/Julia 0.6

# Mathematics

## Mathematical Operators

### Base.:-Method

-(x)

Unary minus operator.

source

### Base.:+Function

+(x, y...)

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

source

### Base.:-Method

-(x, y)

Subtraction operator.

source

### Base.:*Method

*(x, y...)

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

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.

source

### Base.:\Method

\(x, y)

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

julia> 3 \ 6
2.0

julia> inv(3) * 6
2.0

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

julia> A \ x
2-element Array{Float64,1}:
-4.0
4.5

julia> inv(A) * x
2-element Array{Float64,1}:
-4.0
4.5
source

### Base.:^Method

^(x, y)

Exponentiation operator. If x is a matrix, computes matrix exponentiation.

If y is an Int literal (e.g. 2 in x^2 or -3 in x^-3), the Julia code x^y is transformed by the compiler to Base.literal_pow(^, x, Val{y}), to enable compile-time specialization on the value of the exponent. (As a default fallback we have Base.literal_pow(^, x, Val{y}) = ^(x,y), where usually ^ == Base.^ unless ^ has been defined in the calling namespace.)

julia> 3^5
243

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> A^3
2×2 Array{Int64,2}:
37   54
81  118
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 in an efficient manner. This may on some systems be equivalent to x*y+z, or to fma(x,y,z). muladd is used to improve performance. See fma.

Example

julia> muladd(3, 2, 1)
7

julia> 3 * 2 + 1
7
source

### Base.divFunction

div(x, y)
÷(x, y)

The quotient from Euclidean division. Computes x/y, truncated to an integer.

julia> 9 ÷ 4
2

julia> -5 ÷ 3
-1
source

### Base.fldFunction

fld(x, y)

Largest integer less than or equal to x/y.

julia> fld(7.3,5.5)
1.0
source

### Base.cldFunction

cld(x, y)

Smallest integer larger than or equal to x/y.

julia> cld(5.5,2.2)
3.0
source

### Base.modFunction

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.

julia> 129 % Int8
-127
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.

julia> x = 15; y = 4;

julia> x % y
3

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 2π, with the quotient rounded according to the rounding mode r. In other words, the quantity

x - 2π*round(x/(2π),r)

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

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

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

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

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

Example

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

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

### Base.Math.mod2piFunction

mod2pi(x)

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

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

Example

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

### Base.divremFunction

divrem(x, y)

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

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

julia> divrem(7,3)
(2, 1)
source

### Base.fldmodFunction

fldmod(x, y)

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

source

### Base.fld1Function

fld1(x, y)

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

See also: mod1.

julia> x = 15; y = 4;

julia> fld1(x, y)
4

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

julia> x == (fld1(x, y) - 1) * y + mod1(x, y)
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.

julia> mod1(4, 2)
2

julia> mod1(4, 3)
1
source

### Base.fldmod1Function

fldmod1(x, y)

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

See also: fld1, mod1.

source

### Base.://Function

//(num, den)

Divide two integers or rational numbers, giving a Rational result.

julia> 3 // 5
3//5

julia> (3 // 5) // (2 // 1)
3//10
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. If T is not provided, it defaults to Int.

julia> rationalize(5.6)
28//5

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

julia> typeof(numerator(a))
BigInt
source

### Base.numeratorFunction

numerator(x)

Numerator of the rational representation of x.

julia> numerator(2//3)
2

julia> numerator(4)
4
source

### Base.denominatorFunction

denominator(x)

Denominator of the rational representation of x.

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.

julia> Int8(3) << 2
12

julia> bits(Int8(3))
"00000011"

julia> bits(Int8(12))
"00001100"

See also >>, >>>.

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

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

Examples

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

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

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

### Base.:>>Function

>>(x, n)

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

julia> Int8(13) >> 2
3

julia> bits(Int8(13))
"00001101"

julia> bits(Int8(3))
"00000011"

julia> Int8(-14) >> 2
-4

julia> bits(Int8(-14))
"11110010"

julia> bits(Int8(-4))
"11111100"

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.

Example

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

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

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

### Base.:>>>Function

>>>(x, n)

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

julia> Int8(-14) >>> 2
60

julia> bits(Int8(-14))
"11110010"

julia> bits(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.colonFunction

colon(start, [step], stop)

Called by : syntax for constructing ranges.

julia> colon(1, 2, 5)
1:2:5
source
:(start, [step], stop)

Range operator. a:b constructs a range from a to b with a step size of 1, and a:s:b is similar but uses a step size of s. These syntaxes call the function colon. The colon is also used in indexing to select whole dimensions.

source

### Base.rangeFunction

range(start, [step], length)

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

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.

