# Mathematics¶

## Mathematical Operators¶

-(x)

Unary minus operator.

+(x, y...)

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

-(x, y)

Subtraction operator.

*(x, y...)

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

/(x, y)

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

\(x, y)

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

^(x, y)

Exponentiation operator.

.+(x, y)

.-(x, y)

Element-wise subtraction operator.

.*(x, y)

Element-wise multiplication operator.

./(x, y)

Element-wise right division operator.

.\(x, y)

Element-wise left division operator.

.^(x, y)

Element-wise exponentiation operator.

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.

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.

div(x, y)
÷(x, y)

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

fld(x, y)

Largest integer less than or equal to x/y.

cld(x, y)

Smallest integer larger than or equal to x/y.

mod(x, y)

Modulus after flooring division, returning in the range $$[0,y)$$, if y is positive, or $$(y,0]$$ if y is negative.

x == fld(x,y)*y + mod(x,y)

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.

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.

x == div(x,y)*y + rem(x,y)

divrem(x, y)

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

fldmod(x, y)

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

fld1(x, y)

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

x == fld(x,y)*y + mod(x,y)
x == (fld1(x,y)-1)*y + mod1(x,y)

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.

fldmod1(x, y)

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

//(num, den)

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

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(num(a))
BigInt

num(x)

Numerator of the rational representation of x.

den(x)

Denominator of the rational representation of x.

<<(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 >>(), >>>().

>>(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 >>>(), <<().

>>>(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 >>(), <<().

:(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.

colon(start, [step, ]stop)

Called by : syntax for constructing ranges.

range(start, [step, ]length)

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

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.

==(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.

!=(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.

===(x, y)
≡(x, y)

See the is() operator.

!==(x, y)
≢(x, y)

Equivalent to !is(x, y).

<(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.

<=(x, y)
≤(x, y)

Less-than-or-equals comparison operator.

>(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.

>=(x, y)
≥(x, y)

Greater-than-or-equals comparison operator.

.==(x, y)

Element-wise equality comparison operator.

.!=(x, y)
.≠(x, y)

Element-wise not-equals comparison operator.

.<(x, y)

Element-wise less-than comparison operator.

.<=(x, y)
.≤(x, y)

Element-wise less-than-or-equals comparison operator.

.>(x, y)

Element-wise greater-than comparison operator.

.>=(x, y)
.≥(x, y)

Element-wise greater-than-or-equals comparison operator.

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.

~(x)

Bitwise not.

&(x, y)

Bitwise and.

|(x, y)

Bitwise or.

\$(x, y)

Bitwise exclusive or.

!(x)

Boolean not.

x && y

Short-circuiting boolean AND.

x || y

Short-circuiting boolean OR.

## Mathematical Functions¶

isapprox(x, y; rtol::Real=sqrt(eps), atol::Real=0)

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.

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).

sin(x)

Compute sine of x, where x is in radians.

cos(x)

Compute cosine of x, where x is in radians.

tan(x)

Compute tangent of x, where x is in radians.

sind(x)

Compute sine of x, where x is in degrees.

cosd(x)

Compute cosine of x, where x is in degrees.

tand(x)

Compute tangent of x, where x is in degrees.

sinpi(x)

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

cospi(x)

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

sinh(x)

Compute hyperbolic sine of x.

cosh(x)

Compute hyperbolic cosine of x.

tanh(x)

Compute hyperbolic tangent of x.

asin(x)

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

acos(x)

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

atan(x)

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

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.

asind(x)

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

acosd(x)

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

atand(x)

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

sec(x)

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

csc(x)

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

cot(x)

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

secd(x)

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

cscd(x)

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

cotd(x)

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

asec(x)

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

acsc(x)

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

acot(x)

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

asecd(x)

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

acscd(x)

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

acotd(x)

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

sech(x)

Compute the hyperbolic secant of x

csch(x)

Compute the hyperbolic cosecant of x.

coth(x)

Compute the hyperbolic cotangent of x.

asinh(x)

Compute the inverse hyperbolic sine of x.

acosh(x)

Compute the inverse hyperbolic cosine of x.

atanh(x)

Compute the inverse hyperbolic tangent of x.

asech(x)

Compute the inverse hyperbolic secant of x.

acsch(x)

Compute the inverse hyperbolic cosecant of x.

acoth(x)

Compute the inverse hyperbolic cotangent of x.

sinc(x)

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

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).

deg2rad(x)

Convert x from degrees to radians.

rad2deg(x)

Convert x from radians to degrees.

hypot(x, y)

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

hypot(x...)

