Mathematics

Mathematical Operators

Base.:-Method
-(x)

Unary minus operator.

See also: abs, flipsign.

Examples

julia> -1
-1

julia> -(2)
-2

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

julia> -(true)  # promotes to Int
-1

julia> -(0x003)
0xfffd
source
Base.:+Function
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.

+(x, y...)

Addition operator.

Infix x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...), which by default then calls (x+y) + z + ... starting from the left.

Note that overflow is possible for most integer types, including the default Int, when adding large numbers.

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25

julia> [1,2] + [3,4]
2-element Vector{Int64}:
 4
 6

julia> typemax(Int) + 1 < 0
true
source
Base.:-Method
-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1

julia> -(2, 4.5)
-2.5
source
Base.:*Method
*(x, y...)

Multiplication operator.

Infix x*y*z*... calls this function with all arguments, i.e. *(x, y, z, ...), which by default then calls (x*y) * z * ... starting from the left.

Juxtaposition such as 2pi also calls *(2, pi). Note that this operation has higher precedence than a literal *. Note also that juxtaposition "0x..." (integer zero times a variable whose name starts with x) is forbidden as it clashes with unsigned integer literals: 0x01 isa UInt8.

Note that overflow is possible for most integer types, including the default Int, when multiplying large numbers.

Examples

julia> 2 * 7 * 8
112

julia> *(2, 7, 8)
112

julia> [2 0; 0 3] * [1, 10]  # matrix * vector
2-element Vector{Int64}:
  2
 30

julia> 1/2pi, 1/2*pi  # juxtaposition has higher precedence
(0.15915494309189535, 1.5707963267948966)

julia> x = [1, 2]; x'x  # adjoint vector * vector
5
source
Base.:/Function
/(x, y)

Right division operator: multiplication of x by the inverse of y on the right.

Gives floating-point results for integer arguments. See ÷ for integer division, or // for Rational results.

Examples

julia> 1/2
0.5

julia> 4/2
2.0

julia> 4.5/2
2.25
source
A / B

Matrix 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
Base.:\Method
\(x, y)

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

Examples

julia> 3 \ 6
2.0

julia> inv(3) * 6
2.0

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

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

julia> inv(A) * x
2-element Vector{Float64}:
  6.5
 -7.0
source
Base.:^Method
^(x, y)

Exponentiation operator.

If x and y are integers, the result may overflow. To enter numbers in scientific notation, use Float64 literals such as 1.2e3 rather than 1.2 * 10^3.

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.

See also exp2, <<.

Examples

julia> 3^5
243

julia> 3^-1  # uses Base.literal_pow
0.3333333333333333

julia> p = -1;

julia> 3^p
ERROR: DomainError with -1:
Cannot raise an integer x to a negative power -1.
[...]

julia> 3.0^p
0.3333333333333333

julia> 10^19 > 0  # integer overflow
false

julia> big(10)^19 == 1e19
true
source
Base.fmaFunction
fma(x, y, z)

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

source
Base.muladdFunction
muladd(x, y, z)

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

Examples

julia> muladd(3, 2, 1)
7

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

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

Julia 1.6

These methods require Julia 1.6 or later.

Examples

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

julia> muladd(A, B, z)
2×2 Matrix{Float64}:
   3.0    3.0
 107.0  107.0
Base.invMethod
inv(x)

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

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

Examples

julia> inv(2)
0.5

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

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

julia> inv(2//3)
3//2
Julia 1.2

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

source
Base.divFunction
div(x, y)
÷(x, y)

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

See also: cld, fld, rem, divrem.

Examples

julia> 9 ÷ 4
2

julia> -5 ÷ 3
-1

julia> 5.0 ÷ 2
2.0

julia> div.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 -1  -1  -1  0  0  0  0  0  1  1  1
source
Base.divMethod
div(x, y, r::RoundingMode=RoundToZero)

The quotient from Euclidean (integer) division. Computes x / y, rounded to an integer according to the rounding mode r. In other words, the quantity

round(x / y, r)

without any intermediate rounding.

Julia 1.4

The three-argument method taking a RoundingMode requires Julia 1.4 or later.

See also fld and cld, which are special cases of this function.

Julia 1.9

RoundFromZero requires at least Julia 1.9.

Examples:

julia> div(4, 3, RoundToZero) # Matches div(4, 3)
1
julia> div(4, 3, RoundDown) # Matches fld(4, 3)
1
julia> div(4, 3, RoundUp) # Matches cld(4, 3)
2
julia> div(5, 2, RoundNearest)
2
julia> div(5, 2, RoundNearestTiesAway)
3
julia> div(-5, 2, RoundNearest)
-2
julia> div(-5, 2, RoundNearestTiesAway)
-3
julia> div(-5, 2, RoundNearestTiesUp)
-2
julia> div(4, 3, RoundFromZero)
2
julia> div(-4, 3, RoundFromZero)
-2
source
Base.fldFunction
fld(x, y)

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

See also div, cld, fld1.