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### Base.StepRangeLenType

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

A range r where r[i] produces values of type T, parametrized by a 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.

source

### Base.:==Function

==(x, y)

Generic equality operator, giving a single Bool result. Falls back to ===. Should be implemented for all types with a notion of equality, based on the abstract value that an instance represents. For example, all numeric types are compared by numeric value, ignoring type. Strings are compared as sequences of characters, ignoring encoding.

Follows IEEE semantics for floating-point numbers.

Collections should generally implement == by calling == recursively on all contents.

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

source

### Base.:!=Function

!=(x, y)
≠(x,y)

Not-equals comparison operator. Always gives the opposite answer as ==. New types should generally not implement this, and rely on the fallback definition !=(x,y) = !(x==y) instead.

julia> 3 != 2
true

julia> "foo" ≠ "foo"
false
source

### Base.:!==Function

!==(x, y)
≢(x,y)

Equivalent to !(x === y).

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

julia> a ≢ b
true

julia> a ≢ a
false
source

### Base.:<Function

<(x, y)

Less-than comparison operator. New numeric types should implement this function for two arguments of the new type. Because of the behavior of floating-point NaN values, < implements a partial order. Types with a canonical partial order should implement <, and types with a canonical total order should implement isless.

julia> 'a' < 'b'
true

julia> "abc" < "abd"
true

julia> 5 < 3
false
source

### Base.:<=Function

<=(x, y)
≤(x,y)

Less-than-or-equals comparison operator.

julia> 'a' <= 'b'
true

julia> 7 ≤ 7 ≤ 9
true

julia> "abc" ≤ "abc"
true

julia> 5 <= 3
false
source

### Base.:>Function

>(x, y)

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

julia> 'a' > 'b'
false

julia> 7 > 3 > 1
true

julia> "abc" > "abd"
false

julia> 5 > 3
true
source

### Base.:>=Function

>=(x, y)
≥(x,y)

Greater-than-or-equals comparison operator.

julia> 'a' >= 'b'
false

julia> 7 ≥ 7 ≥ 3
true

julia> "abc" ≥ "abc"
true

julia> 5 >= 3
true
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. For floating-point numbers, uses < but throws an error for unordered arguments.

julia> cmp(1, 2)
-1

julia> cmp(2, 1)
1

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

### Base.:~Function

~(x)

Bitwise not.

Examples

julia> ~4
-5

julia> ~10
-11

julia> ~true
false
source

### Base.:&Function

&(x, y)

Bitwise and.

Examples

julia> 4 & 10
0

julia> 4 & 12
4
source

### Base.:|Function

|(x, y)

Bitwise or.

Examples

julia> 4 | 10
14

julia> 4 | 1
5
source

### Base.xorFunction

xor(x, y)
⊻(x, y)

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

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

### Base.:!Function

!(x)

Boolean not.

julia> !true
false

julia> !false
true

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. Example:

julia> str = "∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"
"∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"

julia> filter(isalpha, str)
"εδxyδfxfyε"

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

### &&Keyword

x && y

Short-circuiting boolean AND.

source

### ||Keyword

x || y

Short-circuiting boolean OR.

source

## Mathematical Functions

### Base.isapproxFunction

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

Inexact equality comparison: true if norm(x-y) <= atol + rtol*max(norm(x), norm(y)). The default atol is zero and the default rtol depends on the types of x and y. The keyword argument nans determines whether or not NaN values are considered equal (defaults to false).

For real or complex floating-point values, rtol defaults to sqrt(eps(typeof(real(x-y)))). This corresponds to requiring equality of about half of the significand digits. For other types, rtol defaults to zero.

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

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

julia> 0.1 ≈ (0.1 - 1e-10)
true

julia> isapprox(10, 11; atol = 2)
true

julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0])
true
source

### Base.sinFunction

sin(x)

Compute sine of x, where x is in radians.

source

### Base.cosFunction

cos(x)

Compute cosine of x, where x is in radians.

source

### Base.tanFunction

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.

source

### Base.Math.cosdFunction

cosd(x)

Compute cosine of x, where x is in degrees.

source

### Base.Math.tandFunction

tand(x)

Compute tangent of x, where x is in degrees.

source

### Base.Math.sinpiFunction

sinpi(x)

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

source

### Base.Math.cospiFunction

cospi(x)

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

source

### Base.sinhFunction

sinh(x)

Compute hyperbolic sine of x.

source

### Base.coshFunction

cosh(x)

Compute hyperbolic cosine of x.

source

### Base.tanhFunction

tanh(x)

Compute hyperbolic tangent of x.

source

### Base.asinFunction

asin(x)

