Mathematics
Mathematical Operators
#
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)
25dt::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.
#
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#
Base.:/ — Function
A / BMatrix right-division: A / B is equivalent to (B' \ A')' where \ is the left-division operator. For square matrices, the result X is such that A == X*B.
See also: rdiv!.
Examples
julia> A = Float64[1 4 5; 3 9 2]; B = Float64[1 4 2; 3 4 2; 8 7 1];
julia> X = A / B
2×3 Matrix{Float64}:
-0.65 3.75 -1.2
3.25 -2.75 1.0
julia> isapprox(A, X*B)
true
julia> isapprox(X, A*pinv(B))
true/(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#
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#
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#
Base.muladd — Function
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.
|
Compatibility: 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
107.0 107.0muladd(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#
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|
Compatibility: Julia 1.2
|
#
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.
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#
Base.fld — Function
fld(x, y)Largest integer less than or equal to x / y. Equivalent to div(x, y, RoundDown).
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 1Because 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)
59.99999999999999666933092612453056361837965690217069245739573412231113406246995What 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 of the true value, so the result is 59.0.
#
Base.mod — Function
mod(x::Integer, r::AbstractUnitRange)Find y in the range r such that , where n = length(r), i.e. y = mod(x - first(r), n) + first(r).
See also mod1.
Examples
julia> mod(0, Base.OneTo(3)) # mod1(0, 3)
3
julia> mod(3, 0:2) # mod(3, 3)
0|
Compatibility: Julia 1.3
This method requires at least Julia 1.3. |
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).
|
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 |
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 2rem(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> x = 129 % Int8
-127
julia> typeof(x)
Int8
julia> x = 129 % BigInt
129
julia> typeof(x)
BigInt#
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.
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#
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 ]. This will generally be the most accurate result. See alsoRoundNearest. -
if
r == RoundToZero, then the result is in the interval ] ifxis positive,. or ] otherwise. See alsoRoundToZero. -
if
r == RoundDown, then the result is in the interval ]. See alsoRoundDown. -
if
r == RoundUp, then the result is in the interval ]. See alsoRoundUp.
Examples
julia> rem2pi(7pi/4, RoundNearest)
-0.7853981633974485
julia> rem2pi(7pi/4, RoundDown)
5.497787143782138#
Base.Math.mod2pi — Function
mod2pi(x)Modulus after division by 2π, returning in the range .
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.
|
Depending on the format of the input value, the closest representable value to 2π may be less than 2π. For example, the expression |
Examples
julia> mod2pi(9*pi/4)
0.7853981633974481#
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).
Examples
julia> divrem(3, 7)
(0, 3)
julia> divrem(7, 3)
(2, 1)#
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 ] for positive y and in the range for negative y.
With integer arguments and positive y, this is equal to mod(x, 1:y), and hence natural for 1-based indexing. By comparison, mod(x, y) == mod(x, 0:y-1) is natural for computations with offsets or strides.
Examples
julia> mod1(4, 2)
2
julia> mod1.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
1 2 3 1 2 3 1 2 3 1 2
julia> mod1.([-0.1, 0, 0.1, 1, 2, 2.9, 3, 3.1]', 3)
1×8 Matrix{Float64}:
2.9 3.0 0.1 1.0 2.0 2.9 3.0 0.1#
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#
Base.numerator — Function
numerator(x)Numerator of the rational representation of x.
Examples
julia> numerator(2//3)
2
julia> numerator(4)
4#
Base.denominator — Function
denominator(x)Denominator of the rational representation of x.
Examples
julia> denominator(2//3)
3
julia> denominator(4)
1#
Base.:<< — Function
<<(x, n)Left bit shift operator, x << n. For n >= 0, the result is x shifted left by n bits, filling with 0s. 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"<<(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#
Base.:>> — Function
>>(x, n)Right bit shift operator, x >> n. For n >= 0, the result is x shifted right by n bits, filling with 0s if x >= 0, 1s 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">>(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#
Base.:>>> — Function
>>>(x, n)Unsigned right bit shift operator, x >>> n. For n >= 0, the result is x shifted right by n bits, filling with 0s. 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"BigInts 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.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.
|
Compatibility: Julia 1.5
This function requires Julia 1.5 or later. |
julia> bitrotate(UInt8(114), 2)
0xc9
julia> bitstring(bitrotate(0b01110010, 2))
"11001001"
julia> bitstring(bitrotate(0b01110010, -2))
"10011100"
julia> bitstring(bitrotate(0b01110010, 8))
"01110010"#
Base.:: — Function
:exprQuote an expression expr, returning the abstract syntax tree (AST) of expr. The AST may be of type Expr, Symbol, or a literal value. The syntax :identifier evaluates to a Symbol.
