Mathematics
Mathematical Operators
Base.:-
— Method-(x)
Unary minus operator.
Examples
julia> -1
-1
julia> -(2)
-2
julia> -[1 2; 3 4]
2×2 Array{Int64,2}:
-1 -2
-3 -4
Base.:+
— Function+(x, y...)
Addition operator. x+y+z+...
calls this function with all arguments, i.e. +(x, y, z, ...)
.
Examples
julia> 1 + 20 + 4
25
julia> +(1, 20, 4)
25
dt::Date + t::Time -> DateTime
The addition of a Date
with a Time
produces a DateTime
. The hour, minute, second, and millisecond parts of the Time
are used along with the year, month, and day of the Date
to create the new DateTime
. Non-zero microseconds or nanoseconds in the Time
type will result in an InexactError
being thrown.
Base.:-
— Method-(x, y)
Subtraction operator.
Examples
julia> 2 - 3
-1
julia> -(2, 4.5)
-2.5
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/(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 Array{Float64,1}:
6.5
-7.0
julia> inv(A) * x
2-element Array{Float64,1}:
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.)
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
81 118
Base.fma
— Functionfma(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
— Functionmuladd(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
— Methodinv(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
inv(::Missing)
requires at least Julia 1.2.
Base.div
— Functiondiv(x, y)
÷(x, y)
The quotient from Euclidean division. Computes x/y
, truncated to an integer.
Examples
julia> 9 ÷ 4
2
julia> -5 ÷ 3
-1
julia> 5.0 ÷ 2
2.0
Base.fld
— Functionfld(x, y)
Largest integer less than or equal to x/y
. Equivalent to div(x, y, RoundDown)
.
See also: div
Examples
julia> fld(7.3,5.5)
1.0
Base.cld
— Functioncld(x, y)
Smallest integer larger than or equal to x/y
. Equivalent to div(x, y, RoundUp)
.
See also: div
Examples
julia> cld(5.5,2.2)
3.0
Base.mod
— Functionmod(x::Integer, r::AbstractUnitRange)
Find y
in the range r
such that $x ≡ y (mod n)$, where n = length(r)
, i.e. y = mod(x - first(r), n) + first(r)
.
See also: mod1
.
Examples
julia> mod(0, Base.OneTo(3))
3
julia> mod(3, 0:2)
0
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 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)
3.0
rem(x::Integer, T::Type{<:Integer}) -> T
mod(x::Integer, T::Type{<:Integer}) -> T
%(x::Integer, T::Type{<:Integer}) -> T
Find y::T
such that x
≡ y
(mod n), where n is the number of integers representable in T
, and y
is an integer in [typemin(T),typemax(T)]
. If T
can represent any integer (e.g. T == BigInt
), then this operation corresponds to a conversion to T
.
Examples
julia> 129 % Int8
-127
Base.rem
— Functionrem(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
Base.Math.rem2pi
— Functionrem2pi(x, r::RoundingMode)
Compute the remainder of x
after integer division by 2π
, with the quotient rounded according to the rounding mode r
. In other words, the quantity
x - 2π*round(x/(2π),r)
without any intermediate rounding. This internally uses a high precision approximation of 2π, and so will give a more accurate result than rem(x,2π,r)
if
r == RoundNearest
, then the result is in the interval $[-π, π]$. This will generally be the most accurate result. See alsoRoundNearest
.if
r == RoundToZero
, then the result is in the interval $[0, 2π]$ ifx
is positive,. or $[-2π, 0]$ otherwise. See alsoRoundToZero
.if
r == RoundDown
, then the result is in the interval $[0, 2π]$. See alsoRoundDown
.if
r == RoundUp
, then the result is in the interval $[-2π, 0]$. See alsoRoundUp
.
Examples
julia> rem2pi(7pi/4, RoundNearest)
-0.7853981633974485
julia> rem2pi(7pi/4, RoundDown)
5.497787143782138
Base.Math.mod2pi
— Functionmod2pi(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
.
Depending on the format of the input value, the closest representable value to 2π may be less than 2π. For example, the expression mod2pi(2π)
will not return 0
, because the intermediate value of 2*π
is a Float64
and 2*Float64(π) < 2*big(π)
. See rem2pi
for more refined control of this behavior.