Examples

julia> fld(7.3, 5.5)
1.0

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

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

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

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

source
Base.cldFunction
cld(x, y)

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

See also div, fld.

Examples

julia> cld(5.5, 2.2)
3.0

julia> cld.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 -1  -1  -1  0  0  0  1  1  1  2  2
source
Base.modFunction
mod(x::Integer, r::AbstractUnitRange)

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

See also mod1.

Examples

julia> mod(0, Base.OneTo(3))  # mod1(0, 3)
3

julia> mod(3, 0:2)  # mod(3, 3)
0
Julia 1.3

This method requires at least Julia 1.3.

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

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

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

Note

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

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

julia> mod(8, 3)
2

julia> mod(9, 3)
0

julia> mod(8.9, 3)
2.9000000000000004

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

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

julia> mod.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 1  2  0  1  2  0  1  2  0  1  2
source
rem(x::Integer, T::Type{<:Integer}) -> T
mod(x::Integer, T::Type{<:Integer}) -> T
%(x::Integer, T::Type{<:Integer}) -> T

Find y::T such that xy (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
source
Base.remFunction
rem(x, y)
%(x, y)

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

See also: div, mod, mod1, divrem.

Examples

julia> x = 15; y = 4;

julia> x % y
3

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

julia> rem.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 -2  -1  0  -2  -1  0  1  2  0  1  2
source
Base.remMethod
rem(x, y, r::RoundingMode=RoundToZero)

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

x - y * round(x / y, r)

without any intermediate rounding.

  • if r == RoundNearest, then the result is exact, and in the interval $[-|y| / 2, |y| / 2]$. See also RoundNearest.

  • if r == RoundToZero (default), then the result is exact, and in the interval $[0, |y|)$ if x is positive, or $(-|y|, 0]$ otherwise. See also RoundToZero.

  • if r == RoundDown, then the result is in the interval $[0, y)$ if y is positive, or $(y, 0]$ otherwise. The result may not be exact if x and y have different signs, and abs(x) < abs(y). See also RoundDown.

  • if r == RoundUp, then the result is in the interval $(-y, 0]$ if y is positive, or $[0, -y)$ otherwise. The result may not be exact if x and y have the same sign, and abs(x) < abs(y). See also RoundUp.

  • if r == RoundFromZero, then the result is in the interval $(-y, 0]$ if y is positive, or $[0, -y)$ otherwise. The result may not be exact if x and y have the same sign, and abs(x) < abs(y). See also RoundFromZero.

Julia 1.9

RoundFromZero requires at least Julia 1.9.

Examples:

julia> x = 9; y = 4;

julia> x % y  # same as rem(x, y)
1

julia> x ÷ y  # same as div(x, y)
2

julia> x == div(x, y) * y + rem(x, y)
true
source
Base.Math.rem2piFunction
rem2pi(x, r::RoundingMode)

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

x - 2π*round(x/(2π),r)

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

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

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

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

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

Examples

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

julia> rem2pi(7pi/4, RoundDown)
5.497787143782138
source
Base.Math.mod2piFunction
mod2pi(x)

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

This function computes a floating point representation of the modulus after division by numerically exact , 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 .

Note

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

Examples

julia> mod2pi(9*pi/4)
0.7853981633974481
source
Base.divremFunction
divrem(x, y, r::RoundingMode=RoundToZero)

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

See also: fldmod, cld.

Examples

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

julia> divrem(7, 3)
(2, 1)
source
Base.fldmodFunction
fldmod(x, y)

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

See also: fld, cld, fldmod1.

source
Base.fld1Function
fld1(x, y)

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

See also mod1, fldmod1.

Examples

julia> x = 15; y = 4;

julia> fld1(x, y)
4

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

julia> x == (fld1(x, y) - 1) * y + mod1(x, y)
true
source
Base.mod1Function
mod1(x, y)

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

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.

See also mod, fld1, fldmod1.

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
source
Base.://Function
//(num, den)

Divide two integers or rational numbers, giving a Rational result. More generally, // can be used for exact rational division of other numeric types with integer or rational components, such as complex numbers with integer components.

Note that floating-point (AbstractFloat) arguments are not permitted by // (even if the values are rational). The arguments must be subtypes of Integer, Rational, or composites thereof.