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

source

### Base.acosFunction

acos(x)

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

source

### Base.atanFunction

atan(x)

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

source

### Base.Math.atan2Function

atan2(y, x)

Compute the inverse tangent of y/x, using the signs of both x and y to determine the quadrant of the return value.

source

### Base.Math.asindFunction

asind(x)

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

source

### Base.Math.acosdFunction

acosd(x)

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

source

### Base.Math.atandFunction

atand(x)

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

source

### Base.Math.secFunction

sec(x)

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

source

### Base.Math.cscFunction

csc(x)

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

source

### Base.Math.cotFunction

cot(x)

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

source

### Base.Math.secdFunction

secd(x)

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

source

### Base.Math.cscdFunction

cscd(x)

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

source

### Base.Math.cotdFunction

cotd(x)

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

source

### Base.Math.asecFunction

asec(x)

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

source

### Base.Math.acscFunction

acsc(x)

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

source

### Base.Math.acotFunction

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.

source

### Base.Math.acscdFunction

acscd(x)

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

source

### Base.Math.acotdFunction

acotd(x)

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

source

### Base.Math.sechFunction

sech(x)

Compute the hyperbolic secant of x

source

### Base.Math.cschFunction

csch(x)

Compute the hyperbolic cosecant of x.

source

### Base.Math.cothFunction

coth(x)

Compute the hyperbolic cotangent of x.

source

### Base.asinhFunction

asinh(x)

Compute the inverse hyperbolic sine of x.

source

### Base.acoshFunction

acosh(x)

Compute the inverse hyperbolic cosine of x.

source

### Base.atanhFunction

atanh(x)

Compute the inverse hyperbolic tangent of x.

source

### Base.Math.asechFunction

asech(x)

Compute the inverse hyperbolic secant of x.

source

### Base.Math.acschFunction

acsch(x)

Compute the inverse hyperbolic cosecant of x.

source

### Base.Math.acothFunction

acoth(x)

Compute the inverse hyperbolic cotangent of x.

source

### Base.Math.sincFunction

sinc(x)

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

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).

source

### Base.Math.deg2radFunction

deg2rad(x)

Convert x from degrees to radians.

julia> deg2rad(90)
1.5707963267948966
source

### Base.Math.rad2degFunction

rad2deg(x)

Convert x from radians to degrees.

julia> rad2deg(pi)
180.0
source

### Base.Math.hypotFunction

hypot(x, y)

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

Examples

julia> a = 10^10;

julia> hypot(a, a)
1.4142135623730951e10

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

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

source

### Base.logMethod

log(x)

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

There is an experimental variant in the Base.Math.JuliaLibm module, which is typically faster and more accurate.

source

### Base.logMethod

log(b,x)

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

julia> log(4,8)
1.5

julia> log(4,2)
0.5
Note

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.

Example

julia> log2(4)
2.0

julia> log2(10)
3.321928094887362
source

### Base.log10Function

log10(x)

Compute the logarithm of x to base 10. Throws DomainError for negative Real arguments.

Example

julia> log10(100)
2.0

julia> log10(2)
0.3010299956639812
source

### Base.log1pFunction

log1p(x)

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

There is an experimental variant in the Base.Math.JuliaLibm module, which is typically faster and more accurate.

Examples

julia> log1p(-0.5)
-0.6931471805599453

julia> log1p(0)
0.0
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 $x \times 2^{exp}$.

source

### Base.expFunction

exp(x)

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

source

### Base.exp2Function

exp2(x)

Compute $2^x$.

julia> exp2(5)
32.0
source

### Base.exp10Function

exp10(x)

Compute $10^x$.

Examples

julia> exp10(2)
100.0

julia> exp10(0.2)
1.5848931924611136
source

### Base.Math.ldexpFunction

ldexp(x, n)

Compute $x \times 2^n$.

Example

julia> ldexp(5., 2)
20.0
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.

Example

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

### Base.expm1Function

expm1(x)

Accurately compute $e^x-1$.

source

### Base.roundMethod

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

Rounds x to an integer value according to the provided RoundingMode, returning a value of the same type as x. When not specifying a rounding mode the global mode will be used (see rounding), which by default is round to the nearest integer (RoundNearest mode), with ties (fractional values of 0.5) being rounded to the nearest even integer.

julia> round(1.7)
2.0

julia> round(1.5)
2.0

julia> round(2.5)
2.0

The optional RoundingMode argument will change how the number gets rounded.

round(T, x, [r::RoundingMode]) converts the result to type T, throwing an InexactError if the value is not representable.

round(x, digits) rounds to the specified number of digits after the decimal place (or before if negative). round(x, digits, base) rounds using a base other than 10.