Examples
julia> mod2pi(9*pi/4)
0.7853981633974481
Base.divrem
— Functiondivrem(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.fldmod
— Functionfldmod(x, y)
The floored quotient and modulus after division. A convenience wrapper for divrem(x, y, RoundDown)
. Equivalent to (fld(x,y), mod(x,y))
.
Base.fld1
— Functionfld1(x, y)
Flooring division, returning a value consistent with mod1(x,y)
Examples
julia> x = 15; y = 4;
julia> fld1(x, y)
4
julia> x == fld(x, y) * y + mod(x, y)
true
julia> x == (fld1(x, y) - 1) * y + mod1(x, y)
true
Base.mod1
— Functionmod1(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
.
Examples
julia> mod1(4, 2)
2
julia> mod1(4, 3)
1
Base.fldmod1
— FunctionBase.://
— Function//(num, den)
Divide two integers or rational numbers, giving a Rational
result.
Examples
julia> 3 // 5
3//5
julia> (3 // 5) // (2 // 1)
3//10
Base.rationalize
— Functionrationalize([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
— Functionnumerator(x)
Numerator of the rational representation of x
.
Examples
julia> numerator(2//3)
2
julia> numerator(4)
4
Base.denominator
— Functiondenominator(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 0
s. This is equivalent to x * 2^n
. For n < 0
, this is equivalent to x >> -n
.
Examples
julia> Int8(3) << 2
12
julia> bitstring(Int8(3))
"00000011"
julia> bitstring(Int8(12))
"00001100"
<<(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}:
1
0
1
0
0
julia> B << 1
5-element BitArray{1}:
0
1
0
0
0
julia> B << -1
5-element BitArray{1}:
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, where n >= 0
, filling with 0
s if x >= 0
, 1
s if x < 0
, preserving the sign of x
. This is equivalent to fld(x, 2^n)
. For n < 0
, this is equivalent to x << -n
.
Examples
julia> Int8(13) >> 2
3
julia> bitstring(Int8(13))
"00001101"
julia> bitstring(Int8(3))
"00000011"
julia> Int8(-14) >> 2
-4
julia> bitstring(Int8(-14))
"11110010"
julia> bitstring(Int8(-4))
"11111100"
>>(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 BitArray{1}:
1
0
1
0
0
julia> B >> 1
5-element BitArray{1}:
0
1
0
1
0
julia> B >> -1
5-element BitArray{1}:
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, where n >= 0
, filling with 0
s. For n < 0
, this is equivalent to x << -n
.
For Unsigned
integer types, this is equivalent to >>
. For Signed
integer types, this is equivalent to signed(unsigned(x) >> n)
.
Examples
julia> Int8(-14) >>> 2
60
julia> bitstring(Int8(-14))
"11110010"
julia> bitstring(Int8(60))
"00111100"
BigInt
s are treated as if having infinite size, so no filling is required and this is equivalent to >>
.
>>>(B::BitVector, n) -> BitVector
Unsigned right bitshift operator, B >>> n
. Equivalent to B >> n
. See >>
for details and examples.
Base.bitrotate
— Functionbitrotate(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.
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(:)(I::CartesianIndex, J::CartesianIndex)
Construct CartesianIndices
from two CartesianIndex
.
This method requires at least Julia 1.1.
Examples
julia> I = CartesianIndex(2,1);
julia> J = CartesianIndex(3,3);
julia> I:J
2×3 CartesianIndices{2,Tuple{UnitRange{Int64},UnitRange{Int64}}}:
CartesianIndex(2, 1) CartesianIndex(2, 2) CartesianIndex(2, 3)
CartesianIndex(3, 1) CartesianIndex(3, 2) CartesianIndex(3, 3)
(:)(start, [step], stop)
Range operator. a:b
constructs a range from a
to b
with a step size of 1 (a UnitRange
) , and a:s:b
is similar but uses a step size of s
(a StepRange
).
:
is also used in indexing to select whole dimensions and for Symbol
literals, as in e.g. :hello
.
Base.range
— Functionrange(start[, stop]; length, stop, step=1)
Given a starting value, construct a range either by length or from start
to stop
, optionally with a given step (defaults to 1, a UnitRange
). One of length
or stop
is required. If length
, stop
, and step
are all specified, they must agree.
