initial filter state vector `si`

(defaults to zeros).

`Base.DSP.filt!`

Function
filt!(out, b, a, x, [si])

Same as `filt`

but writes the result into the `out`

argument, which may alias the input `x`

to modify it in-place.

`Base.DSP.deconv`

Function
deconv(b,a) -> c

Construct vector `c`

such that `b = conv(a,c) + r`

. Equivalent to polynomial division.

`Base.DSP.conv`

Function
conv(u,v)

Convolution of two vectors. Uses FFT algorithm.

source`Base.DSP.conv2`

Function
conv2(u,v,A)

2-D convolution of the matrix `A`

with the 2-D separable kernel generated by the vectors `u`

and `v`

. Uses 2-D FFT algorithm.

conv2(B,A)

2-D convolution of the matrix `B`

with the matrix `A`

. Uses 2-D FFT algorithm.

`Base.DSP.xcorr`

Function
xcorr(u,v)

Compute the cross-correlation of two vectors.

sourceThe following functions are defined within the `Base.FFTW`

module.

`Base.DFT.FFTW.r2r`

Function
r2r(A, kind [, dims])

Performs a multidimensional real-input/real-output (r2r) transform of type `kind`

of the array `A`

, as defined in the FFTW manual. `kind`

specifies either a discrete cosine transform of various types (`FFTW.REDFT00`

, `FFTW.REDFT01`

, `FFTW.REDFT10`

, or `FFTW.REDFT11`

), a discrete sine transform of various types (`FFTW.RODFT00`

, `FFTW.RODFT01`

, `FFTW.RODFT10`

, or `FFTW.RODFT11`

), a real-input DFT with halfcomplex-format output (`FFTW.R2HC`

and its inverse `FFTW.HC2R`

), or a discrete Hartley transform (`FFTW.DHT`

). The `kind`

argument may be an array or tuple in order to specify different transform types along the different dimensions of `A`

; `kind[end]`

is used for any unspecified dimensions. See the FFTW manual for precise definitions of these transform types, at http://www.fftw.org/doc.

The optional `dims`

argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. `kind[i]`

is then the transform type for `dims[i]`

, with `kind[end]`

being used for `i > length(kind)`

.

See also `plan_r2r`

to pre-plan optimized r2r transforms.

`Base.DFT.FFTW.r2r!`

Function
r2r!(A, kind [, dims])

Same as `r2r`

, but operates in-place on `A`

, which must be an array of real or complex floating-point numbers.

`Base.DFT.FFTW.plan_r2r`

Function
plan_r2r(A, kind [, dims [, flags [, timelimit]]])

Pre-plan an optimized r2r transform, similar to `plan_fft`

except that the transforms (and the first three arguments) correspond to `r2r`

and `r2r!`

, respectively.

`Base.DFT.FFTW.plan_r2r!`

Function
plan_r2r!(A, kind [, dims [, flags [, timelimit]]])

Similar to `plan_fft`

, but corresponds to `r2r!`

.

© 2009–2016 Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and other contributors

Licensed under the MIT License.

https://docs.julialang.org/en/release-0.6/stdlib/math/