Getting Around¶

exit([code])¶

Quit (or control-D at the prompt). The default exit code is zero, indicating that the processes completed successfully.

whos([Module,] [pattern::Regex])¶

Print information about global variables in a module, optionally restricted to those matching pattern.

edit(file::String[, line])¶

Edit a file optionally providing a line number to edit at. Returns to the julia prompt when you quit the editor. If the file name ends in ”.jl” it is reloaded when the editor closes the file.

edit(function[, types])

Edit the definition of a function, optionally specifying a tuple of types to indicate which method to edit. When the editor exits, the source file containing the definition is reloaded.

require(file::String...)¶

Load source files once, in the context of the Main module, on every active node, searching the system-wide LOAD_PATH for files. require is considered a top-level operation, so it sets the current include path but does not use it to search for files (see help for include). This function is typically used to load library code, and is implicitly called by using to load packages.

reload(file::String)¶

Like require, except forces loading of files regardless of whether they have been loaded before. Typically used when interactively developing libraries.

include(path::String)¶

Evaluate the contents of a source file in the current context. During including, a task-local include path is set to the directory containing the file. Nested calls to include will search relative to that path. All paths refer to files on node 1 when running in parallel, and files will be fetched from node 1. This function is typically used to load source interactively, or to combine files in packages that are broken into multiple source files.

include_string(code::String)¶

Like include, except reads code from the given string rather than from a file. Since there is no file path involved, no path processing or fetching from node 1 is done.

evalfile(path::String)¶

Evaluate all expressions in the given file, and return the value of the last one. No other processing (path searching, fetching from node 1, etc.) is performed.

help(name)¶

Get help for a function. name can be an object or a string.

apropos(string)¶

Search documentation for functions related to string.

which(f, args...)¶

Show which method of f will be called for the given arguments.

methods(f)¶

Show all methods of f with their argument types.

methodswith(typ[, showparents])¶

Show all methods with an argument of type typ. If optional showparents is true, also show arguments with a parent type of typ, excluding type Any.

All Objects¶

is(x, y)¶

Determine whether x and y are identical, in the sense that no program could distinguish them.

isa(x, type)¶

Determine whether x is of the given type.

isequal(x, y)¶

True if and only if x and y have the same contents. Loosely speaking, this means x and y would look the same when printed.

isless(x, y)¶

Test whether x is less than y. Provides a total order consistent with isequal. Values that are normally unordered, such as NaN, are ordered in an arbitrary but consistent fashion. This is the default comparison used by sort. Non-numeric types that can be ordered should implement this function.

typeof(x)¶

Get the concrete type of x.

tuple(xs...)¶

Construct a tuple of the given objects.

ntuple(n, f::Function)¶

Create a tuple of length n, computing each element as f(i), where i is the index of the element.

object_id(x)¶

Get a unique integer id for x. object_id(x)==object_id(y) if and only if is(x,y).

hash(x)¶

Compute an integer hash code such that isequal(x,y) implies hash(x)==hash(y).

finalizer(x, function)¶

Register a function f(x) to be called when there are no program-accessible references to x. The behavior of this function is unpredictable if x is of a bits type.

copy(x)¶

Create a shallow copy of x: the outer structure is copied, but not all internal values. For example, copying an array produces a new array with identically-same elements as the original.

deepcopy(x)¶

Create a deep copy of x: everything is copied recursively, resulting in a fully independent object. For example, deep-copying an array produces a new array whose elements are deep-copies of the original elements.

As a special case, functions can only be actually deep-copied if they are anonymous, otherwise they are just copied. The difference is only relevant in the case of closures, i.e. functions which may contain hidden internal references.

While it isn’t normally necessary, user-defined types can override the default deepcopy behavior by defining a specialized version of the function deepcopy_internal(x::T, dict::ObjectIdDict) (which shouldn’t otherwise be used), where T is the type to be specialized for, and dict keeps track of objects copied so far within the recursion. Within the definition, deepcopy_internal should be used in place of deepcopy, and the dict variable should be updated as appropriate before returning.

convert(type, x)¶

Try to convert x to the given type.

promote(xs...)¶

Convert all arguments to their common promotion type (if any), and return them all (as a tuple).