See also: Expr, Symbol, Meta.parse
Examples
julia> expr = :(a = b + 2*x)
:(a = b + 2x)
julia> sym = :some_identifier
:some_identifier
julia> value = :0xff
0xff
julia> typeof((expr, sym, value))
Tuple{Expr, Symbol, UInt8}#
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
rangewith any three ofstart,step,stop,length. -
Call
rangewith two ofstart,stop,length. In this casestepwill be assumed to be one. If both arguments are Integers, aUnitRangewill be returned. -
Call
rangewith one ofstoporlength.startandstepwill be assumed to be one.
See Extended Help for additional details on the returned type.
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
julia> range(; length = 10)
Base.OneTo(10)
julia> range(; stop = 6)
Base.OneTo(6)
julia> range(; stop = 6.5)
1.0:1.0:6.0If 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)
1.0:2.0:3.0Special care is taken to ensure intermediate values are computed rationally. To avoid this induced overhead, see the LinRange constructor.
|
Compatibility: Julia 1.1
|
|
Compatibility: Julia 1.7
The versions without keyword arguments and |
|
Compatibility: Julia 1.8
The versions with |
Extended Help
range will produce a Base.OneTo when the arguments are Integers and
-
Only
lengthis provided -
Only
stopis provided
range will produce a UnitRange when the arguments are Integers and
-
Only
startandstopare provided -
Only
lengthandstopare provided
A UnitRange is not produced if step is provided even if specified as one.
#
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( ref::R, step::S, len, [offset=1]) where { R,S}
StepRangeLen{T,R,S}( ref::R, step::S, len, [offset=1]) where {T,R,S}
StepRangeLen{T,R,S,L}(ref::R, step::S, len, [offset=1]) where {T,R,S,L}A range r where r[i] produces values of type T (in the first form, T is deduced automatically), parameterized by a reference value, a step, and the length. 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.
|
Compatibility: Julia 1.7
The 4th type parameter |
#
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.
#
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!=(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.
|
Compatibility: Julia 1.2
This functionality requires at least Julia 1.2. |
#
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 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<(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.
|
Compatibility: Julia 1.2
This functionality requires at least Julia 1.2. |
#
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<=(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.
|
Compatibility: Julia 1.2
This functionality requires at least Julia 1.2. |
#
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>(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.
|
Compatibility: Julia 1.2
This functionality requires at least Julia 1.2. |
#
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>=(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.
|
Compatibility: Julia 1.2
This functionality requires at least Julia 1.2. |
#
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})
[...]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.
cmp(a::AbstractString, b::AbstractString) -> IntCompare 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#
Base.:& — Function
x & yBitwise 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).
Examples
julia> 4 & 10
0
julia> 4 & 12
4
julia> true & missing
missing
julia> false & missing
false#
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#
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#
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 and the other is not true.
The infix operation a ⊽ b is a synonym for nor(a,b), and ⊽ can be typed by tab-completing \nor or \barvee 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> false ⊽ missing
missing
julia> [true; true; false] .⊽ [true; false; false]
3-element BitVector:
0
0
1#
Base.:! — Function
!(x)Boolean not. Implements three-valued logic, returning missing if x is missing.
Examples
julia> !true
false
julia> !false
true
julia> !missing
missing
julia> .![true false true]
1×3 BitMatrix:
0 1 0!f::Function
Predicate function negation: when the argument of ! is a function, it returns a composed 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: |-| < ⇒ |()-()| < "|
Compatibility: Julia 1.9
Starting with Julia 1.9, |
#
&& — Keyword
x && yShort-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")
falseMathematical 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 significant 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.
|
Compatibility: Julia 1.6
Passing the |
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`
trueisapprox(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.
|
Compatibility: Julia 1.5
This method requires Julia 1.5 or later. |
#
Base.sin — Method
sin(x)Compute sine of x, where x is in radians.
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)
0.7071067811865476
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#
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.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
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.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
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.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
Base.Math.sincosd — Function
sincosd(x)Simultaneously compute the sine and cosine of x, where x is in degrees.
|
Compatibility: Julia 1.3
This function requires at least Julia 1.3. |
#
Base.Math.cospi — Function
cospi(x)Compute more accurately than cos(pi*x), especially for large x.
#
Base.tanh — Method
tanh(x)Compute hyperbolic tangent of x.
Examples
julia> tanh.(-3:3f0) # Here 3f0 isa Float32
7-element Vector{Float32}:
-0.9950548
-0.9640276
-0.7615942
0.0
0.7615942
0.9640276
0.9950548
julia> tan.(im .* (1:3))
3-element Vector{ComplexF64}:
0.0 + 0.7615941559557649im
0.0 + 0.9640275800758169im
0.0 + 0.9950547536867306im#
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 ].