Examples

julia> 3 // 5
3//5

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

julia> (1+2im) // (3+4im)
11//25 + 2//25*im

julia> 1.0 // 2
ERROR: MethodError: no method matching //(::Float64, ::Int64)
[...]
source
Base.rationalizeFunction
rationalize([T<:Integer=Int,] x; tol::Real=eps(x))

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

Examples

julia> rationalize(5.6)
28//5

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

julia> typeof(numerator(a))
BigInt
source
Base.numeratorFunction
numerator(x)

Numerator of the rational representation of x.

Examples

julia> numerator(2//3)
2

julia> numerator(4)
4
source
Base.denominatorFunction
denominator(x)

Denominator of the rational representation of x.

Examples

julia> denominator(2//3)
3

julia> denominator(4)
1
source
Base.:<<Function
<<(x, n)

Left bit shift operator, x << n. For n >= 0, the result is x shifted left by n bits, filling with 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"

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

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

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

Examples

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

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

julia> B << -1
5-element BitVector:
 0
 1
 0
 1
 0
source
Base.:>>Function
>>(x, n)

Right bit shift operator, x >> n. For n >= 0, the result is x shifted right by n bits, 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"

See also >>>, <<.

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

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

Examples

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

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

julia> B >> -1
5-element BitVector:
 0
 1
 0
 0
 0
source
Base.:>>>Function
>>>(x, n)

Unsigned right bit shift operator, x >>> n. For n >= 0, the result is x shifted right by n bits, 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 >>.

See also >>, <<.

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

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

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

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

Julia 1.5

This function requires Julia 1.5 or later.

See also: <<, circshift, BitArray.

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

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

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

julia> bitstring(bitrotate(0b01110010, 8))
"01110010"
source
Base.::Function
:expr

Quote 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}
source
Base.rangeFunction
range(start, stop, length)
range(start, stop; length, step)
range(start; length, stop, step)
range(;start, length, stop, step)

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

  • Call range with any three of start, step, stop, length.
  • Call range with two of start, stop, length. In this case step will be assumed to be one. If both arguments are Integers, a UnitRange will be returned.
  • Call range with one of stop or length. start and step will be assumed to be one.

See Extended Help for additional details on the returned type. See also logrange for logarithmically spaced points.

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)
1.0:1.0:6.0

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

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

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

Julia 1.1

stop as a positional argument requires at least Julia 1.1.

Julia 1.7

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

Julia 1.8

The versions with stop as a sole keyword argument, or length as a sole keyword argument require at least Julia 1.8.

Extended Help

range will produce a Base.OneTo when the arguments are Integers and

  • Only length is provided
  • Only stop is provided

range will produce a UnitRange when the arguments are Integers and

  • Only start and stop are provided
  • Only length and stop are provided

A UnitRange is not produced if step is provided even if specified as one.

source
Base.OneToType
Base.OneTo(n)

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

source
Base.StepRangeLenType
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. The syntax a:b or a:b:c, where any of a, b, or c are floating-point numbers, creates a StepRangeLen.

Julia 1.7

The 4th type parameter L requires at least Julia 1.7.

source
Base.lograngeFunction
logrange(start, stop, length)
logrange(start, stop; length)

Construct a specialized array whose elements are spaced logarithmically between the given endpoints. That is, the ratio of successive elements is a constant, calculated from the length.

This is similar to geomspace in Python. Unlike PowerRange in Mathematica, you specify the number of elements not the ratio. Unlike logspace in Python and Matlab, the start and stop arguments are always the first and last elements of the result, not powers applied to some base.

Examples

julia> logrange(10, 4000, length=3)
3-element Base.LogRange{Float64, Base.TwicePrecision{Float64}}:
 10.0, 200.0, 4000.0

julia> ans[2] ≈ sqrt(10 * 4000)  # middle element is the geometric mean
true

julia> range(10, 40, length=3)[2] ≈ (10 + 40)/2  # arithmetic mean
true

julia> logrange(1f0, 32f0, 11)
11-element Base.LogRange{Float32, Float64}:
 1.0, 1.41421, 2.0, 2.82843, 4.0, 5.65685, 8.0, 11.3137, 16.0, 22.6274, 32.0

julia> logrange(1, 1000, length=4) ≈ 10 .^ (0:3)
true

See the LogRange type for further details.

See also range for linearly spaced points.

Julia 1.11

This function requires at least Julia 1.11.

source
Base.LogRangeType
LogRange{T}(start, stop, len) <: AbstractVector{T}

A range whose elements are spaced logarithmically between start and stop, with spacing controlled by len. Returned by logrange.

Like LinRange, the first and last elements will be exactly those provided, but intermediate values may have small floating-point errors. These are calculated using the logs of the endpoints, which are stored on construction, often in higher precision than T.