Types¶

subtype(type1, type2)¶

True if and only if all values of type1 are also of type2. Can also be written using the <: infix operator as type1 <: type2.

<:(T1, T2)¶

Subtype operator, equivalent to subtype(T1,T2).

typemin(type)¶

The lowest value representable by the given (real) numeric type.

typemax(type)¶

The highest value representable by the given (real) numeric type.

realmin(type)¶

The smallest in absolute value non-denormal value representable by the given floating-point type

realmax(type)¶

The highest finite value representable by the given floating-point type

maxintfloat(type)¶

The largest integer losslessly representable by the given floating-point type

sizeof(type)¶

Size, in bytes, of the canonical binary representation of the given type, if any.

eps([type])¶

The distance between 1.0 and the next larger representable floating-point value of type. The only types that are sensible arguments are Float32 and Float64. If type is omitted, then eps(Float64) is returned.

eps(x)

The distance between x and the next larger representable floating-point value of the same type as x.

promote_type(type1, type2)¶

Determine a type big enough to hold values of each argument type without loss, whenever possible. In some cases, where no type exists which to which both types can be promoted losslessly, some loss is tolerated; for example, promote_type(Int64,Float64) returns Float64 even though strictly, not all Int64 values can be represented exactly as Float64 values.

getfield(value, name::Symbol)¶

Extract a named field from a value of composite type. The syntax a.b calls getfield(a, :b), and the syntax a.(b) calls getfield(a, b).

setfield(value, name::Symbol, x)¶

Assign x to a named field in value of composite type. The syntax a.b = c calls setfield(a, :b, c), and the syntax a.(b) = c calls setfield(a, b, c).

fieldtype(value, name::Symbol)¶

Determine the declared type of a named field in a value of composite type.

Generic Functions¶

method_exists(f, tuple) → Bool¶

Determine whether the given generic function has a method matching the given tuple of argument types.

Example: method_exists(length, (Array,)) = true

applicable(f, args...)¶

Determine whether the given generic function has a method applicable to the given arguments.

invoke(f, (types...), args...)¶

Invoke a method for the given generic function matching the specified types (as a tuple), on the specified arguments. The arguments must be compatible with the specified types. This allows invoking a method other than the most specific matching method, which is useful when the behavior of a more general definition is explicitly needed (often as part of the implementation of a more specific method of the same function).

|(x, f)¶

Applies a function to the preceding argument which allows for easy function chaining.

Example: [1:5] | x->x.^2 | sum | inv

Iteration¶

Sequential iteration is implemented by the methods start, done, and next. The general for loop:

for i = I
  # body
end

is translated to:

state = start(I)
while !done(I, state)
  (i, state) = next(I, state)
  # body
end

The state object may be anything, and should be chosen appropriately for each iterable type.

start(iter) → state¶

Get initial iteration state for an iterable object

done(iter, state) → Bool¶

Test whether we are done iterating

next(iter, state) → item, state¶

For a given iterable object and iteration state, return the current item and the next iteration state

zip(iters...)¶

For a set of iterable objects, returns an iterable of tuples, where the ith tuple contains the ith component of each input iterable.

Note that zip is it’s own inverse: [zip(zip(a...)...)...] == [a...].

enumerate(iter)¶

Return an iterator that yields (i, x) where i is an index starting at 1, and x is the ith value from the given iterator.

Fully implemented by: Range, Range1, NDRange, Tuple, Real, AbstractArray, IntSet, ObjectIdDict, Dict, WeakKeyDict, EachLine, String, Set, Task.

General Collections¶

isempty(collection) → Bool¶

Determine whether a collection is empty (has no elements).

empty!(collection) → collection¶

Remove all elements from a collection.

length(collection) → Integer¶

For ordered, indexable collections, the maximum index i for which getindex(collection, i) is valid. For unordered collections, the number of elements.

endof(collection) → Integer¶

Returns the last index of the collection.

Example: endof([1,2,4]) = 3

Fully implemented by: Range, Range1, Tuple, Number, AbstractArray, IntSet, Dict, WeakKeyDict, String, Set.