julia> round(pi, 2)
3.14

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

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

julia> x = 1.15
1.15

julia> @sprintf "%.20f" x
"1.14999999999999991118"

julia> x < 115//100
true

julia> round(x, 1)
1.2
source

### Base.Rounding.RoundingModeType

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

### 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.RoundNearestTiesAwayConstant

RoundNearestTiesAway

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

source

### Base.Rounding.RoundNearestTiesUpConstant

RoundNearestTiesUp

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

source

### Base.Rounding.RoundToZeroConstant

RoundToZero

round using this rounding mode is an alias for trunc.

source

### Base.Rounding.RoundUpConstant

RoundUp

round using this rounding mode is an alias for ceil.

source

### Base.Rounding.RoundDownConstant

RoundDown

round using this rounding mode is an alias for floor.

source

### Base.roundMethod

round(z, RoundingModeReal, RoundingModeImaginary)

Returns the nearest integral value of the same type as the complex-valued z to z, breaking ties using the specified RoundingModes. The first RoundingMode is used for rounding the real components while the second is used for rounding the imaginary components.

source

### Base.ceilFunction

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

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

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

digits and base work as for round.

source

### Base.floorFunction

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

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

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

digits and base work as for round.

source

### Base.truncFunction

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

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

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

digits and base work as for round.

source

### Base.unsafe_truncFunction

unsafe_trunc(T, x)

unsafe_trunc(T, x) returns the nearest integral value of type T whose absolute value is less than or equal to x. If the value is not representable by T, an arbitrary value will be returned.

source

### Base.signifFunction

signif(x, digits, [base])

Rounds (in the sense of round) x so that there are digits significant digits, under a base base representation, default 10. E.g., signif(123.456, 2) is 120.0, and signif(357.913, 4, 2) is 352.0.

source

### Base.minFunction

min(x, y, ...)

Return the minimum of the arguments. See also the minimum function to take the minimum element from a collection.

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

### Base.maxFunction

max(x, y, ...)

Return the maximum of the arguments. See also the maximum function to take the maximum element from a collection.

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

### Base.minmaxFunction

minmax(x, y)

Return (min(x,y), max(x,y)). See also: extrema that returns (minimum(x), maximum(x)).

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

### Base.Math.clampFunction

clamp(x, lo, hi)

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

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

### Base.Math.clamp!Function

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

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

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.

julia> abs(-3)
3

julia> abs(1 + im)
1.4142135623730951

julia> abs(typemin(Int64))
-9223372036854775808
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.add_with_overflowFunction

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

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

source

### Base.Checked.sub_with_overflowFunction

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

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

source

### Base.Checked.mul_with_overflowFunction

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

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

source

### Base.abs2Function

abs2(x)

Squared absolute value of x.

julia> abs2(-3)
9
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).

source

### Base.signbitFunction

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

### Base.flipsignFunction

flipsign(x, y)

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

julia> flipsign(5, 3)
5

julia> flipsign(5, -3)
-5
source

### Base.sqrtFunction

sqrt(x)

Return $\sqrt{x}$. Throws DomainError for negative Real arguments. Use complex negative arguments instead. The prefix operator √ is equivalent to sqrt.

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.cbrtFunction

cbrt(x::Real)

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

The prefix operator ∛ is equivalent to cbrt.

julia> cbrt(big(27))
3.000000000000000000000000000000000000000000000000000000000000000000000000000000
source

### Base.realMethod

real(z)

Return the real part of the complex number z.

julia> real(1 + 3im)
1
source

### Base.imagFunction

imag(z)

Return the imaginary part of the complex number z.

julia> imag(1 + 3im)
3
source

### Base.reimFunction

reim(z)

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

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

### Base.conjFunction

conj(z)

Compute the complex conjugate of a complex number z.

julia> conj(1 + 3im)
1 - 3im
source
conj(v::RowVector)

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

Example

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

julia> conj(v)
1×2 RowVector{Complex{Int64},ConjArray{Complex{Int64},1,Array{Complex{Int64},1}}}:
1-1im  1+1im
source

### Base.angleFunction

angle(z)

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

source

### Base.cisFunction

cis(z)

Return $\exp(iz)$.

source

### Base.binomialFunction

binomial(n, k)

Number of ways to choose k out of n items.