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 ]. This corresponds to a standard atan2 function. Note that by convention atan(0.0,x) is defined as and atan(-0.0,x) is defined as 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#
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.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
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.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
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.
|
Compatibility: Julia 1.7
The one-argument method supports square matrix arguments as of Julia 1.7. |
#
Base.Math.acsc — Method
acsc(x)Compute the inverse cosecant of x, where the output is in radians.
#
Base.Math.acot — Method
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. If x is a matrix, x needs to be a square matrix.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
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.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
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.
|
Compatibility: Julia 1.7
Matrix arguments require Julia 1.7 or later. |
#
Base.Math.hypot — Function
hypot(x, y)Compute the hypotenuse 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 avoiding overflow and underflow.
See also norm in the LinearAlgebra standard library.
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
julia> using LinearAlgebra
julia> norm([a, a, a, a]) == hypot(a, a, a, a)
true#
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.
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
[...]
julia> log.(exp.(-1:1))
3-element Vector{Float64}:
-1.0
0.0
1.0#
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
[...]#
Base.log2 — Function
log2(x)Compute the logarithm of x to base 2. Throws DomainError for negative Real arguments.
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
[...]
julia> log2.(2.0 .^ (-1:1))
3-element Vector{Float64}:
-1.0
0.0
1.0#
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
[...]#
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
[...]#
Base.Math.frexp — Function
frexp(val)Return (x,exp) such that x has a magnitude in the interval or 0, and val is equal to .
Examples
julia> frexp(12.8)
(0.8, 4)#
Base.exp10 — Function
exp10(x)Compute the base 10 exponential of x, in other words .
Examples
julia> exp10(2)
100.0
julia> 10^2
100#
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)#
Base.expm1 — Function
expm1(x)Accurately compute . 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#
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|
Rounding to specified digits in bases other than 2 can be inexact when operating on binary floating point numbers. For example, the |
Extensions
To extend round to new numeric types, it is typically sufficient to define Base.round(x::NewType, r::RoundingMode).
#
Base.Rounding.RoundingMode — Type
RoundingModeA 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:
|
Compatibility: Julia 1.9
|
#
Base.Rounding.RoundNearest — Constant
RoundNearestThe default rounding mode. Rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer.
#
Base.Rounding.RoundFromZero — Constant
RoundFromZeroRounds away from zero.
|
Compatibility: Julia 1.9
|
Examples
julia> BigFloat("1.0000000000000001", 5, RoundFromZero)
1.06#
Base.round — Method
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]])
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; digits=0, 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 RoundingModes. The first RoundingMode is used for rounding the real components while the second is used for rounding the imaginary components.
RoundingModeReal and RoundingModeImaginary default to RoundNearest, which rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer.
Example
julia> round(3.14 + 4.5im)
3.0 + 4.0im
julia> round(3.14 + 4.5im, RoundUp, RoundNearestTiesUp)
4.0 + 5.0im
julia> round(3.14159 + 4.512im; digits = 1)
3.1 + 4.5im
julia> round(3.14159 + 4.512im; sigdigits = 3)
3.14 + 4.51im#
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.
#
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.
#
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#
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#
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.
|
Compatibility: Julia 1.3
|
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
10clamp(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)
39clamp(x::Integer, r::AbstractUnitRange)
Clamp x to lie within range r.
|
Compatibility: Julia 1.6
This method requires at least Julia 1.6. |
#
Base.Math.clamp! — Function
clamp!(array::AbstractArray, lo, hi)Restrict values in array to the specified range, in-place. See also clamp.
|
Compatibility: Julia 1.3
|
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#
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.
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#
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.
#
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.
#
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.
#
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.
#
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.
#
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.
#
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.
#
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.
#
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.
#
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.
#
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.copysign — Function
copysign(x, y) -> zReturn z which has the magnitude of x and the same sign as y.
Examples
julia> copysign(1, -2)
-1
julia> copysign(-1, 2)
1#
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#
Base.sqrt — Method
sqrt(x)Return . 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#
Base.isqrt — Function
isqrt(n::Integer)Integer square root: the largest integer m such that m*m <= n.
julia> isqrt(5)
2#
Base.Math.cbrt — Function
cbrt(x::Real)Return the cube root of x, i.e. . Negative values are accepted (returning the negative real root when ).
The prefix operator ∛ is equivalent to cbrt.
Examples
julia> cbrt(big(27))
3.0
julia> cbrt(big(-27))
-3.0#
Base.real — Function
real(z)Return the real part of the complex number z.
Examples
julia> real(1 + 3im)
1real(T::Type)
Return the type that represents the real part of a value of type T. e.g: for T == Complex{R}, returns R. Equivalent to typeof(real(zero(T))).
Examples
julia> real(Complex{Int})
Int64
julia> real(Float64)
Float64real(A::AbstractArray)
Return an array containing the real part of each entry in array A.