Iterable Collections¶

contains(itr, x) → Bool¶

Determine whether a collection contains the given value, x.

findin(a, b)¶

Returns the indices of elements in collection a that appear in collection b

unique(itr)¶

Returns an array containing only the unique elements of the iterable itr.

reduce(op, v0, itr)¶

Reduce the given collection with the given operator, i.e. accumulate v = op(v,elt) for each element, where v starts as v0. Reductions for certain commonly-used operators are available in a more convenient 1-argument form: max(itr), min(itr), sum(itr), prod(itr), any(itr), all(itr).

max(itr)¶

Returns the largest element in a collection

min(itr)¶

Returns the smallest element in a collection

indmax(itr) → Integer¶

Returns the index of the maximum element in a collection

indmin(itr) → Integer¶

Returns the index of the minimum element in a collection

findmax(itr) -> (x, index)¶

Returns the maximum element and its index

findmin(itr) -> (x, index)¶

Returns the minimum element and its index

sum(itr)¶

Returns the sum of all elements in a collection

prod(itr)¶

Returns the product of all elements of a collection

any(itr) → Bool¶

Test whether any elements of a boolean collection are true

all(itr) → Bool¶

Test whether all elements of a boolean collection are true

count(itr) → Integer¶

Count the number of boolean elements in itr which are true.

countp(p, itr) → Integer¶

Count the number of elements in itr for which predicate p is true.

any(p, itr) → Bool

Determine whether any element of itr satisfies the given predicate.

all(p, itr) → Bool

Determine whether all elements of itr satisfy the given predicate.

map(f, c) → collection¶

Transform collection c by applying f to each element.

Example: map((x) -> x * 2, [1, 2, 3]) = [2, 4, 6]

map!(function, collection)¶

In-place version of map().

mapreduce(f, op, itr)¶

Applies function f to each element in itr and then reduces the result using the binary function op.

Example: mapreduce(x->x^2, +, [1:3]) == 1 + 4 + 9 == 14

first(coll)¶

Get the first element of an ordered collection.

last(coll)¶

Get the last element of an ordered collection.

Indexable Collections¶

getindex(collection, key...)¶

Retrieve the value(s) stored at the given key or index within a collection. The syntax a[i,j,...] is converted by the compiler to getindex(a, i, j, ...).

setindex!(collection, value, key...)¶

Store the given value at the given key or index within a collection. The syntax a[i,j,...] = x is converted by the compiler to setindex!(a, x, i, j, ...).

Fully implemented by: Array, DArray, AbstractArray, SubArray, ObjectIdDict, Dict, WeakKeyDict, String.

Partially implemented by: Range, Range1, Tuple.

Associative Collections¶

Dict is the standard associative collection. Its implementation uses the hash(x) as the hashing function for the key, and isequal(x,y) to determine equality. Define these two functions for custom types to override how they are stored in a hash table.

ObjectIdDict is a special hash table where the keys are always object identities. WeakKeyDict is a hash table implementation where the keys are weak references to objects, and thus may be garbage collected even when referenced in a hash table.

Dicts can be created using a literal syntax: {"A"=>1, "B"=>2}. Use of curly brackets will create a Dict of type Dict{Any,Any}. Use of square brackets will attempt to infer type information from the keys and values (i.e. ["A"=>1, "B"=>2] creates a Dict{ASCIIString, Int64}). To explicitly specify types use the syntax: (KeyType=>ValueType)[...]. For example, (ASCIIString=>Int32)["A"=>1, "B"=>2].

As with arrays, Dicts may be created with comprehensions. For example, {i => f(i) for i = 1:10}.

Dict{K,V}()¶

Construct a hashtable with keys of type K and values of type V

has(collection, key)¶

Determine whether a collection has a mapping for a given key.

get(collection, key, default)¶

Return the value stored for the given key, or the given default value if no mapping for the key is present.

getkey(collection, key, default)¶

Return the key matching argument key if one exists in collection, otherwise return default.

delete!(collection, key)¶

Delete the mapping for the given key in a collection.

keys(collection)¶

Return an array of all keys in a collection.

values(collection)¶

Return an array of all values in a collection.

collect(collection)¶

Return an array of all items in a collection. For associative collections, returns (key, value) tuples.

merge(collection, others...)¶

Construct a merged collection from the given collections.

merge!(collection, others...)¶

Update collection with pairs from the other collections

filter(function, collection)¶

Return a copy of collection, removing (key, value) pairs for which function is false.

filter!(function, collection)¶

Update collection, removing (key, value) pairs for which function is false.

eltype(collection)¶

Returns the type tuple of the (key,value) pairs contained in collection.

sizehint(s, n)¶

Suggest that collection s reserve capacity for at least n elements. This can improve performance.