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

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.

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

log2(x)

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

log10(x)

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

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.

frexp(val)

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

exp(x)

Compute $$e^x$$.

exp2(x)

Compute $$2^x$$.

exp10(x)

Compute $$10^x$$.

ldexp(x, n)

Compute $$x \times 2^n$$.

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.

expm1(x)

Accurately compute $$e^x-1$$.

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

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:

RoundNearest

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

RoundNearestTiesAway

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

RoundNearestTiesUp

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

RoundToZero

round() using this rounding mode is an alias for trunc().

RoundUp

round() using this rounding mode is an alias for ceil().

RoundDown

round() using this rounding mode is an alias for floor().

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.

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().

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().

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().

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.

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.

min(x, y, ...)

Return the minimum of the arguments. Operates elementwise over arrays.

max(x, y, ...)

Return the maximum of the arguments. Operates elementwise over arrays.

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')

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. Operates elementwise over x if x is an array.

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

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

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

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.

Base.checked_abs(x)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_neg(x)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_add(x, y)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_sub(x, y)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_mul(x, y)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_div(x, y)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_rem(x, y)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_fld(x, y)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_mod(x, y)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_cld(x, y)

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

The overflow protection may impose a perceptible performance penalty.

abs2(x)

Squared absolute value of x.

copysign(x, y)

Return x such that it has the same sign as y

sign(x)

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

signbit(x)

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

flipsign(x, y)

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

sqrt(x)

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

isqrt(n)

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

cbrt(x)

Return $$x^{1/3}$$. The prefix operator ∛ is equivalent to cbrt.

erf(x)

Compute the error function of x, defined by $$\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt$$ for arbitrary complex x.

erfc(x)

Compute the complementary error function of x, defined by $$1 - \operatorname{erf}(x)$$.

erfcx(x)

Compute the scaled complementary error function of x, defined by $$e^{x^2} \operatorname{erfc}(x)$$. Note also that $$\operatorname{erfcx}(-ix)$$ computes the Faddeeva function $$w(x)$$.

erfi(x)

Compute the imaginary error function of x, defined by $$-i \operatorname{erf}(ix)$$.

dawson(x)

Compute the Dawson function (scaled imaginary error function) of x, defined by $$\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)$$.

erfinv(x)

Compute the inverse error function of a real x, defined by $$\operatorname{erf}(\operatorname{erfinv}(x)) = x$$.

erfcinv(x)

Compute the inverse error complementary function of a real x, defined by $$\operatorname{erfc}(\operatorname{erfcinv}(x)) = x$$.

real(z)

Return the real part of the complex number z.

imag(z)

Return the imaginary part of the complex number z.

reim(z)

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

conj(z)

Compute the complex conjugate of a complex number z.

angle(z)

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

cis(z)

Return $$\exp(iz)$$.

binomial(n, k)

Number of ways to choose k out of n items.

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).

gcd(x, y)

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

lcm(x, y)

Least common (non-negative) multiple.

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 $$(d,u,v)$$.

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.

ispow2(n) → Bool

Test whether n is a power of two.

nextpow2(n)

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

prevpow2(n)

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

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.

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.

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

Next integer not less than n that can be written as $$\prod k_i^{p_i}$$ for integers $$p_1$$, $$p_2$$, etc.

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 $$gcd(x,m) \neq 1$$.

powermod(x, p, m)

Compute $$x^p \pmod m$$.

gamma(x)

Compute the gamma function of x.

lgamma(x)

Compute the logarithm of the absolute value of gamma() for Real x, while for Complex x it computes the logarithm of gamma(x).

lfact(x)

Compute the logarithmic factorial of x

digamma(x)

Compute the digamma function of x (the logarithmic derivative of gamma(x))

invdigamma(x)

Compute the inverse digamma function of x.

trigamma(x)

Compute the trigamma function of x (the logarithmic second derivative of gamma(x)).

polygamma(m, x)

Compute the polygamma function of order m of argument x (the (m+1)th derivative of the logarithm of gamma(x))

airy(k, x)

The kth derivative of the Airy function $$\operatorname{Ai}(x)$$.

airyai(x)

Airy function $$\operatorname{Ai}(x)$$.

airyprime(x)

Airy function derivative $$\operatorname{Ai}'(x)$$.

airyaiprime(x)