Equivalent to real.(A), except that when eltype(A) <: Real A is returned without copying, and that when A has zero dimensions, a 0-dimensional array is returned (rather than a scalar).
Examples
julia> real([1, 2im, 3 + 4im])
3-element Vector{Int64}:
1
0
3
julia> real(fill(2 - im))
0-dimensional Array{Int64, 0}:
2#
Base.imag — Function
imag(z)Return the imaginary part of the complex number z.
Examples
julia> imag(1 + 3im)
3imag(A::AbstractArray)
Return an array containing the imaginary part of each entry in array A.
Equivalent to imag.(A), except that when A has zero dimensions, a 0-dimensional array is returned (rather than a scalar).
Examples
julia> imag([1, 2im, 3 + 4im])
3-element Vector{Int64}:
0
2
4
julia> imag(fill(2 - im))
0-dimensional Array{Int64, 0}:
-1#
Base.reim — Function
reim(z)Return a tuple of the real and imaginary parts of the complex number z.
Examples
julia> reim(1 + 3im)
(1, 3)reim(A::AbstractArray)
Return a tuple of two arrays containing respectively the real and the imaginary part of each entry in A.
Equivalent to (real.(A), imag.(A)), except that when eltype(A) <: Real A is returned without copying to represent the real part, and that when A has zero dimensions, a 0-dimensional array is returned (rather than a scalar).
Examples
julia> reim([1, 2im, 3 + 4im])
([1, 0, 3], [0, 2, 4])
julia> reim(fill(2 - im))
(fill(2), fill(-1))#
Base.conj — Function
conj(z)Compute the complex conjugate of a complex number z.
Examples
julia> conj(1 + 3im)
1 - 3imconj(A::AbstractArray)
Return an array containing the complex conjugate of each entry in array A.
Equivalent to conj.(A), except that when eltype(A) <: Real A is returned without copying, and that when A has zero dimensions, a 0-dimensional array is returned (rather than a scalar).
Examples
julia> conj([1, 2im, 3 + 4im])
3-element Vector{Complex{Int64}}:
1 + 0im
0 - 2im
3 - 4im
julia> conj(fill(2 - im))
0-dimensional Array{Complex{Int64}, 0}:
2 + 1im#
Base.cispi — Function
cispi(x)More accurate method for cis(pi*x) (especially for large x).
Examples
julia> cispi(10000)
1.0 + 0.0im
julia> cispi(0.25 + 1im)
0.030556854645954562 + 0.030556854645954562im|
Compatibility: Julia 1.6
This function requires Julia 1.6 or later. |
#
Base.binomial — Function
binomial(n::Integer, k::Integer)The binomial coefficient , being the coefficient of the th term in the polynomial expansion of .
If is non-negative, then it is the number of ways to choose k out of n items:
where is the factorial function.
If is negative, then it is defined in terms of the identity
See also factorial.
Examples
julia> binomial(5, 3)
10
julia> factorial(5) ÷ (factorial(5-3) * factorial(3))
10
julia> binomial(-5, 3)
-35External links
-
Binomial coefficient on Wikipedia.
#
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))
51090942171709440000External links
-
Factorial on Wikipedia.
#
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.
|
Compatibility: 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#
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.
|
Compatibility: 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#
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 . returns .
The arguments may be integer and rational numbers.
|
Compatibility: 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)|
Bézout coefficients are not uniquely defined. |
#
Base.ispow2 — Function
ispow2(n::Number) -> BoolTest 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|
Compatibility: Julia 1.6
Support for non- |
#
Base.nextprod — Function
nextprod(factors::Union{Tuple,AbstractVector}, n)Next integer greater than or equal to n that can be written as for integers , , etcetera, for factors in factors.
Examples
julia> nextprod((2, 3), 105)
108
julia> 2^2 * 3^3
108|
Compatibility: Julia 1.6
The method that accepts a tuple requires Julia 1.6 or later. |
#
Base.invmod — Function
invmod(n, m)Take the inverse of n modulo m: y such that , and . This will throw an error if , or if .
Examples
julia> invmod(2, 5)
3
julia> invmod(2, 3)
2
julia> invmod(5, 6)
5#
Base.powermod — Function
powermod(x::Integer, p::Integer, m)Compute .
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#
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(0)
1
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#
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#
Base.Math.evalpoly — Function
evalpoly(x, p)Evaluate the polynomial ] 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 [1] algorithm if x is complex.
|
Compatibility: Julia 1.4
This function requires Julia 1.4 or later. |
Example
julia> evalpoly(2, (1, 2, 3))
17#
Base.Math.@evalpoly — Macro
@evalpoly(z, c...)Evaluate the polynomial ] 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#
Base.FastMath.@fastmath — Macro
@fastmath exprExecute 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))
0.1411200080598672Customizable 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.