Fully implemented by: ObjectIdDict, Dict, WeakKeyDict.

Partially implemented by: IntSet, Set, EnvHash, Array.

Set-Like Collections¶

add!(collection, key)¶

Add an element to a set-like collection.

add_each!(collection, iterable)¶

Adds each element in iterable to the collection.

Set(x...)¶

Construct a Set with the given elements. Should be used instead of IntSet for sparse integer sets.

IntSet(i...)¶

Construct an IntSet of the given integers. Implemented as a bit string, and therefore good for dense integer sets.

union(s1, s2...)¶

Construct the union of two or more sets. Maintains order with arrays.

union!(s1, s2)¶

Constructs the union of IntSets s1 and s2, stores the result in s1.

intersect(s1, s2...)¶

Construct the intersection of two or more sets. Maintains order with arrays.

setdiff(s1, s2)¶

Construct the set of elements in s1 but not s2. Maintains order with arrays.

symdiff(s1, s2...)¶

Construct the symmetric difference of elements in the passed in sets or arrays. Maintains order with arrays.

symdiff!(s, n)¶

IntSet s is destructively modified to toggle the inclusion of integer n.

symdiff!(s, itr)

For each element in itr, destructively toggle its inclusion in set s.

symdiff!(s1, s2)

Construct the symmetric difference of IntSets s1 and s2, storing the result in s1.

complement(s)¶

Returns the set-complement of IntSet s.

complement!(s)¶

Mutates IntSet s into its set-complement.

del_each!(s, itr)¶

Deletes each element of itr in set s in-place.

intersect!(s1, s2)¶

Intersects IntSets s1 and s2 and overwrites the set s1 with the result. If needed, s1 will be expanded to the size of s2.

Fully implemented by: IntSet, Set.

Partially implemented by: Array.

Dequeues¶

push!(collection, item) → collection¶

Insert an item at the end of a collection.

pop!(collection) → item¶

Remove the last item in a collection and return it.

unshift!(collection, item) → collection¶

Insert an item at the beginning of a collection.

shift!(collection) → item¶

Remove the first item in a collection.

insert!(collection, index, item)¶

Insert an item at the given index.

delete!(collection, index) → item

Remove the item at the given index, and return the deleted item.

delete!(collection, range) → items

Remove items at specified range, and return a collection containing the deleted items.

resize!(collection, n) → collection¶

Resize collection to contain n elements.

append!(collection, items) → collection¶

Add the elements of items to the end of a collection.

Fully implemented by: Vector (aka 1-d Array).

Strings¶

length(s)

The number of characters in string s.

*(s, t)¶

Concatenate strings.

Example: "Hello " * "world" == "Hello world"

^(s, n)¶

Repeat string s n times.

Example: "Julia "^3 == "Julia Julia Julia "

string(xs...)¶

Create a string from any values using the print function.

repr(x)¶

Create a string from any value using the show function.

bytestring(::Ptr{Uint8})¶

Create a string from the address of a C (0-terminated) string. A copy is made; the ptr can be safely freed.

bytestring(s)

Convert a string to a contiguous byte array representation appropriate for passing it to C functions.

ascii(::Array{Uint8, 1})¶

Create an ASCII string from a byte array.

ascii(s)

Convert a string to a contiguous ASCII string (all characters must be valid ASCII characters).

utf8(::Array{Uint8, 1})¶

Create a UTF-8 string from a byte array.

utf8(s)

Convert a string to a contiguous UTF-8 string (all characters must be valid UTF-8 characters).

is_valid_ascii(s) → Bool¶

Returns true if the string or byte vector is valid ASCII, false otherwise.

is_valid_utf8(s) → Bool¶

Returns true if the string or byte vector is valid UTF-8, false otherwise.

is_valid_char(c) → Bool¶

Returns true if the given char or integer is a valid Unicode code point.

ismatch(r::Regex, s::String)¶

Test whether a string contains a match of the given regular expression.

lpad(string, n, p)¶

Make a string at least n characters long by padding on the left with copies of p.

