Mathematics

# Mathematics

## Mathematical Operators

-(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
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.

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+(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
source
-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1

julia> -(2, 4.5)
-2.5
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*(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
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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
source
\(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
source
^(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
source
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.

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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
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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
source
div(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
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fld(x, y)

Largest integer less than or equal to x/y.

Examples

julia> fld(7.3,5.5)
1.0
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cld(x, y)

Smallest integer larger than or equal to x/y.

Examples

julia> cld(5.5,2.2)
3.0
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mod(x, y)
rem(x, y, RoundDown)

The reduction of x modulo y, or equivalently, the remainder of x after floored division by y, i.e.

x - y*fld(x,y)

if computed without intermediate rounding.

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

Note

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

julia> mod(8, 3)
2

julia> mod(9, 3)
0

julia> mod(8.9, 3)
2.9000000000000004

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

julia> mod(-eps(), 3)
3.0
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> 129 % Int8
-127
source
rem(x, y)
%(x, y)

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

Examples

julia> x = 15; y = 4;

julia> x % y
3

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

Compute the remainder of x after integer division by 2π, with the quotient rounded according to the rounding mode r. In other words, the quantity

x - 2π*round(x/(2π),r)

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

• if r == RoundNearest, then the result is in the interval $[-π, π]$. This will generally be the most accurate result. 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
mod2pi(x)

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

This function computes a floating point representation of the modulus after division by numerically exact 2π, and is therefore not exactly the same as mod(x,2π), which would compute the modulus of x relative to division by the floating-point number 2π.

Examples

julia> mod2pi(9*pi/4)
0.7853981633974481
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divrem(x, y)

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

Examples

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

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

The floored quotient and modulus after division. Equivalent to (fld(x,y), mod(x,y)).

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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
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.

See also: fld1, fldmod1.

Examples

julia> mod1(4, 2)
2

julia> mod1(4, 3)
1
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fldmod1(x, y)

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

See also: fld1, mod1.

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//(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
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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
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numerator(x)

Numerator of the rational representation of x.

Examples

julia> numerator(2//3)
2

julia> numerator(4)
4
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denominator(x)

Denominator of the rational representation of x.

Examples

julia> denominator(2//3)
3

julia> denominator(4)
1
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<<(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 >>, >>>.

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

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

Examples

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

julia> B << 1
5-element BitArray{1}:
false
true
false
false
false

julia> B << -1
5-element BitArray{1}:
false
true
false
true
false
source
>>(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 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 >>>, <<.

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>>(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}:
true
false
true
false
false

julia> B >> 1
5-element BitArray{1}:
false
true
false
true
false

julia> B >> -1
5-element BitArray{1}:
false
true
false
false
false
source
>>>(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 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 >>, <<.

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

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

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Base.::Function.
(:)(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.

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range(start; 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 (a LinRange).

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 (a StepRange).

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
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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.

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StepRangeLen{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 reference value, a step, and the length. By default ref is the starting value r[1], but alternatively you can supply it as the value of r[offset] for some other index 1 <= offset <= len. In conjunction with TwicePrecision this can be used to implement ranges that are free of roundoff error.

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==(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.

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==(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.

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==(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
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!=(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, 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
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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
source
<=(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
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, 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
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.

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.

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.

Examples

julia> 4 | 10
14

julia> 4 | 1
5

julia> true | missing
true

julia> false | missing
missing
source
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 BitArray{1}:
false
true
false
source
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}:
false  true  false
source
!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: |-| <  ⇒ |()-()| < "
source
&&Keyword.
x && y

Short-circuiting boolean AND.

source
||Keyword.
x || y

Short-circuiting boolean OR.

source

## Mathematical Functions

isapprox(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 vecnorm but may be changed by passing a norm::Function keyword argument. (For numbers, norm is the same thing as abs.) When x and y are arrays, if norm(x-y) is not finite (i.e. ±Inf or NaN), the comparison falls back to checking whether all elements of x and y are approximately equal component-wise.

The binary operator ≈ is equivalent to isapprox with the default arguments, and x ≉ y is equivalent to !isapprox(x,y).

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
source
sin(x)

Compute sine of x, where x is in radians.

source
cos(x)

Compute cosine of x, where x is in radians.

source
sincos(x)

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

source
tan(x)

Compute tangent of x, where x is in radians.

source
sind(x)

Compute sine of x, where x is in degrees.

source
cosd(x)

Compute cosine of x, where x is in degrees.

source
tand(x)

Compute tangent of x, where x is in degrees.