Airy function derivative $$\operatorname{Ai}'(x)$$.

airybi(x)

Airy function $$\operatorname{Bi}(x)$$.

airybiprime(x)

Airy function derivative $$\operatorname{Bi}'(x)$$.

airyx(k, x)

scaled kth derivative of the Airy function, return $$\operatorname{Ai}(x) e^{\frac{2}{3} x \sqrt{x}}$$ for k == 0 || k == 1, and $$\operatorname{Ai}(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$$ for k == 2 || k == 3.

besselj0(x)

Bessel function of the first kind of order 0, $$J_0(x)$$.

besselj1(x)

Bessel function of the first kind of order 1, $$J_1(x)$$.

besselj(nu, x)

Bessel function of the first kind of order nu, $$J_\nu(x)$$.

besseljx(nu, x)

Scaled Bessel function of the first kind of order nu, $$J_\nu(x) e^{- | \operatorname{Im}(x) |}$$.

bessely0(x)

Bessel function of the second kind of order 0, $$Y_0(x)$$.

bessely1(x)

Bessel function of the second kind of order 1, $$Y_1(x)$$.

bessely(nu, x)

Bessel function of the second kind of order nu, $$Y_\nu(x)$$.

besselyx(nu, x)

Scaled Bessel function of the second kind of order nu, $$Y_\nu(x) e^{- | \operatorname{Im}(x) |}$$.

hankelh1(nu, x)

Bessel function of the third kind of order nu, $$H^{(1)}_\nu(x)$$.

hankelh1x(nu, x)

Scaled Bessel function of the third kind of order nu, $$H^{(1)}_\nu(x) e^{-x i}$$.

hankelh2(nu, x)

Bessel function of the third kind of order nu, $$H^{(2)}_\nu(x)$$.

hankelh2x(nu, x)

Scaled Bessel function of the third kind of order nu, $$H^{(2)}_\nu(x) e^{x i}$$.

besselh(nu, [k=1, ]x)

Bessel function of the third kind of order nu (the Hankel function). k is either 1 or 2, selecting hankelh1() or hankelh2(), respectively. k defaults to 1 if it is omitted. (See also besselhx() for an exponentially scaled variant.)

besselhx(nu, [k=1, ]z)

Compute the scaled Hankel function $$\exp(∓iz) H_ν^{(k)}(z)$$, where $$k$$ is 1 or 2, $$H_ν^{(k)}(z)$$ is besselh(nu, k, z), and $$∓$$ is $$-$$ for $$k=1$$ and $$+$$ for $$k=2$$. k defaults to 1 if it is omitted.

The reason for this function is that $$H_ν^{(k)}(z)$$ is asymptotically proportional to $$\exp(∓iz)/\sqrt{z}$$ for large $$|z|$$, and so the besselh() function is susceptible to overflow or underflow when z has a large imaginary part. The besselhx function cancels this exponential factor (analytically), so it avoids these problems.

besseli(nu, x)

Modified Bessel function of the first kind of order nu, $$I_\nu(x)$$.

besselix(nu, x)

Scaled modified Bessel function of the first kind of order nu, $$I_\nu(x) e^{- | \operatorname{Re}(x) |}$$.

besselk(nu, x)

Modified Bessel function of the second kind of order nu, $$K_\nu(x)$$.

besselkx(nu, x)

Scaled modified Bessel function of the second kind of order nu, $$K_\nu(x) e^x$$.

beta(x, y)

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

lbeta(x, y)

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

eta(x)

Dirichlet eta function $$\eta(s) = \sum^\infty_{n=1}(-1)^{n-1}/n^{s}$$.

zeta(s)

Riemann zeta function $$\zeta(s)$$.

zeta(s, z)

Generalized zeta function $$\zeta(s, z)$$, defined by the sum $$\sum_{k=0}^\infty ((k+z)^2)^{-s/2}$$, where any term with $$k+z=0$$ is excluded. For $$\Re z > 0$$, this definition is equivalent to the Hurwitz zeta function $$\sum_{k=0}^\infty (k+z)^{-s}$$. For $$z=1$$, it yields the Riemann zeta function $$\zeta(s)$$.

ndigits(n, b = 10)

Compute the number of digits in number n written in base b.

widemul(x, y)

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

@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.

## Statistics¶

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.

mean(f::Function, v)

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

mean!(r, v)

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

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.