Examples

julia> logrange(1, 4, length=5)
5-element Base.LogRange{Float64, Base.TwicePrecision{Float64}}:
 1.0, 1.41421, 2.0, 2.82843, 4.0

julia> Base.LogRange{Float16}(1, 4, 5)
5-element Base.LogRange{Float16, Float64}:
 1.0, 1.414, 2.0, 2.828, 4.0

julia> logrange(1e-310, 1e-300, 11)[1:2:end]
6-element Vector{Float64}:
 1.0e-310
 9.999999999999974e-309
 9.999999999999981e-307
 9.999999999999988e-305
 9.999999999999994e-303
 1.0e-300

julia> prevfloat(1e-308, 5) == ans[2]
true

Note that integer eltype T is not allowed. Use for instance round.(Int, xs), or explicit powers of some integer base:

julia> xs = logrange(1, 512, 4)
4-element Base.LogRange{Float64, Base.TwicePrecision{Float64}}:
 1.0, 8.0, 64.0, 512.0

julia> 2 .^ (0:3:9) |> println
[1, 8, 64, 512]
Julia 1.11

This type requires at least Julia 1.11.

source
Base.:==Function
==(x, y)

Generic equality operator. Falls back to ===. Should be implemented for all types with a notion of equality, based on the abstract value that an instance represents. For example, all numeric types are compared by numeric value, ignoring type. Strings are compared as sequences of characters, ignoring encoding. Collections of the same type generally compare their key sets, and if those are ==, then compare the values for each of those keys, returning true if all such pairs are ==. Other properties are typically not taken into account (such as the exact type).

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). Collections generally implement three-valued logic akin to all, returning missing if any operands contain missing values and all other pairs are equal. Use isequal or === to always get a Bool result.

Implementation

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

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

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

source
Base.:!=Function
!=(x, y)
≠(x,y)

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

Implementation

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

Examples

julia> 3 != 2
true

julia> "foo" ≠ "foo"
false
source
!=(x)

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

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:!==Function
!==(x, y)
≢(x,y)

Always gives the opposite answer as ===.

Examples

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

julia> a ≢ b
true

julia> a ≢ a
false
source
Base.:<Function
<(x, y)

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

Implementation

New 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.

See also isunordered.

Examples

julia> 'a' < 'b'
true

julia> "abc" < "abd"
true

julia> 5 < 3
false
source
<(x)

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

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:<=Function
<=(x, y)
≤(x,y)

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

Examples

julia> 'a' <= 'b'
true

julia> 7 ≤ 7 ≤ 9
true

julia> "abc" ≤ "abc"
true

julia> 5 <= 3
false
source
<=(x)

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

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:>Function
>(x, y)

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

Implementation

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

Examples

julia> 'a' > 'b'
false

julia> 7 > 3 > 1
true

julia> "abc" > "abd"
false

julia> 5 > 3
true
source
>(x)

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

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:>=Function
>=(x, y)
≥(x,y)

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

Examples

julia> 'a' >= 'b'
false

julia> 7 ≥ 7 ≥ 3
true

julia> "abc" ≥ "abc"
true

julia> 5 >= 3
true
source
>=(x)

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

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.cmpFunction
cmp(x,y)

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

Examples

julia> cmp(1, 2)
-1

julia> cmp(2, 1)
1

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

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

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

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

Examples

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

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

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

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

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

julia> cmp("α", "a")
1

julia> cmp("b", "β")
-1
source
Base.:~Function
~(x)

Bitwise not.

See also: !, &, |.

Examples

julia> ~4
-5

julia> ~10
-11

julia> ~true
false
source
Base.:&Function
x & y

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

See also: |, xor, &&.

Examples

julia> 4 & 10
0

julia> 4 & 12
4

julia> true & missing
missing

julia> false & missing
false
source
Base.:|Function
x | y

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

See also: &, xor, ||.

Examples

julia> 4 | 10
14

julia> 4 | 1
5

julia> true | missing
true

julia> false | missing
missing
source
Base.xorFunction
xor(x, y)
⊻(x, y)

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

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

Examples

julia> xor(true, false)
true

julia> xor(true, true)
false

julia> xor(true, missing)
missing

julia> false ⊻ false
false

julia> [true; true; false] .⊻ [true; false; false]
3-element BitVector:
 0
 1
 0
source
Base.nandFunction
nand(x, y)
⊼(x, y)

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

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

Examples

julia> nand(true, false)
true

julia> nand(true, true)
false

julia> nand(true, missing)
missing

julia> false ⊼ false
true

julia> [true; true; false] .⊼ [true; false; false]
3-element BitVector:
 0
 1
 1
source
Base.norFunction
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
source
Base.:!Function
!(x)

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

See also ~ for bitwise not.