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sinpi(x)

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

source
cospi(x)

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

source
sinh(x)

Compute hyperbolic sine of x.

source
cosh(x)

Compute hyperbolic cosine of x.

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tanh(x)

Compute hyperbolic tangent of x.

source
asin(x)

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

source
acos(x)

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

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atan(y)
atan(y, x)

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

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

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.

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asind(x)

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

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acosd(x)

Compute the inverse cosine of x, where the output is in degrees.

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atand(y)
atand(y,x)

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

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sec(x)

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

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csc(x)

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

source
cot(x)

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

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secd(x)

Compute the secant of x, where x is in degrees.

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cscd(x)

Compute the cosecant of x, where x is in degrees.

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cotd(x)

Compute the cotangent of x, where x is in degrees.

source
asec(x)

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

source
acsc(x)

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

source
acot(x)

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

source
asecd(x)

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

source
acscd(x)

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

source
acotd(x)

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

source
sech(x)

Compute the hyperbolic secant of x.

source
csch(x)

Compute the hyperbolic cosecant of x.

source
coth(x)

Compute the hyperbolic cotangent of x.

source
asinh(x)

Compute the inverse hyperbolic sine of x.

source
acosh(x)

Compute the inverse hyperbolic cosine of x.

source
atanh(x)

Compute the inverse hyperbolic tangent of x.

source
asech(x)

Compute the inverse hyperbolic secant of x.

source
acsch(x)

Compute the inverse hyperbolic cosecant of x.

source
acoth(x)

Compute the inverse hyperbolic cotangent of x.

source
sinc(x)

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

source
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).

source
deg2rad(x)

Convert x from degrees to radians.

Examples

julia> deg2rad(90)
1.5707963267948966
source
rad2deg(x)

Convert x from radians to degrees.

Examples

julia> rad2deg(pi)
180.0
source
hypot(x, y)

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

Examples

julia> a = 10^10;

julia> hypot(a, a)
1.4142135623730951e10

julia> √(a^2 + a^2) # a^2 overflows
ERROR: DomainError with -2.914184810805068e18:
sqrt will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
Stacktrace:
[...]
source
hypot(x...)

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

source
log(x)

Compute the natural logarithm of x. Throws DomainError for negative Real arguments. Use complex negative arguments to obtain complex results.

Examples

julia> log(2)
0.6931471805599453

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

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

Examples

julia> log(4,8)
1.5

julia> log(4,2)
0.5

julia> log(-2, 3)
ERROR: DomainError with -2.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

julia> log(2, -3)
ERROR: DomainError with -3.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
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
log2(x)

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

Examples

julia> log2(4)
2.0

julia> log2(10)
3.321928094887362

julia> log2(-2)
ERROR: DomainError with -2.0:
NaN result for non-NaN input.
Stacktrace:
[1] nan_dom_err at ./math.jl:325 [inlined]
[...]
source
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:
NaN result for non-NaN input.
Stacktrace:
[1] nan_dom_err at ./math.jl:325 [inlined]
[...]
source
log1p(x)

Accurate natural logarithm of 1+x. Throws DomainError for Real arguments less than -1.

Examples

julia> log1p(-0.5)
-0.6931471805599453

julia> log1p(0)
0.0

julia> log1p(-2)
ERROR: DomainError with -2.0:
log1p will only return a complex result if called with a complex argument. Try log1p(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
source
frexp(val)

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

source
exp(x)

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

Examples

julia> exp(1.0)
2.718281828459045
source
exp2(x)

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

Examples

julia> exp2(5)
32.0
source
exp10(x)

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

Examples

julia> exp10(2)
100.0
source
exp10(x)

Compute $10^x$.

Examples

julia> exp10(2)
100.0

julia> exp10(0.2)
1.5848931924611136
source
ldexp(x, n)

Compute $x \times 2^n$.

Examples

julia> ldexp(5., 2)
20.0
source
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
expm1(x)

Accurately compute $e^x-1$.

source
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 RoundingModer 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.

# 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).

source
RoundingMode

A type used for controlling the rounding mode of floating point operations (via rounding/setrounding functions), or as optional arguments for rounding to the nearest integer (via the round function).