If length
and stop
are provided and step
is not, the step size will be computed automatically such that there are length
linearly spaced elements in the range.
If step
and stop
are provided and length
is not, the overall range length will be computed automatically such that the elements are step
spaced.
Special care is taken to ensure intermediate values are computed rationally. To avoid this induced overhead, see the LinRange
constructor.
stop
may be specified as either a positional or keyword argument.
stop
as a positional argument requires at least Julia 1.1.
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
Base.OneTo
— TypeBase.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
— TypeStepRangeLen{T,R,S}(ref::R, step::S, len, [offset=1]) where {T,R,S}
StepRangeLen( ref::R, step::S, len, [offset=1]) where { R,S}
A range r
where r[i]
produces values of type T
(in the second form, T
is deduced automatically), parameterized by a ref
erence value, a step
, and the len
gth. By default ref
is the starting value r[1]
, but alternatively you can supply it as the value of r[offset]
for some other index 1 <= offset <= len
. In conjunction with TwicePrecision
this can be used to implement ranges that are free of roundoff error.
Base.:==
— Function==(x, y)
Generic equality operator. Falls back to ===
. Should be implemented for all types with a notion of equality, based on the abstract value that an instance represents. For example, all numeric types are compared by numeric value, ignoring type. Strings are compared as sequences of characters, ignoring encoding. For collections, ==
is generally called recursively on all contents, though other properties (like the shape for arrays) may also be taken into account.
This operator follows IEEE semantics for floating-point numbers: 0.0 == -0.0
and NaN != NaN
.
The result is of type Bool
, except when one of the operands is missing
, in which case missing
is returned (three-valued logic). For collections, missing
is returned if at least one of the operands contains a missing
value and all non-missing values are equal. Use isequal
or ===
to always get a Bool
result.
Implementation
New numeric types should implement this function for two arguments of the new type, and handle comparison to other types via promotion rules where possible.
isequal
falls back to ==
, so new methods of ==
will be used by the Dict
type to compare keys. If your type will be used as a dictionary key, it should therefore also implement hash
.
==(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.
==(a::AbstractString, b::AbstractString) -> Bool
Test whether two strings are equal character by character (technically, Unicode code point by code point).
Examples
julia> "abc" == "abc"
true
julia> "abc" == "αβγ"
false
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.
This functionality requires at least Julia 1.2.
Base.:!==
— Function!==(x, y)
≢(x,y)
Always gives the opposite answer as ===
.
Examples
julia> a = [1, 2]; b = [1, 2];
julia> a ≢ b
true
julia> a ≢ a
false
Base.:<
— Function<(x, y)
Less-than comparison operator. Falls back to isless
. Because of the behavior of floating-point NaN values, this operator implements a partial order.
Implementation
New numeric types with a canonical partial order should implement this function for two arguments of the new type. Types with a canonical total order should implement isless
instead. (x < y) | (x == y)
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.
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.
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.
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.
This functionality requires at least Julia 1.2.
Base.cmp
— Functioncmp(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) -> Int
Compare two strings. Return 0
if both strings have the same length and the character at each index is the same in both strings. Return -1
if a
is a prefix of b
, or if a
comes before b
in alphabetical order. Return 1
if b
is a prefix of a
, or if b
comes before a
in alphabetical order (technically, lexicographical order by Unicode code points).
Examples
julia> cmp("abc", "abc")
0
julia> cmp("ab", "abc")
-1
julia> cmp("abc", "ab")
1
julia> cmp("ab", "ac")
-1
julia> cmp("ac", "ab")
1
julia> cmp("α", "a")
1
julia> cmp("b", "β")
-1
Base.:~
— Function~(x)
Bitwise not.
Examples
julia> ~4
-5
julia> ~10
-11
julia> ~true
false
Base.:&
— Functionx & y
Bitwise and. Implements three-valued logic, returning missing
if one operand is missing
and the other is true
. Add parentheses for function application form: (&)(x, y)
.
Examples
julia> 4 & 10
0
julia> 4 & 12
4
julia> true & missing
missing
julia> false & missing
false
Base.:|
— Functionx | y
Bitwise or. Implements three-valued logic, returning missing
if one operand is missing
and the other is false
.