rpad(string, n, p)¶

Make a string at least n characters long by padding on the right with copies of p.

search(string, chars[, start])¶

Search for the given characters within the given string. The second argument may be a single character, a vector or a set of characters, a string, or a regular expression (though regular expressions are only allowed on contiguous strings, such as ASCII or UTF-8 strings). The third argument optionally specifies a starting index. The return value is a range of indexes where the matching sequence is found, such that s[search(s,x)] == x. The return value is 0:-1 if there is no match.

replace(string, pat, r[, n])¶

Search for the given pattern pat, and replace each occurance with r. If n is provided, replace at most n occurances. As with search, the second argument may be a single character, a vector or a set of characters, a string, or a regular expression. If r is a function, each occurrence is replaced with r(s) where s is the matched substring.

split(string, [chars, [limit,] [include_empty]])¶

Return an array of strings by splitting the given string on occurrences of the given character delimiters, which may be specified in any of the formats allowed by search‘s second argument (i.e. a single character, collection of characters, string, or regular expression). If chars is omitted, it defaults to the set of all space characters, and include_empty is taken to be false. The last two arguments are also optional: they are are a maximum size for the result and a flag determining whether empty fields should be included in the result.

strip(string[, chars])¶

Return string with any leading and trailing whitespace removed. If a string chars is provided, instead remove characters contained in that string.

lstrip(string[, chars])¶

Return string with any leading whitespace removed. If a string chars is provided, instead remove characters contained in that string.

rstrip(string[, chars])¶

Return string with any trailing whitespace removed. If a string chars is provided, instead remove characters contained in that string.

beginswith(string, prefix)¶

Returns true if string starts with prefix.

endswith(string, suffix)¶

Returns true if string ends with suffix.

uppercase(string)¶

Returns string with all characters converted to uppercase.

lowercase(string)¶

Returns string with all characters converted to lowercase.

join(strings, delim)¶

Join an array of strings into a single string, inserting the given delimiter between adjacent strings.

chop(string)¶

Remove the last character from a string

chomp(string)¶

Remove a trailing newline from a string

ind2chr(string, i)¶

Convert a byte index to a character index

chr2ind(string, i)¶

Convert a character index to a byte index

isvalid(str, i)¶

Tells whether index i is valid for the given string

nextind(str, i)¶

Get the next valid string index after i. Returns endof(str)+1 at the end of the string.

prevind(str, i)¶

Get the previous valid string index before i. Returns 0 at the beginning of the string.

thisind(str, i)¶

Adjust i downwards until it reaches a valid index for the given string.

randstring(len)¶

Create a random ASCII string of length len, consisting of upper- and lower-case letters and the digits 0-9

charwidth(c)¶

Gives the number of columns needed to print a character.

strwidth(s)¶

Gives the number of columns needed to print a string.

isalnum(c::Char)¶

Tests whether a character is alphanumeric.

isalpha(c::Char)¶

Tests whether a character is alphabetic.

isascii(c::Char)¶

Tests whether a character belongs to the ASCII character set.

isblank(c::Char)¶

Tests whether a character is a tab or space.

iscntrl(c::Char)¶

Tests whether a character is a control character.

isdigit(c::Char)¶

Tests whether a character is a numeric digit (0-9).

isgraph(c::Char)¶

Tests whether a character is printable, and not a space.

islower(c::Char)¶

Tests whether a character is a lowercase letter.

isprint(c::Char)¶

Tests whether a character is printable, including space.

ispunct(c::Char)¶

Tests whether a character is printable, and not a space or alphanumeric.

isspace(c::Char)¶

Tests whether a character is any whitespace character.

isupper(c::Char)¶

Tests whether a character is an uppercase letter.

isxdigit(c::Char)¶

Tests whether a character is a valid hexadecimal digit.

I/O¶

STDOUT¶

Global variable referring to the standard out stream.

STDERR¶

Global variable referring to the standard error stream.

STDIN¶

Global variable referring to the standard input stream.