Example

julia> binomial(5, 3)
10

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

### Base.factorialFunction

factorial(n)

Factorial of n. If n is an Integer, the factorial is computed as an integer (promoted to at least 64 bits). Note that this may overflow if n is not small, but you can use factorial(big(n)) to compute the result exactly in arbitrary precision. If n is not an Integer, factorial(n) is equivalent to gamma(n+1).

julia> factorial(6)
720

julia> factorial(21)
ERROR: OverflowError()
[...]

julia> factorial(21.0)
5.109094217170944e19

julia> factorial(big(21))
51090942171709440000
source

### Base.gcdFunction

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

### Base.lcmFunction

lcm(x,y)

Least common (non-negative) multiple.

Examples

julia> lcm(2,3)
6

julia> lcm(-2,3)
6
source

### Base.gcdxFunction

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

### Base.ispow2Function

ispow2(n::Integer) -> Bool

Test whether n is a power of two.

Examples

julia> ispow2(4)
true

julia> ispow2(5)
false
source

### Base.nextpow2Function

nextpow2(n::Integer)

The smallest power of two not less than n. Returns 0 for n==0, and returns -nextpow2(-n) for negative arguments.

Examples

julia> nextpow2(16)
16

julia> nextpow2(17)
32
source

### Base.prevpow2Function

prevpow2(n::Integer)

The largest power of two not greater than n. Returns 0 for n==0, and returns -prevpow2(-n) for negative arguments.

Examples

julia> prevpow2(5)
4

julia> prevpow2(0)
0
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.

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

### 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.

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

### Base.nextprodFunction

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.

Example

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

julia> 2^2 * 3^3
108
source

### Base.invmodFunction

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

### 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.Math.gammaFunction

gamma(x)

Compute the gamma function of x.

source

### Base.Math.lgammaFunction

lgamma(x)

Compute the logarithm of the absolute value of gamma for Real x, while for Complex x compute the principal branch cut of the logarithm of gamma(x) (defined for negative real(x) by analytic continuation from positive real(x)).

source

### Base.Math.lfactFunction

lfact(x)

Compute the logarithmic factorial of a nonnegative integer x. Equivalent to lgamma of x + 1, but lgamma extends this function to non-integer x.

source

### Base.Math.betaFunction

beta(x, y)

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

source

### Base.Math.lbetaFunction

lbeta(x, y)

Natural logarithm of the absolute value of the beta function $\log(|\operatorname{B}(x,y)|)$.

source

### Base.ndigitsFunction

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

Compute the number of digits in integer n written in base b. The base b must not be in [-1, 0, 1].

Examples

julia> ndigits(12345)
5

julia> ndigits(1022, 16)
3

julia> base(16, 1022)
"3fe"
source

### Base.widemulFunction

widemul(x, y)

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

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

### Base.Math.@evalpolyMacro

@evalpoly(z, c...)

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

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

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

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

## Statistics

### Base.meanFunction

mean(f::Function, v)

Apply the function f to each element of v and take the mean.

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

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

Compute the mean of whole array v, or optionally along the dimensions in region.

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended.

source

### Base.mean!Function

mean!(r, v)

Compute the mean of v over the singleton dimensions of r, and write results to r.

source

### Base.stdFunction

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

Compute the sample standard deviation of a vector or array v, optionally along dimensions in region. The algorithm returns an estimator of the generative distribution's standard deviation under the assumption that each entry of v is an IID drawn from that generative distribution. This computation is equivalent to calculating sqrt(sum((v - mean(v)).^2) / (length(v) - 1)). A pre-computed mean may be provided. If corrected is true, then the sum is scaled with n-1, whereas the sum is scaled with n if corrected is false where n = length(x).

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended.

source

### Base.stdmFunction

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

Compute the sample standard deviation of a vector v with known mean m. If corrected is true, then the sum is scaled with n-1, whereas the sum is scaled with n if corrected is false where n = length(x).

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended.

source

### Base.varFunction

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

Compute the sample variance of a vector or array v, optionally along dimensions in region. The algorithm will return an estimator of the generative distribution's variance under the assumption that each entry of v is an IID drawn from that generative distribution. This computation is equivalent to calculating sum(abs2, v - mean(v)) / (length(v) - 1). If corrected is true, then the sum is scaled with n-1, whereas the sum is scaled with n if corrected is false where n = length(x). The mean mean over the region may be provided.

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended.

source

### Base.varmFunction

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

Compute the sample variance of a collection v with known mean(s) m, optionally over region. m may contain means for each dimension of v. If corrected is true, then the sum is scaled with n-1, whereas the sum is scaled with n if corrected is false where n = length(x).

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended.

source

### Base.middleFunction

middle(x)

Compute the middle of a scalar value, which is equivalent to x itself, but of the type of middle(x, x) for consistency.

source
middle(x, y)

Compute the middle of two reals x and y, which is equivalent in both value and type to computing their mean ((x + y) / 2).

source
middle(range)

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

julia> middle(1:10)
5.5
source
middle(a)

Compute the middle of an array a, which consists of finding its extrema and then computing their mean.