Examples
julia> 4 | 10
14
julia> 4 | 1
5
julia> true | missing
true
julia> false | missing
missing
Base.xor
— Functionxor(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 BitArray{1}:
0
1
0
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 BitArray{2}:
0 1 0
!f::Function
Predicate function negation: when the argument of !
is a function, it returns a function which computes the boolean negation of f
.
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: |-| < ⇒ |()-()| < "
&&
— Keywordx && y
Short-circuiting boolean AND.
||
— Keywordx || y
Short-circuiting boolean OR.
Mathematical Functions
Base.isapprox
— Functionisapprox(x, y; rtol::Real=atol>0 ? 0 : √eps, atol::Real=0, nans::Bool=false, norm::Function)
Inexact equality comparison: true
if norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))
. The default atol
is zero and the default rtol
depends on the types of x
and y
. The keyword argument nans
determines whether or not NaN values are considered equal (defaults to false).
For real or complex floating-point values, if an atol > 0
is not specified, rtol
defaults to the square root of eps
of the type of x
or y
, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significand digits. Otherwise, e.g. for integer arguments or if an atol > 0
is supplied, rtol
defaults to zero.
x
and y
may also be arrays of numbers, in which case norm
defaults to the usual norm
function in LinearAlgebra, 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)
.
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.
Examples
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])
true
julia> 1e-10 ≈ 0
false
julia> isapprox(1e-10, 0, atol=1e-8)
true
isapprox(x; kwargs...) / ≈(x; kwargs...)
Create a function that compares its argument to x
using ≈
, i.e. a function equivalent to y -> y ≈ x
.
The keyword arguments supported here are the same as those in the 2-argument isapprox
.
Base.sin
— Methodsin(x)
Compute sine of x
, where x
is in radians.
Base.cos
— Methodcos(x)
Compute cosine of x
, where x
is in radians.
Base.Math.sincos
— Methodsincos(x)
Simultaneously compute the sine and cosine of x
, where the x
is in radians.
Base.tan
— Methodtan(x)
Compute tangent of x
, where x
is in radians.
Base.Math.sind
— Functionsind(x)
Compute sine of x
, where x
is in degrees.
Base.Math.cosd
— Functioncosd(x)
Compute cosine of x
, where x
is in degrees.
Base.Math.tand
— Functiontand(x)
Compute tangent of x
, where x
is in degrees.
Base.Math.sinpi
— Functionsinpi(x)
Compute $\sin(\pi x)$ more accurately than sin(pi*x)
, especially for large x
.
Base.Math.cospi
— Functioncospi(x)
Compute $\cos(\pi x)$ more accurately than cos(pi*x)
, especially for large x
.
Base.sinh
— Methodsinh(x)
Compute hyperbolic sine of x
.
Base.cosh
— Methodcosh(x)
Compute hyperbolic cosine of x
.
Base.tanh
— Methodtanh(x)
Compute hyperbolic tangent of x
.
Base.asin
— Methodasin(x)
Compute the inverse sine of x
, where the output is in radians.
Base.acos
— Methodacos(x)
Compute the inverse cosine of x
, where the output is in radians
Base.atan
— Methodatan(y)
atan(y, x)
Compute the inverse tangent of y
or y/x
, respectively.
For one argument, this is the angle in radians between the positive x-axis and the point (1, y), returning a value in the interval $[-\pi/2, \pi/2]$.
For two arguments, this is the angle in radians between the positive x-axis and the point (x, y), returning a value in the interval $[-\pi, \pi]$. This corresponds to a standard atan2
function.
Base.Math.asind
— Functionasind(x)
Compute the inverse sine of x
, where the output is in degrees.
Base.Math.acosd
— Functionacosd(x)
Compute the inverse cosine of x
, where the output is in degrees.
Base.Math.atand
— Functionatand(y)
atand(y,x)
Compute the inverse tangent of y
or y/x
, respectively, where the output is in degrees.
Base.Math.sec
— Methodsec(x)
Compute the secant of x
, where x
is in radians.
Base.Math.csc
— Methodcsc(x)
Compute the cosecant of x
, where x
is in radians.
Base.Math.cot
— Methodcot(x)
Compute the cotangent of x
, where x
is in radians.
Base.Math.secd
— Functionsecd(x)
Compute the secant of x
, where x
is in degrees.
Base.Math.cscd
— Functioncscd(x)
Compute the cosecant of x
, where x
is in degrees.