OUTPUT_STREAM¶

The default stream used for text output, e.g. in the print and show functions.

open(file_name[, read, write, create, truncate, append]) → IOStream¶

Open a file in a mode specified by five boolean arguments. The default is to open files for reading only. Returns a stream for accessing the file.

open(file_name[, mode]) → IOStream

Alternate syntax for open, where a string-based mode specifier is used instead of the five booleans. The values of mode correspond to those from fopen(3) or Perl open, and are equivalent to setting the following boolean groups:

r	read
r+	read, write
w	write, create, truncate
w+	read, write, create, truncate
a	write, create, append
a+	read, write, create, append

open(f::function, args...)

Apply the function f to the result of open(args...) and close the resulting file descriptor upon completion.

Example: open(readall, "file.txt")

memio([size[, finalize::Bool]]) → IOStream¶

Create an in-memory I/O stream, optionally specifying how much initial space is needed.

fdio([name::String], fd::Integer[, own::Bool]) → IOStream¶

Create an IOStream object from an integer file descriptor. If own is true, closing this object will close the underlying descriptor. By default, an IOStream is closed when it is garbage collected. name allows you to associate the descriptor with a named file.

flush(stream)¶

Commit all currently buffered writes to the given stream.

close(stream)¶

Close an I/O stream. Performs a flush first.

write(stream, x)¶

Write the canonical binary representation of a value to the given stream.

read(stream, type)¶

Read a value of the given type from a stream, in canonical binary representation.

read(stream, type, dims)

Read a series of values of the given type from a stream, in canonical binary representation. dims is either a tuple or a series of integer arguments specifying the size of Array to return.

position(s)¶

Get the current position of a stream.

seek(s, pos)¶

Seek a stream to the given position.

seek_end(s)¶

Seek a stream to the end.

skip(s, offset)¶

Seek a stream relative to the current position.

eof(stream)¶

Tests whether an I/O stream is at end-of-file. If the stream is not yet exhausted, this function will block to wait for more data if necessary, and then return false. Therefore it is always safe to read one byte after seeing eof return false.

Text I/O¶

show(x)¶

Write an informative text representation of a value to the current output stream. New types should overload show(io, x) where the first argument is a stream.

print(x)¶

Write (to the default output stream) a canonical (un-decorated) text representation of a value if there is one, otherwise call show.

println(x)¶

Print (using print()) x followed by a newline

@printf([io::IOStream], "%Fmt", args...)¶

Print arg(s) using C printf() style format specification string. Optionally, an IOStream may be passed as the first argument to redirect output.

@sprintf("%Fmt", args...)¶

Return @printf formatted output as string.

showall(x)¶

Show x, printing all elements of arrays

dump(x)¶

Write a thorough text representation of a value to the current output stream.

readall(stream)¶

Read the entire contents of an I/O stream as a string.

readline(stream)¶

Read a single line of text, including a trailing newline character (if one is reached before the end of the input).

readuntil(stream, delim)¶

Read a string, up to and including the given delimiter byte.

readlines(stream)¶

Read all lines as an array.

eachline(stream)¶

Create an iterable object that will yield each line from a stream.

readdlm(filename, delim::Char)¶

Read a matrix from a text file where each line gives one row, with elements separated by the given delimeter. If all data is numeric, the result will be a numeric array. If some elements cannot be parsed as numbers, a cell array of numbers and strings is returned.

readdlm(filename, delim::Char, T::Type)

Read a matrix from a text file with a given element type. If T is a numeric type, the result is an array of that type, with any non-numeric elements as NaN for floating-point types, or zero. Other useful values of T include ASCIIString, String, and Any.

writedlm(filename, array, delim::Char)¶

Write an array to a text file using the given delimeter (defaults to comma).

readcsv(filename[, T::Type])¶

Equivalent to readdlm with delim set to comma.

writecsv(filename, array)¶

Equivalent to writedlm with delim set to comma.

Memory-mapped I/O¶

mmap_array(type, dims, stream[, offset])¶

Create an array whose values are linked to a file, using memory-mapping. This provides a convenient way of working with data too large to fit in the computer’s memory.

The type determines how the bytes of the array are interpreted (no format conversions are possible), and dims is a tuple containing the size of the array.

The file is specified via the stream. When you initialize the stream, use “r” for a “read-only” array, and “w+” to create a new array used to write values to disk. Optionally, you can specify an offset (in bytes) if, for example, you want to skip over a header in the file.