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.

var(v[, region])

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 sumabs2(v - mean(v)) / (length(v) - 1). Note: Julia does not ignore NaN values in the computation. For applications requiring the handling of missing data, the DataArray package is recommended.

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.

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.

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).

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

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

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.

median!(v)

Like median, but may overwrite the input vector.

midpoints(e)

Compute the midpoints of the bins with edges e. The result is a vector/range of length length(e) - 1. Note: Julia does not ignore NaN values in the computation.

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
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
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).

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).

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

Compute the covariance between the vectors x and y. 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) = length(y).

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).

cor(x)

Return the number one.

cor(X[, vardim=1])

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

cor(x, y)

Compute the Pearson correlation between the vectors x and y.

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

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

## Signal Processing¶

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

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.
fft!(A[, dims])

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

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.

ifft!(A[, dims])

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

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.)

$\operatorname{BDFT}(A)[k] = \operatorname{length}(A) \operatorname{IDFT}(A)[k]$
bfft!(A[, dims])

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

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.

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

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

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

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

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

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

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

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

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

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

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.

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.)

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.

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().

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().

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().

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.

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.

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.

idct!(A[, dims])

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

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().

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

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

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().

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

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

fftshift(x)

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

fftshift(x, dim)

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

ifftshift(x[, dim])

Undoes the effect of fftshift.

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).

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.

deconv(b, a)

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

conv(u, v)

Convolution of two vectors. Uses FFT algorithm.

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.

conv2(B, A)

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

xcorr(u, v)

Compute the cross-correlation of two vectors.

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

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.

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.

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

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

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

Similar to Base.plan_fft(), but corresponds to r2r!().

## Numerical Integration¶

Although several external packages are available for numeric integration and solution of ordinary differential equations, we also provide some built-in integration support in Julia.

quadgk(f, a, b, c...; reltol=sqrt(eps), abstol=0, maxevals=10^7, order=7, norm=vecnorm)

Numerically integrate the function f(x) from a to b, and optionally over additional intervals b to c and so on. Keyword options include a relative error tolerance reltol (defaults to sqrt(eps) in the precision of the endpoints), an absolute error tolerance abstol (defaults to 0), a maximum number of function evaluations maxevals (defaults to 10^7), and the order of the integration rule (defaults to 7).

Returns a pair (I,E) of the estimated integral I and an estimated upper bound on the absolute error E. If maxevals is not exceeded then E <= max(abstol, reltol*norm(I)) will hold. (Note that it is useful to specify a positive abstol in cases where norm(I) may be zero.)

The endpoints a et cetera can also be complex (in which case the integral is performed over straight-line segments in the complex plane). If the endpoints are BigFloat, then the integration will be performed in BigFloat precision as well.

Note

It is advisable to increase the integration order in rough proportion to the precision, for smooth integrands.

More generally, the precision is set by the precision of the integration endpoints (promoted to floating-point types).

The integrand f(x) can return any numeric scalar, vector, or matrix type, or in fact any type supporting +, -, multiplication by real values, and a norm (i.e., any normed vector space). Alternatively, a different norm can be specified by passing a norm-like function as the norm keyword argument (which defaults to vecnorm).

Note

Only one-dimensional integrals are provided by this function. For multi-dimensional integration (cubature), there are many different algorithms (often much better than simple nested 1d integrals) and the optimal choice tends to be very problem-dependent. See the Julia external-package listing for available algorithms for multidimensional integration or other specialized tasks (such as integrals of highly oscillatory or singular functions).

The algorithm is an adaptive Gauss-Kronrod integration technique: the integral in each interval is estimated using a Kronrod rule (2*order+1 points) and the error is estimated using an embedded Gauss rule (order points). The interval with the largest error is then subdivided into two intervals and the process is repeated until the desired error tolerance is achieved.

These quadrature rules work best for smooth functions within each interval, so if your function has a known discontinuity or other singularity, it is best to subdivide your interval to put the singularity at an endpoint. For example, if f has a discontinuity at x=0.7 and you want to integrate from 0 to 1, you should use quadgk(f, 0,0.7,1) to subdivide the interval at the point of discontinuity. The integrand is never evaluated exactly at the endpoints of the intervals, so it is possible to integrate functions that diverge at the endpoints as long as the singularity is integrable (for example, a log(x) or 1/sqrt(x) singularity).

For real-valued endpoints, the starting and/or ending points may be infinite. (A coordinate transformation is performed internally to map the infinite interval to a finite one.)