Base.Math.cotd
— Functioncotd(x)
Compute the cotangent of x
, where x
is in degrees.
Base.Math.asec
— Methodasec(x)
Compute the inverse secant of x
, where the output is in radians.
Base.Math.acsc
— Methodacsc(x)
Compute the inverse cosecant of x
, where the output is in radians.
Base.Math.acot
— Methodacot(x)
Compute the inverse cotangent of x
, where the output is in radians.
Base.Math.asecd
— Functionasecd(x)
Compute the inverse secant of x
, where the output is in degrees.
Base.Math.acscd
— Functionacscd(x)
Compute the inverse cosecant of x
, where the output is in degrees.
Base.Math.acotd
— Functionacotd(x)
Compute the inverse cotangent of x
, where the output is in degrees.
Base.Math.sech
— Methodsech(x)
Compute the hyperbolic secant of x
.
Base.Math.csch
— Methodcsch(x)
Compute the hyperbolic cosecant of x
.
Base.Math.coth
— Methodcoth(x)
Compute the hyperbolic cotangent of x
.
Base.asinh
— Methodasinh(x)
Compute the inverse hyperbolic sine of x
.
Base.acosh
— Methodacosh(x)
Compute the inverse hyperbolic cosine of x
.
Base.atanh
— Methodatanh(x)
Compute the inverse hyperbolic tangent of x
.
Base.Math.asech
— Methodasech(x)
Compute the inverse hyperbolic secant of x
.
Base.Math.acsch
— Methodacsch(x)
Compute the inverse hyperbolic cosecant of x
.
Base.Math.acoth
— Methodacoth(x)
Compute the inverse hyperbolic cotangent of x
.
Base.Math.sinc
— Functionsinc(x)
Compute $\sin(\pi x) / (\pi x)$ if $x \neq 0$, and $1$ if $x = 0$.
Base.Math.cosc
— Functioncosc(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
— Functiondeg2rad(x)
Convert x
from degrees to radians.
Examples
julia> deg2rad(90)
1.5707963267948966
Base.Math.rad2deg
— Functionrad2deg(x)
Convert x
from radians to degrees.
Examples
julia> rad2deg(pi)
180.0
Base.Math.hypot
— Functionhypot(x, y)
Compute the hypotenuse $\sqrt{|x|^2+|y|^2}$ avoiding overflow and underflow.
This code is an implementation of the algorithm described in: An Improved Algorithm for hypot(a,b)
by Carlos F. Borges The article is available online at ArXiv at the link https://arxiv.org/abs/1904.09481
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
hypot(x...)
Compute the hypotenuse $\sqrt{\sum |x_i|^2}$ avoiding overflow and underflow.
Examples
julia> hypot(-5.7)
5.7
julia> hypot(3, 4im, 12.0)
13.0
Base.log
— Methodlog(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
[...]
Base.log
— Methodlog(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
— Functionlog2(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:
NaN result for non-NaN input.
Stacktrace:
[1] nan_dom_err at ./math.jl:325 [inlined]
[...]
Base.log10
— Functionlog10(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:
NaN result for non-NaN input.
Stacktrace:
[1] nan_dom_err at ./math.jl:325 [inlined]
[...]
Base.log1p
— Functionlog1p(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
— Functionfrexp(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
— Methodexp(x)
Compute the natural base exponential of x
, in other words $e^x$.
Examples
julia> exp(1.0)
2.718281828459045
Base.exp2
— Functionexp2(x)
Compute the base 2 exponential of x
, in other words $2^x$.
Examples
julia> exp2(5)
32.0
Base.exp10
— Functionexp10(x)
Compute the base 10 exponential of x
, in other words $10^x$.
Examples
julia> exp10(2)
100.0
exp10(x)
Compute $10^x$.
Examples
julia> exp10(2)
100.0
julia> exp10(0.2)
1.5848931924611136
Base.Math.ldexp
— Functionldexp(x, n)
Compute $x \times 2^n$.
Examples
julia> ldexp(5., 2)
20.0
Base.Math.modf
— Functionmodf(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
— Functionexpm1(x)
Accurately compute $e^x-1$.
Base.round
— Methodround([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 Float64
value represented by 1.15
is actually less than 1.15, yet will be rounded to 1.2.