Examples

julia> !true
false

julia> !false
true

julia> !missing
missing

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

Predicate function negation: when the argument of ! is a function, it returns a 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: |-| <  ⇒ |()-()| < "
Julia 1.9

Starting with Julia 1.9, !f returns a ComposedFunction instead of an anonymous function.

source
&&Keyword
x && y

Short-circuiting boolean AND.

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

Examples

julia> x = 3;

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

julia> x < 0 && error("expected positive x")
false
source
||Keyword
x || y

Short-circuiting boolean OR.

See also: |, xor, &&.

Examples

julia> pi < 3 || ℯ < 3
true

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

Mathematical Functions

Base.isapproxFunction
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 (absolute tolerance) is zero and the default rtol (relative tolerance) 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.

Julia 1.6

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

Examples

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

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

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

julia> 0.1 + 1e-10 ≈ 0.1
true

julia> 1e-10 ≈ 0
false

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

julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`
true
source
isapprox(x; kwargs...) / ≈(x; kwargs...)

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

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

Julia 1.5

This method requires Julia 1.5 or later.

source
Base.sinMethod
sin(x)

Compute sine of x, where x is in radians.

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

Examples

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

julia> sind(45)
0.7071067811865476

julia> sinpi(1/4)
0.7071067811865475

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

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

julia> round(exp(im*pi/6), digits=3)
0.866 + 0.5im
source
Base.tanMethod
tan(x)

Compute tangent of x, where x is in radians.

source
Base.Math.sindFunction
sind(x)

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

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.cosdFunction
cosd(x)

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

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.tandFunction
tand(x)

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

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.sincosdFunction
sincosd(x)

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

Julia 1.3

This function requires at least Julia 1.3.

source
Base.Math.cospiFunction
cospi(x)

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

source
Base.tanhMethod
tanh(x)

Compute hyperbolic tangent of x.

See also tan, atanh.

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
source
Base.asinMethod
asin(x)

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

See also asind for output in degrees.

Examples

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

julia> asind.((0, 1/2, 1))
(0.0, 30.000000000000004, 90.0)
source
Base.acosMethod
acos(x)

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

source
Base.atanMethod
atan(y)
atan(y, x)

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

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

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

See also atand for degrees.

Examples

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

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

julia> rad2deg(atan(1, -√3))
150.0
source
Base.Math.asindFunction
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.

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.acosdFunction
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.

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.atandFunction
atand(y)
atand(y,x)

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

Julia 1.7

The one-argument method supports square matrix arguments as of Julia 1.7.

source
Base.Math.asecMethod
asec(x)

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

source
Base.Math.acscMethod
acsc(x)

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

source
Base.Math.acotMethod
acot(x)

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

source
Base.Math.asecdFunction
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.

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.acscdFunction
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.

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.acotdFunction
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.

Julia 1.7

Matrix arguments require Julia 1.7 or later.

source
Base.Math.sincFunction
sinc(x)

Compute normalized sinc function $\operatorname{sinc}(x) = \sin(\pi x) / (\pi x)$ if $x \neq 0$, and $1$ if $x = 0$.

See also cosc, its derivative.

source
Base.Math.coscFunction
cosc(x)

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

See also sinc.

source
Base.Math.hypotFunction
hypot(x, y)

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

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

hypot(x...)

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

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 was called with a negative real argument but 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
source
Base.logMethod
log(x)

Compute the natural logarithm of x.

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

See also , log1p, log2, log10.

Examples

julia> log(2)
0.6931471805599453

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

julia> log(-3 + 0im)
1.0986122886681098 + 3.141592653589793im

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

julia> log.(exp.(-1:1))
3-element Vector{Float64}:
 -1.0
  0.0
  1.0
source
Base.logMethod
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 was called with a negative real argument but will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

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

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

julia> log(100,1000000)
2.9999999999999996

julia> log10(1000000)/2
3.0
source
Base.log2Function
log2(x)

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

See also: exp2, ldexp, ispow2.

Examples

julia> log2(4)
2.0

julia> log2(10)
3.321928094887362

julia> log2(-2)
ERROR: DomainError with -2.0:
log2 was called with a negative real argument but 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
source
Base.log10Function
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 was called with a negative real argument but will only return a complex result if called with a complex argument. Try log10(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(f::Symbol, x::Float64) at ./math.jl:31
[...]
source
Base.log1pFunction
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 was called with a real argument < -1 but will only return a complex result if called with a complex argument. Try log1p(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
source
Base.Math.frexpFunction
frexp(val)

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

See also significand, exponent, ldexp.