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

julia> middle(a)
5.95
source

### Base.medianFunction

median(v[, region])

Compute the median of an entire array v, or, optionally, along the dimensions in region. For an even number of elements no exact median element exists, so the result is equivalent to calculating mean of two median elements.

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended.

source

### Base.median!Function

median!(v)

Like median, but may overwrite the input vector.

source

### Base.quantileFunction

quantile(v, p; sorted=false)

Compute the quantile(s) of a vector v at a specified probability or vector p. The keyword argument sorted indicates whether v can be assumed to be sorted.

The p should be on the interval [0,1], and v should not have any NaN values.

Quantiles are computed via linear interpolation between the points ((k-1)/(n-1), v[k]), for k = 1:n where n = length(v). This corresponds to Definition 7 of Hyndman and Fan (1996), and is the same as the R default.

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended. quantile will throw an ArgumentError in the presence of NaN values in the data array.

• Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages", The American Statistician, Vol. 50, No. 4, pp. 361-365

source

### Base.quantile!Function

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

Compute the quantile(s) of a vector v at the probabilities p, with optional output into array q (if not provided, a new output array is created). The keyword argument sorted indicates whether v can be assumed to be sorted; if false (the default), then the elements of v may be partially sorted.

The elements of p should be on the interval [0,1], and v should not have any NaN values.

Quantiles are computed via linear interpolation between the points ((k-1)/(n-1), v[k]), for k = 1:n where n = length(v). This corresponds to Definition 7 of Hyndman and Fan (1996), and is the same as the R default.

Note

Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArrays.jl package is recommended. quantile! will throw an ArgumentError in the presence of NaN values in the data array.

• Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages", The American Statistician, Vol. 50, No. 4, pp. 361-365

source

### Base.covFunction

cov(x[, corrected=true])

Compute the variance of the vector x. If corrected is true (the default) then the sum is scaled with n-1, whereas the sum is scaled with n if corrected is false where n = length(x).

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

Compute the covariance matrix of the matrix X along the dimension vardim. If corrected is true (the default) then the sum is scaled with n-1, whereas the sum is scaled with n if corrected is false where n = size(X, vardim).

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

Compute the covariance between the vectors x and y. If corrected is true (the default), computes $\frac{1}{n-1}\sum_{i=1}^n (x_i-\bar x) (y_i-\bar y)^*$ where $*$ denotes the complex conjugate and n = length(x) = length(y). If corrected is false, computes $rac{1}{n}sum_{i=1}^n (x_i-\bar x) (y_i-\bar y)^*$.

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

Compute the covariance between the vectors or matrices X and Y along the dimension vardim. If corrected is true (the default) then the sum is scaled with n-1, whereas the sum is scaled with n if corrected is false where n = size(X, vardim) = size(Y, vardim).

source

### Base.corFunction

cor(x)

Return the number one.

source
cor(X[, vardim=1])

Compute the Pearson correlation matrix of the matrix X along the dimension vardim.

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cor(x, y)

Compute the Pearson correlation between the vectors x and y.

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

Compute the Pearson correlation between the vectors or matrices X and Y along the dimension vardim.

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## Signal Processing

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

### Base.DFT.fftFunction

fft(A [, dims])

Performs a multidimensional FFT of the array A. The optional dims argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. Most efficient if the size of A along the transformed dimensions is a product of small primes; see nextprod(). See also plan_fft() for even greater efficiency.

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

$\operatorname{DFT}(A)[k] = \sum_{n=1}^{\operatorname{length}(A)} \exp\left(-i\frac{2\pi (n-1)(k-1)}{\operatorname{length}(A)} \right) A[n].$

A multidimensional FFT simply performs this operation along each transformed dimension of A.

Note
• Julia starts FFTW up with 1 thread by default. Higher performance is usually possible by increasing number of threads. Use FFTW.set_num_threads(Sys.CPU_CORES) to use as many threads as cores on your system.

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

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### Base.DFT.fft!Function

fft!(A [, dims])

Same as fft, but operates in-place on A, which must be an array of complex floating-point numbers.

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### Base.DFT.ifftFunction

ifft(A [, dims])

Multidimensional inverse FFT.

A one-dimensional inverse FFT computes

$\operatorname{IDFT}(A)[k] = \frac{1}{\operatorname{length}(A)} \sum_{n=1}^{\operatorname{length}(A)} \exp\left(+i\frac{2\pi (n-1)(k-1)} {\operatorname{length}(A)} \right) A[n].$

A multidimensional inverse FFT simply performs this operation along each transformed dimension of A.