Examples
julia> x = 1.15
1.15
julia> @sprintf "%.20f" x
"1.14999999999999991118"
julia> x < 115//100
true
julia> round(x, digits=1)
1.2
Extensions
To extend round
to new numeric types, it is typically sufficient to define Base.round(x::NewType, r::RoundingMode)
.
Base.Rounding.RoundingMode
— TypeRoundingMode
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:
Base.Rounding.RoundNearest
— ConstantRoundNearest
The default rounding mode. Rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer.
Base.Rounding.RoundNearestTiesAway
— ConstantRoundNearestTiesAway
Rounds to nearest integer, with ties rounded away from zero (C/C++ round
behaviour).
Base.Rounding.RoundNearestTiesUp
— ConstantRoundNearestTiesUp
Rounds to nearest integer, with ties rounded toward positive infinity (Java/JavaScript round
behaviour).
Base.Rounding.RoundToZero
— ConstantBase.Rounding.RoundFromZero
— ConstantRoundFromZero
Rounds away from zero. This rounding mode may only be used with T == BigFloat
inputs to round
.
Examples
julia> BigFloat("1.0000000000000001", 5, RoundFromZero)
1.06
Base.Rounding.RoundUp
— ConstantBase.Rounding.RoundDown
— ConstantBase.round
— Methodround(z::Complex[, RoundingModeReal, [RoundingModeImaginary]])
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; digits=, base=10)
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; sigdigits=, base=10)
Return the nearest integral value of the same type as the complex-valued z
to z
, breaking ties using the specified RoundingMode
s. The first RoundingMode
is used for rounding the real components while the second is used for rounding the imaginary components.
Example
julia> round(3.14 + 4.5im)
3.0 + 4.0im
Base.ceil
— Functionceil([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.
digits
, sigdigits
and base
work as for round
.
Base.floor
— Functionfloor([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.
digits
, sigdigits
and base
work as for round
.
Base.trunc
— Functiontrunc([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 x
.
trunc(T, x)
converts the result to type T
, throwing an InexactError
if the value is not representable.
digits
, sigdigits
and base
work as for round
.
Base.unsafe_trunc
— Functionunsafe_trunc(T, x)
Return 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.min
— Functionmin(x, y, ...)
Return the minimum of the arguments. See also the minimum
function to take the minimum element from a collection.
Examples
julia> min(2, 5, 1)
1
Base.max
— Functionmax(x, y, ...)
Return the maximum of the arguments. See also the maximum
function to take the maximum element from a collection.
Examples
julia> max(2, 5, 1)
5
Base.minmax
— Functionminmax(x, y)
Return (min(x,y), max(x,y))
. See also: extrema
that returns (minimum(x), maximum(x))
.
Examples
julia> minmax('c','b')
('b', 'c')
Base.Math.clamp
— Functionclamp(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.
Examples
julia> clamp.([pi, 1.0, big(10.)], 2., 9.)
3-element Array{BigFloat,1}:
3.141592653589793238462643383279502884197169399375105820974944592307816406286198
2.0
9.0
julia> clamp.([11,8,5],10,6) # an example where lo > hi
3-element Array{Int64,1}:
6
6
10
clamp(x, T)::T
Clamp x
between typemin(T)
and typemax(T)
and convert the result to type T
.
Examples
julia> clamp(200, Int8)
127
julia> clamp(-200, Int8)
-128
Base.Math.clamp!
— Functionclamp!(array::AbstractArray, lo, hi)
Restrict values in array
to the specified range, in-place. See also clamp
.
Base.abs
— Functionabs(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(typemin(Int64))
-9223372036854775808
Base.Checked.checked_abs
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.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
— FunctionBase.mul_with_overflow(x, y) -> (r, f)
Calculates r = x*y
, with the flag f
indicating whether overflow has occurred.
Base.abs2
— Functionabs2(x)
Squared absolute value of x
.
Examples
julia> abs2(-3)
9
Base.copysign
— Functioncopysign(x, y) -> z
Return z
which has the magnitude of x
and the same sign as y
.
Examples
julia> copysign(1, -2)
-1
julia> copysign(-1, 2)
1
Base.sign
— Functionsign(x)
Return zero if x==0
and $x/|x|$ otherwise (i.e., ±1 for real x
).