Examples

julia> frexp(6.0)
(0.75, 3)

julia> significand(6.0), exponent(6.0)  # interval [1, 2) instead
(1.5, 2)

julia> frexp(0.0), frexp(NaN), frexp(-Inf)  # exponent would give an error
((0.0, 0), (NaN, 0), (-Inf, 0))
source
Base.expMethod
exp(x)

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

See also exp2, exp10 and cis.

Examples

julia> exp(1.0)
2.718281828459045

julia> exp(im * pi) ≈ cis(pi)
true
source
Base.exp2Function
exp2(x)

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

See also ldexp, <<.

Examples

julia> exp2(5)
32.0

julia> 2^5
32

julia> exp2(63) > typemax(Int)
true
source
Base.exp10Function
exp10(x)

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

Examples

julia> exp10(2)
100.0

julia> 10^2
100
source
Base.Math.modfFunction
modf(x)

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

Examples

julia> modf(3.5)
(0.5, 3.0)

julia> modf(-3.5)
(-0.5, -3.0)
source
Base.expm1Function
expm1(x)

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

Examples

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

julia> exp(1e-16) - 1
0.0
source
Base.roundFunction
round([T,] x, [r::RoundingMode])
round(x, [r::RoundingMode]; digits::Integer=0, base = 10)
round(x, [r::RoundingMode]; sigdigits::Integer, base = 10)

Rounds the number x.

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

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

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

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

Examples

julia> round(1.7)
2.0

julia> round(Int, 1.7)
2

julia> round(1.5)
2.0

julia> round(2.5)
2.0

julia> round(pi; digits=2)
3.14

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

julia> round(123.456; sigdigits=2)
120.0

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

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

julia> x = 1.15
1.15

julia> big(1.15)
1.149999999999999911182158029987476766109466552734375

julia> x < 115//100
true

julia> round(x, digits=1)
1.2

Extensions

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

source
Base.Rounding.RoundNearestConstant
RoundNearest

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

source
Base.Rounding.RoundFromZeroConstant
RoundFromZero

Rounds away from zero.

Julia 1.9

RoundFromZero requires at least Julia 1.9. Prior versions support RoundFromZero for BigFloats only.

Examples

julia> BigFloat("1.0000000000000001", 5, RoundFromZero)
1.06
source
Base.roundMethod
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.

Examples

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
source
Base.ceilFunction
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 ceiled value is not representable as a T.

Keywords digits, sigdigits and base work as for round.

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

source
Base.floorFunction
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 floored value is not representable a T.

Keywords digits, sigdigits and base work as for round.

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

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

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

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

Keywords digits, sigdigits and base work as for round.

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

See also: %, floor, unsigned, unsafe_trunc.

Examples

julia> trunc(2.22)
2.0

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

julia> trunc(Int, -2.22)
-2
source
Base.unsafe_truncFunction
unsafe_trunc(T, x)

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

Examples

julia> unsafe_trunc(Int, -2.2)
-2

julia> unsafe_trunc(Int, NaN)
-9223372036854775808
source
Base.minFunction
min(x, y, ...)

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

Examples

julia> min(2, 5, 1)
1

julia> min(4, missing, 6)
missing
source
Base.maxFunction
max(x, y, ...)

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

Examples

julia> max(2, 5, 1)
5

julia> max(5, missing, 6)
missing
source
Base.minmaxFunction
minmax(x, y)

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

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

Examples

julia> minmax('c','b')
('b', 'c')
source
Base.Math.clampFunction
clamp(x, lo, hi)

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

See also clamp!, min, max.

Julia 1.3

missing as the first argument requires at least 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
 10
source
clamp(x, T)::T

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

See also trunc.

Examples

julia> clamp(200, Int8)
127

julia> clamp(-200, Int8)
-128

julia> trunc(Int, 4pi^2)
39
source
clamp(x::Integer, r::AbstractUnitRange)

Clamp x to lie within range r.

Julia 1.6

This method requires at least Julia 1.6.

source
Base.Math.clamp!Function
clamp!(array::AbstractArray, lo, hi)

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

Julia 1.3

missing entries in array require at least 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
source
Base.absFunction
abs(x)

The absolute value of x.

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

See also: abs2, unsigned, sign.