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### Base.DFT.ifft!Function

ifft!(A [, dims])

Same as ifft, but operates in-place on A.

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### Base.DFT.bfftFunction

bfft(A [, dims])

Similar to ifft, but computes an unnormalized inverse (backward) transform, which must be divided by the product of the sizes of the transformed dimensions in order to obtain the inverse. (This is slightly more efficient than ifft because it omits a scaling step, which in some applications can be combined with other computational steps elsewhere.)

$\operatorname{BDFT}(A)[k] = \operatorname{length}(A) \operatorname{IDFT}(A)[k]$source

### Base.DFT.bfft!Function

bfft!(A [, dims])

Same as bfft, but operates in-place on A.

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### Base.DFT.plan_fftFunction

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

Pre-plan an optimized FFT along given dimensions (dims) of arrays matching the shape and type of A. (The first two arguments have the same meaning as for fft.) Returns an object P which represents the linear operator computed by the FFT, and which contains all of the information needed to compute fft(A, dims) quickly.

To apply P to an array A, use P * A; in general, the syntax for applying plans is much like that of matrices. (A plan can only be applied to arrays of the same size as the A for which the plan was created.) You can also apply a plan with a preallocated output array Â by calling A_mul_B!(Â, plan, A). (For A_mul_B!, however, the input array A must be a complex floating-point array like the output Â.) You can compute the inverse-transform plan by inv(P) and apply the inverse plan with P \ Â (the inverse plan is cached and reused for subsequent calls to inv or \), and apply the inverse plan to a pre-allocated output array A with A_ldiv_B!(A, P, Â).

The flags argument is a bitwise-or of FFTW planner flags, defaulting to FFTW.ESTIMATE. e.g. passing FFTW.MEASURE or FFTW.PATIENT will instead spend several seconds (or more) benchmarking different possible FFT algorithms and picking the fastest one; see the FFTW manual for more information on planner flags. The optional timelimit argument specifies a rough upper bound on the allowed planning time, in seconds. Passing FFTW.MEASURE or FFTW.PATIENT may cause the input array A to be overwritten with zeros during plan creation.

plan_fft! is the same as plan_fft but creates a plan that operates in-place on its argument (which must be an array of complex floating-point numbers). plan_ifft and so on are similar but produce plans that perform the equivalent of the inverse transforms ifft and so on.

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### Base.DFT.plan_ifftFunction

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

Same as plan_fft, but produces a plan that performs inverse transforms ifft.

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### Base.DFT.plan_bfftFunction

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

Same as plan_fft, but produces a plan that performs an unnormalized backwards transform bfft.

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### Base.DFT.plan_fft!Function

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

Same as plan_fft, but operates in-place on A.

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### Base.DFT.plan_ifft!Function

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

Same as plan_ifft, but operates in-place on A.

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### Base.DFT.plan_bfft!Function

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

Same as plan_bfft, but operates in-place on A.

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### Base.DFT.rfftFunction

rfft(A [, dims])

Multidimensional FFT of a real array A, exploiting the fact that the transform has conjugate symmetry in order to save roughly half the computational time and storage costs compared with fft. If A has size (n_1, ..., n_d), the result has size (div(n_1,2)+1, ..., n_d).

The optional dims argument specifies an iterable subset of one or more dimensions of A to transform, similar to fft. Instead of (roughly) halving the first dimension of A in the result, the dims[1] dimension is (roughly) halved in the same way.

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### Base.DFT.irfftFunction

irfft(A, d [, dims])

Inverse of rfft: for a complex array A, gives the corresponding real array whose FFT yields A in the first half. As for rfft, dims is an optional subset of dimensions to transform, defaulting to 1:ndims(A).

d is the length of the transformed real array along the dims[1] dimension, which must satisfy div(d,2)+1 == size(A,dims[1]). (This parameter cannot be inferred from size(A) since both 2*size(A,dims[1])-2 as well as 2*size(A,dims[1])-1 are valid sizes for the transformed real array.)

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### Base.DFT.brfftFunction

brfft(A, d [, dims])

Similar to irfft but computes an unnormalized inverse transform (similar to bfft), which must be divided by the product of the sizes of the transformed dimensions (of the real output array) in order to obtain the inverse transform.

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### Base.DFT.plan_rfftFunction

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

Pre-plan an optimized real-input FFT, similar to plan_fft except for rfft instead of fft. The first two arguments, and the size of the transformed result, are the same as for rfft.