Base.signbit
— Functionsignbit(x)
Returns true
if the value of the sign of x
is negative, otherwise false
.
Examples
julia> signbit(-4)
true
julia> signbit(5)
false
julia> signbit(5.5)
false
julia> signbit(-4.1)
true
Base.flipsign
— Functionflipsign(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
— Methodsqrt(x)
Return $\sqrt{x}$. Throws DomainError
for negative Real
arguments. Use complex negative arguments instead. The prefix operator √
is equivalent to sqrt
.
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
Base.isqrt
— Functionisqrt(n::Integer)
Integer square root: the largest integer m
such that m*m <= n
.
julia> isqrt(5)
2
Base.Math.cbrt
— Functioncbrt(x::Real)
Return the cube root of x
, i.e. $x^{1/3}$. Negative values are accepted (returning the negative real root when $x < 0$).
The prefix operator ∛
is equivalent to cbrt
.
Examples
julia> cbrt(big(27))
3.0
julia> cbrt(big(-27))
-3.0
Base.real
— Methodreal(z)
Return the real part of the complex number z
.
Examples
julia> real(1 + 3im)
1
Base.imag
— Functionimag(z)
Return the imaginary part of the complex number z
.
Examples
julia> imag(1 + 3im)
3
Base.reim
— Functionreim(z)
Return both the real and imaginary parts of the complex number z
.
Examples
julia> reim(1 + 3im)
(1, 3)
Base.conj
— Functionconj(z)
Compute the complex conjugate of a complex number z
.
Examples
julia> conj(1 + 3im)
1 - 3im
Base.angle
— Functionangle(z)
Compute the phase angle in radians of a complex number z
.
Examples
julia> rad2deg(angle(1 + im))
45.0
julia> rad2deg(angle(1 - im))
-45.0
julia> rad2deg(angle(-1 - im))
-135.0
Base.cis
— Functioncis(z)
Return $\exp(iz)$.
Examples
julia> cis(π) ≈ -1
true
Base.binomial
— Functionbinomial(n::Integer, k::Integer)
The binomial coefficient $\binom{n}{k}$, being the coefficient of the $k$th term in the polynomial expansion of $(1+x)^n$.
If $n$ is non-negative, then it is the number of ways to choose k
out of n
items:
where $n!$ is the factorial
function.
If $n$ is negative, then it is defined in terms of the identity
Examples
julia> binomial(5, 3)
10
julia> factorial(5) ÷ (factorial(5-3) * factorial(3))
10
julia> binomial(-5, 3)
-35
See also
External links
- Binomial coefficient on Wikipedia.
Base.factorial
— Functionfactorial(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.
Examples
julia> factorial(6)
720
julia> factorial(21)
ERROR: OverflowError: 21 is too large to look up in the table; consider using `factorial(big(21))` instead
Stacktrace:
[...]
julia> factorial(big(21))
51090942171709440000
See also
External links
- Factorial on Wikipedia.
Base.gcd
— Functiongcd(x,y)
Greatest common (positive) divisor (or zero if x
and y
are both zero). The arguments may be integer and rational numbers.
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
Base.lcm
— Functionlcm(x,y)
Least common (non-negative) multiple. The arguments may be integer and rational numbers.
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
Base.gcdx
— Functiongcdx(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 $(d,u,v)$.
The arguments may be integer and rational numbers.
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. 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
— Functionispow2(n::Integer) -> Bool
Test whether n
is a power of two.
Examples
julia> ispow2(4)
true
julia> ispow2(5)
false
Base.nextpow
— Functionnextpow(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.
Examples
julia> nextpow(2, 7)
8
julia> nextpow(2, 9)
16
julia> nextpow(5, 20)
25
julia> nextpow(4, 16)
16
See also prevpow
.
Base.prevpow
— Functionprevpow(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.
Examples
julia> prevpow(2, 7)
4
julia> prevpow(2, 9)
8
julia> prevpow(5, 20)
5
julia> prevpow(4, 16)
16
See also nextpow
.
Base.nextprod
— Functionnextprod([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.
Examples
julia> nextprod([2, 3], 105)
108
julia> 2^2 * 3^3
108
Base.invmod
— Functioninvmod(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 $gcd(x,m) \neq 1$.