Examples

julia> abs(-3)
3

julia> abs(1 + im)
1.4142135623730951

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

julia> maximum(abs, [1, -2, 3, -4])
4
source
Base.CheckedModule
Checked

The Checked module provides arithmetic functions for the built-in signed and unsigned Integer types which throw an error when an overflow occurs. They are named like checked_sub, checked_div, etc. In addition, add_with_overflow, sub_with_overflow, mul_with_overflow return both the unchecked results and a boolean value denoting the presence of an overflow.

source
Base.Checked.checked_absFunction
Base.checked_abs(x)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_negFunction
Base.checked_neg(x)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_addFunction
Base.checked_add(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_subFunction
Base.checked_sub(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_mulFunction
Base.checked_mul(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_divFunction
Base.checked_div(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_remFunction
Base.checked_rem(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_fldFunction
Base.checked_fld(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_modFunction
Base.checked_mod(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_cldFunction
Base.checked_cld(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_powFunction
Base.checked_pow(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.abs2Function
abs2(x)

Squared absolute value of x.

This can be faster than abs(x)^2, especially for complex numbers where abs(x) requires a square root via hypot.

See also abs, conj, real.

Examples

julia> abs2(-3)
9

julia> abs2(3.0 + 4.0im)
25.0

julia> sum(abs2, [1+2im, 3+4im])  # LinearAlgebra.norm(x)^2
30
source
Base.copysignFunction
copysign(x, y) -> z

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

Examples

julia> copysign(1, -2)
-1

julia> copysign(-1, 2)
1
source
Base.signFunction
sign(x)

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

See also signbit, zero, copysign, flipsign.

Examples

julia> sign(-4.0)
-1.0

julia> sign(99)
1

julia> sign(-0.0)
-0.0

julia> sign(0 + im)
0.0 + 1.0im
source
Base.signbitFunction
signbit(x)

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

See also sign and copysign.

Examples

julia> signbit(-4)
true

julia> signbit(5)
false

julia> signbit(5.5)
false

julia> signbit(-4.1)
true
source
Base.flipsignFunction
flipsign(x, y)

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

Examples

julia> flipsign(5, 3)
5

julia> flipsign(5, -3)
-5
source
Base.sqrtMethod
sqrt(x)

Return $\sqrt{x}$.

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

The prefix operator is equivalent to sqrt.

See also: hypot.

Examples

julia> sqrt(big(81))
9.0

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

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

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

julia> .√(1:4)
4-element Vector{Float64}:
 1.0
 1.4142135623730951
 1.7320508075688772
 2.0
source
Base.isqrtFunction
isqrt(n::Integer)

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

julia> isqrt(5)
2
source
Base.Math.cbrtMethod
cbrt(x::Real)

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

The prefix operator is equivalent to cbrt.

Examples

julia> cbrt(big(27))
3.0

julia> cbrt(big(-27))
-3.0
source
Base.realFunction
real(z)

Return the real part of the complex number z.

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

Examples

julia> real(1 + 3im)
1
source
real(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)
Float64
source
real(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
source
Base.imagFunction
imag(z)

Return the imaginary part of the complex number z.

See also: conj, reim, adjoint, angle.

Examples

julia> imag(1 + 3im)
3
source
imag(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
source
Base.reimFunction
reim(z)

Return a tuple of the real and imaginary parts of the complex number z.

Examples

julia> reim(1 + 3im)
(1, 3)
source
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))
source
Base.conjFunction
conj(z)

Compute the complex conjugate of a complex number z.

See also: angle, adjoint.

Examples

julia> conj(1 + 3im)
1 - 3im
source
conj(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
source
Base.angleFunction
angle(z)

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

Returns a number -pi ≤ angle(z) ≤ pi, and is thus discontinuous along the negative real axis.

See also: atan, cis, rad2deg.

Examples

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

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

julia> rad2deg(angle(-1 + 1e-20im))
180.0

julia> rad2deg(angle(-1 - 1e-20im))
-180.0
source
Base.cisFunction
cis(x)

More efficient method for exp(im*x) by using Euler's formula: $\cos(x) + i \sin(x) = \exp(i x)$.

See also cispi, sincos, exp, angle.

Examples

julia> cis(π) ≈ -1
true
source
Base.cispiFunction
cispi(x)

More accurate method for cis(pi*x) (especially for large x).

See also cis, sincospi, exp, angle.

Examples

julia> cispi(10000)
1.0 + 0.0im

julia> cispi(0.25 + 1im)
0.030556854645954562 + 0.03055685464595456im
Julia 1.6

This function requires Julia 1.6 or later.

source
Base.binomialFunction
binomial(n::Integer, k::Integer)

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

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

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

where $n!$ is the factorial function.

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

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

See also factorial.

Examples

julia> binomial(5, 3)
10

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

julia> binomial(-5, 3)
-35

External links

source
binomial(x::Number, k::Integer)

The generalized binomial coefficient, defined for k ≥ 0 by the polynomial

\[\frac{1}{k!} \prod_{j=0}^{k-1} (x - j)\]

When k < 0 it returns zero.