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### Base.DFT.plan_brfftFunction

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

Pre-plan an optimized real-input unnormalized transform, similar to plan_rfft except for brfft instead of rfft. The first two arguments and the size of the transformed result, are the same as for brfft.

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### Base.DFT.plan_irfftFunction

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

Pre-plan an optimized inverse real-input FFT, similar to plan_rfft except for irfft and brfft, respectively. The first three arguments have the same meaning as for irfft.

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### Base.DFT.FFTW.dctFunction

dct(A [, dims])

Performs a multidimensional type-II discrete cosine transform (DCT) of the array A, using the unitary normalization of the DCT. The optional dims argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. Most efficient if the size of A along the transformed dimensions is a product of small primes; see nextprod. See also plan_dct for even greater efficiency.

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### Base.DFT.FFTW.dct!Function

dct!(A [, dims])

Same as dct!, except that it operates in-place on A, which must be an array of real or complex floating-point values.

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### Base.DFT.FFTW.idctFunction

idct(A [, dims])

Computes the multidimensional inverse discrete cosine transform (DCT) of the array A (technically, a type-III DCT with the unitary normalization). The optional dims argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. Most efficient if the size of A along the transformed dimensions is a product of small primes; see nextprod. See also plan_idct for even greater efficiency.

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### Base.DFT.FFTW.idct!Function

idct!(A [, dims])

Same as idct!, but operates in-place on A.

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### Base.DFT.FFTW.plan_dctFunction

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

Pre-plan an optimized discrete cosine transform (DCT), similar to plan_fft except producing a function that computes dct. The first two arguments have the same meaning as for dct.

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### Base.DFT.FFTW.plan_dct!Function

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

Same as plan_dct, but operates in-place on A.

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### Base.DFT.FFTW.plan_idctFunction

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

Pre-plan an optimized inverse discrete cosine transform (DCT), similar to plan_fft except producing a function that computes idct. The first two arguments have the same meaning as for idct.

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### Base.DFT.FFTW.plan_idct!Function

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

Same as plan_idct, but operates in-place on A.

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### Base.DFT.fftshiftMethod

fftshift(x)

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

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### Base.DFT.fftshiftMethod

fftshift(x,dim)

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

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### Base.DFT.ifftshiftFunction

ifftshift(x, [dim])

Undoes the effect of fftshift.

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### Base.DSP.filtFunction

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

Apply filter described by vectors a and b to vector x, with an optional initial filter state vector si (defaults to zeros).

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### Base.DSP.filt!Function

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

Same as filt but writes the result into the out argument, which may alias the input x to modify it in-place.

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### Base.DSP.deconvFunction

deconv(b,a) -> c

Construct vector c such that b = conv(a,c) + r. Equivalent to polynomial division.

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### Base.DSP.convFunction

conv(u,v)

Convolution of two vectors. Uses FFT algorithm.

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### Base.DSP.conv2Function

conv2(u,v,A)

2-D convolution of the matrix A with the 2-D separable kernel generated by the vectors u and v. Uses 2-D FFT algorithm.

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conv2(B,A)

2-D convolution of the matrix B with the matrix A. Uses 2-D FFT algorithm.

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### Base.DSP.xcorrFunction

xcorr(u,v)

Compute the cross-correlation of two vectors.

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The following functions are defined within the Base.FFTW module.

### Base.DFT.FFTW.r2rFunction

r2r(A, kind [, dims])

Performs a multidimensional real-input/real-output (r2r) transform of type kind of the array A, as defined in the FFTW manual. kind specifies either a discrete cosine transform of various types (FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFT10, or FFTW.REDFT11), a discrete sine transform of various types (FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFT10, or FFTW.RODFT11), a real-input DFT with halfcomplex-format output (FFTW.R2HC and its inverse FFTW.HC2R), or a discrete Hartley transform (FFTW.DHT). The kind argument may be an array or tuple in order to specify different transform types along the different dimensions of A; kind[end] is used for any unspecified dimensions. See the FFTW manual for precise definitions of these transform types, at http://www.fftw.org/doc.

The optional dims argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. kind[i] is then the transform type for dims[i], with kind[end] being used for i > length(kind).

See also plan_r2r to pre-plan optimized r2r transforms.

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### Base.DFT.FFTW.r2r!Function

r2r!(A, kind [, dims])

Same as r2r, but operates in-place on A, which must be an array of real or complex floating-point numbers.

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### Base.DFT.FFTW.plan_r2rFunction

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

Pre-plan an optimized r2r transform, similar to plan_fft except that the transforms (and the first three arguments) correspond to r2r and r2r!, respectively.

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### Base.DFT.FFTW.plan_r2r!Function

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

Similar to plan_fft, but corresponds to r2r!.

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