Examples
julia> invmod(2,5)
3
julia> invmod(2,3)
2
julia> invmod(5,6)
5
Base.powermod
— Functionpowermod(x::Integer, p::Integer, m)
Compute $x^p \pmod m$.
Examples
julia> powermod(2, 6, 5)
4
julia> mod(2^6, 5)
4
julia> powermod(5, 2, 20)
5
julia> powermod(5, 2, 19)
6
julia> powermod(5, 3, 19)
11
Base.ndigits
— Functionndigits(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
).
Examples
julia> ndigits(12345)
5
julia> ndigits(1022, base=16)
3
julia> string(1022, base=16)
"3fe"
julia> ndigits(123, pad=5)
5
Base.widemul
— Functionwidemul(x, y)
Multiply x
and y
, giving the result as a larger type.
Examples
julia> widemul(Float32(3.), 4.)
12.0
Base.Math.evalpoly
— Functionevalpoly(x, p)
Evaluate the polynomial $\sum_k x^{k-1} p[k]$ for the coefficients p[1]
, p[2]
, ...; that is, the coefficients are given in ascending order by power of x
. Loops are unrolled at compile time if the number of coefficients is statically known, i.e. when p
is a Tuple
. This function generates efficient code using Horner's method if x
is real, or using a Goertzel-like [DK62] algorithm if x
is complex.
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 $\sum_k z^{k-1} c[k]$ for the coefficients c[1]
, c[2]
, ...; that is, the coefficients are given in ascending order by power of z
. This macro expands to efficient inline code that uses either Horner's method or, for complex z
, a more efficient Goertzel-like algorithm.
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 expr
Execute a transformed version of the expression, which calls functions that may violate strict IEEE semantics. This allows the fastest possible operation, but results are undefined – be careful when doing this, as it may change numerical results.
This sets the LLVM Fast-Math flags, and corresponds to the -ffast-math
option in clang. See the notes on performance annotations for more details.
Examples
julia> @fastmath 1+2
3
julia> @fastmath(sin(3))
0.1411200080598672
Customizable binary operators
Some unicode characters can be used to define new binary operators that support infix notation. For example ⊗(x,y) = kron(x,y)
defines the ⊗
(otimes) function to be the Kronecker product, and one can call it as binary operator using infix syntax: C = A ⊗ B
as well as with the usual prefix syntax C = ⊗(A,B)
.
Other characters that support such extensions include \odot ⊙
and \oplus ⊕
The complete list is in the parser code: https://github.com/JuliaLang/julia/blob/master/src/julia-parser.scm
Those that are parsed like *
(in terms of precedence) include * / ÷ % & ⋅ ∘ × |\\| ∩ ∧ ⊗ ⊘ ⊙ ⊚ ⊛ ⊠ ⊡ ⊓ ∗ ∙ ∤ ⅋ ≀ ⊼ ⋄ ⋆ ⋇ ⋉ ⋊ ⋋ ⋌ ⋏ ⋒ ⟑ ⦸ ⦼ ⦾ ⦿ ⧶ ⧷ ⨇ ⨰ ⨱ ⨲ ⨳ ⨴ ⨵ ⨶ ⨷ ⨸ ⨻ ⨼ ⨽ ⩀ ⩃ ⩄ ⩋ ⩍ ⩎ ⩑ ⩓ ⩕ ⩘ ⩚ ⩜ ⩞ ⩟ ⩠ ⫛ ⊍ ▷ ⨝ ⟕ ⟖ ⟗
and those that are parsed like +
include + - |\|| ⊕ ⊖ ⊞ ⊟ |++| ∪ ∨ ⊔ ± ∓ ∔ ∸ ≏ ⊎ ⊻ ⊽ ⋎ ⋓ ⧺ ⧻ ⨈ ⨢ ⨣ ⨤ ⨥ ⨦ ⨧ ⨨ ⨩ ⨪ ⨫ ⨬ ⨭ ⨮ ⨹ ⨺ ⩁ ⩂ ⩅ ⩊ ⩌ ⩏ ⩐ ⩒ ⩔ ⩖ ⩗ ⩛ ⩝ ⩡ ⩢ ⩣
There are many others that are related to arrows, comparisons, and powers.
- DK62Donald Knuth, Art of Computer Programming, Volume 2: Seminumerical Algorithms, Sec. 4.6.4.