For the case of integer x, this is equivalent to the ordinary integer binomial coefficient

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

Further generalizations to non-integer k are mathematically possible, but involve the Gamma function and/or the beta function, which are not provided by the Julia standard library but are available in external packages such as SpecialFunctions.jl.

External links

source
Base.factorialFunction
factorial(n::Integer)

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

See also binomial.

Examples

julia> factorial(6)
720

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

julia> factorial(big(21))
51090942171709440000

External links

source
Base.gcdFunction
gcd(x, y...)

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

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

julia> gcd(6, 9)
3

julia> gcd(6, -9)
3

julia> gcd(6, 0)
6

julia> gcd(0, 0)
0

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

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

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

julia> gcd(0, 0, 10, 15)
5
source
Base.lcmFunction
lcm(x, y...)

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

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

julia> lcm(2, 3)
6

julia> lcm(-2, 3)
6

julia> lcm(0, 3)
0

julia> lcm(0, 0)
0

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

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

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

julia> lcm(1, 3, 5, 7)
105
source
Base.gcdxFunction
gcdx(a, b)

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

The arguments may be integer and rational numbers.

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

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

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

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

source
Base.ispow2Function
ispow2(n::Number) -> Bool

Test whether n is an integer power of two.

See also count_ones, prevpow, nextpow.

Examples

julia> ispow2(4)
true

julia> ispow2(5)
false

julia> ispow2(4.5)
false

julia> ispow2(0.25)
true

julia> ispow2(1//8)
true
Julia 1.6

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

source
Base.nextpowFunction
nextpow(a, x)

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

See also prevpow.

Examples

julia> nextpow(2, 7)
8

julia> nextpow(2, 9)
16

julia> nextpow(5, 20)
25

julia> nextpow(4, 16)
16
source
Base.prevpowFunction
prevpow(a, x)

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

See also nextpow, isqrt.

Examples

julia> prevpow(2, 7)
4

julia> prevpow(2, 9)
8

julia> prevpow(5, 20)
5

julia> prevpow(4, 16)
16
source
Base.nextprodFunction
nextprod(factors::Union{Tuple,AbstractVector}, n)

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

Examples

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

julia> 2^2 * 3^3
108
Julia 1.6

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

source
Base.invmodFunction
invmod(n::Integer, m::Integer)

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

Examples

julia> invmod(2, 5)
3

julia> invmod(2, 3)
2

julia> invmod(5, 6)
5
source
invmod(n::Integer, T) where {T <: Base.BitInteger}
invmod(n::T) where {T <: Base.BitInteger}

Compute the modular inverse of n in the integer ring of type T, i.e. modulo 2^N where N = 8*sizeof(T) (e.g. N = 32 for Int32). In other words these methods satisfy the following identities:

n * invmod(n) == 1
(n * invmod(n, T)) % T == 1
(n % T) * invmod(n, T) == 1

Note that * here is modular multiplication in the integer ring, T.

Specifying the modulus implied by an integer type as an explicit value is often inconvenient since the modulus is by definition too big to be represented by the type.

The modular inverse is computed much more efficiently than the general case using the algorithm described in https://arxiv.org/pdf/2204.04342.pdf.

Julia 1.11

The invmod(n) and invmod(n, T) methods require Julia 1.11 or later.

source
Base.powermodFunction
powermod(x::Integer, p::Integer, m)

Compute $x^p \pmod m$.

Examples

julia> powermod(2, 6, 5)
4

julia> mod(2^6, 5)
4

julia> powermod(5, 2, 20)
5

julia> powermod(5, 2, 19)
6

julia> powermod(5, 3, 19)
11
source
Base.ndigitsFunction
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
source
Base.add_sumFunction
Base.add_sum(x, y)

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

source
Base.widemulFunction
widemul(x, y)

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

See also promote, Base.add_sum.

Examples

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

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

julia> widemul(typemax(Int8), typemax(Int8))  # == 127^2
16129
source
Base.Math.evalpolyFunction
evalpoly(x, p)

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

Julia 1.4

This function requires Julia 1.4 or later.

Examples

julia> evalpoly(2, (1, 2, 3))
17
source
Base.Math.@evalpolyMacro
@evalpoly(z, c...)

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

See also evalpoly.

Examples

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

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

julia> @evalpoly(2, 1, 1, 1)
7
source
Base.FastMath.@fastmathMacro
@fastmath expr

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

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

Examples

julia> @fastmath 1+2
3

julia> @fastmath(sin(3))
0.1411200080598672
source

Customizable binary operators

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

Other characters that support such extensions include \odot and \oplus

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

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

  • DK62Donald Knuth, Art of Computer Programming, Volume 2: Seminumerical Algorithms, Sec. 4.6.4.