Currently supported rounding modes are:

source
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
RoundNearestTiesAway

Rounds to nearest integer, with ties rounded away from zero (C/C++ round behaviour).

source
RoundNearestTiesUp

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

source
RoundToZero

round using this rounding mode is an alias for trunc.

source
RoundFromZero

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
source
RoundUp

round using this rounding mode is an alias for ceil.

source
RoundDown

round using this rounding mode is an alias for floor.

source
round(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 RoundingModes. 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
source
ceil([T,] x)
ceil(x; digits::Integer= [, base = 10])
ceil(x; sigdigits::Integer= [, base = 10])

ceil(x) returns the nearest integral value of the same type as x that is greater than or equal to x.

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

digits, sigdigits and base work as for round.

source
floor([T,] x)
floor(x; digits::Integer= [, base = 10])
floor(x; sigdigits::Integer= [, base = 10])

floor(x) returns the nearest integral value of the same type as x that is less than or equal to x.

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

digits, sigdigits and base work as for round.

source
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 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.

source
unsafe_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.

source
min(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
source
max(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
source
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
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.

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

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

source
abs(x)

The absolute value of x.

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

Examples

julia> abs(-3)
3

julia> abs(1 + im)
1.4142135623730951

julia> abs(typemin(Int64))
-9223372036854775808
source
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_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_add(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
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_mul(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
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_rem(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
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_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_cld(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
Base.add_with_overflow(x, y) -> (r, f)

Calculates r = x+y, with the flag f indicating whether overflow has occurred.

source
Base.sub_with_overflow(x, y) -> (r, f)

Calculates r = x-y, with the flag f indicating whether overflow has occurred.

source
Base.mul_with_overflow(x, y) -> (r, f)

Calculates r = x*y, with the flag f indicating whether overflow has occurred.

source
abs2(x)

Squared absolute value of x.

Examples

julia> abs2(-3)
9
source
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
sign(x)

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

source
signbit(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
source
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
sqrt(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 -8.1e+01:
NaN result for non-NaN input.
Stacktrace:
[1] sqrt(::BigFloat) at ./mpfr.jl:501
[...]

julia> sqrt(big(complex(-81)))
0.0 + 9.0im
source
isqrt(n::Integer)

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

julia> isqrt(5)
2
source
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
real(z)

Return the real part of the complex number z.

Examples

julia> real(1 + 3im)
1
source
imag(z)

Return the imaginary part of the complex number z.

Examples

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

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

Examples

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

Compute the complex conjugate of a complex number z.

Examples

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

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

Examples

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

-45.0

-135.0
source
cis(z)

Return $\exp(iz)$.

Examples

julia> cis(π) ≈ -1
true
source
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

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

Examples

julia> binomial(5, 3)
10

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

julia> binomial(-5, 3)
-35

source
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.

Examples

julia> factorial(6)
720

julia> factorial(21)
ERROR: OverflowError: 21 is too large to look up in the table
Stacktrace:
[...]

julia> factorial(big(21))
51090942171709440000

source
gcd(x,y)

Greatest common (positive) divisor (or zero if x and y are both zero).

Examples

julia> gcd(6,9)
3

julia> gcd(6,-9)
3
source
lcm(x,y)

Least common (non-negative) multiple.

Examples

julia> lcm(2,3)
6

julia> lcm(-2,3)
6
source
gcdx(x,y)

Computes the greatest common (positive) divisor of x and y and their Bézout coefficients, i.e. the integer coefficients u and v that satisfy $ux+vy = d = gcd(x,y)$. $gcdx(x,y)$ returns $(d,u,v)$.

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 $|u| < |y/d|$ 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.

source
ispow2(n::Integer) -> Bool

Test whether n is a power of two.

Examples

julia> ispow2(4)
true

julia> ispow2(5)
false
source
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.

Examples

julia> nextpow(2, 7)
8

julia> nextpow(2, 9)
16

julia> nextpow(5, 20)
25

julia> nextpow(4, 16)
16

See also prevpow.

source
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.

Examples

julia> prevpow(2, 7)
4

julia> prevpow(2, 9)
8

julia> prevpow(5, 20)
5

julia> prevpow(4, 16)
16

See also nextpow.

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

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

Examples

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

julia> 2^2 * 3^3
108
source
invmod(x,m)

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

Examples

julia> invmod(2,5)
3

julia> invmod(2,3)
2

julia> invmod(5,6)
5
source
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
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).

Examples

julia> ndigits(12345)
5

julia> ndigits(1022, base=16)
3

julia> string(1022, base=16)
"3fe"

5
source
widemul(x, y)

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

Examples

julia> widemul(Float32(3.), 4.)
1.2e+01
source
@evalpoly(z, c...)

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

Examples

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

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

julia> @evalpoly(2, 1, 1, 1)
7
source
@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