Example: A = mmap_array(Int64, (25,30000), s)

This would create a 25-by-30000 array of Int64s, linked to the file associated with stream s.

msync(array)¶

Forces synchronization between the in-memory version of a memory-mapped array and the on-disk version. You may not need to call this function, because synchronization is performed at intervals automatically by the operating system. Hower, you can call this directly if, for example, you are concerned about losing the result of a long-running calculation.

mmap(len, prot, flags, fd, offset)¶

Low-level interface to the mmap system call. See the man page.

munmap(pointer, len)¶

Low-level interface for unmapping memory (see the man page). With mmap_array you do not need to call this directly; the memory is unmapped for you when the array goes out of scope.

Standard Numeric Types¶

Bool Int8 Uint8 Int16 Uint16 Int32 Uint32 Int64 Uint64 Float32 Float64 Complex64 Complex128

Mathematical Functions¶

-(x)¶

Unary minus operator.

+(x, y)¶

Binary addition operator.

-(x, y)

Binary subtraction operator.

*(x, y)

Binary multiplication operator.

/(x, y)¶

Binary left-division operator.

\(x, y)¶

Binary right-division operator.

^(x, y)

Binary exponentiation operator.

.+(x, y)¶

Element-wise binary addition operator.

.-(x, y)¶

Element-wise binary subtraction operator.

.*(x, y)¶

Element-wise binary multiplication operator.

./(x, y)¶

Element-wise binary left division operator.

.\(x, y)¶

Element-wise binary right division operator.

.^(x, y)¶

Element-wise binary exponentiation operator.

div(a, b)¶

Compute a/b, truncating to an integer

fld(a, b)¶

Largest integer less than or equal to a/b

mod(x, m)¶

Modulus after division, returning in the range [0,m)

rem(x, m)¶

Remainder after division

%(x, m)¶

Remainder after division. The operator form of rem.

mod1(x, m)¶

Modulus after division, returning in the range (0,m]

//(num, den)¶

Rational division

num(x)¶

Numerator of the rational representation of x

den(x)¶

Denominator of the rational representation of x

<<(x, n)¶

Left shift operator.

>>(x, n)¶

Right shift operator.

>>>(x, n)¶

Unsigned right shift operator.

:(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.

==(x, y)¶

Equality comparison operator.

!=(x, y)¶

Not-equals comparison operator.

<(x, y)¶

Less-than comparison operator.

<=(x, y)¶

Less-than-or-equals comparison operator.

>(x, y)¶

Greater-than comparison operator.

>=(x, y)¶

Greater-than-or-equals comparison operator.

.==(x, y)¶

Element-wise equality comparison operator.

.!=(x, y)¶

Element-wise not-equals comparison operator.

.<(x, y)¶

Element-wise less-than comparison operator.

.<=(x, y)¶

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

.>(x, y)¶

Element-wise greater-than comparison operator.

.>=(x, y)¶

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

cmp(x, y)¶

Return -1, 0, or 1 depending on whether x<y, x==y, or x>y, respectively

!(x)¶

Boolean not

~(x)¶

Bitwise not

&(x, y)¶

Bitwise and

|(x, y)

Bitwise or

$(x, y)¶

Bitwise exclusive or

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

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 cotangent 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).

degrees2radians(x)¶

Convert x from degrees to radians

radians2degrees(x)¶

Convert x from radians to degrees

hypot(x, y)¶

Compute the \(\sqrt{x^2+y^2}\) without undue overflow or underflow

log(x)¶

Compute the natural logarithm of x

log2(x)¶

Compute the natural logarithm of x to base 2

log10(x)¶

Compute the natural logarithm of x to base 10

log1p(x)¶

Accurate natural logarithm of 1+x

frexp(val, exp)¶

Return a number x such that it 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 \(2^x\)

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 \(e^x-1\)

square(x)¶

Compute \(x^2\)

round(x[, digits[, base]]) → FloatingPoint¶

round(x) returns the nearest integer to x. round(x, digits) rounds to the specified number of digits after the decimal place, or before if negative, e.g., round(pi,2) is 3.14. round(x, digits, base) rounds using a different base, defaulting to 10, e.g., round(pi, 3, 2) is 3.125.

ceil(x[, digits[, base]]) → FloatingPoint¶

Returns the nearest integer not less than x. digits and base work as above.

floor(x[, digits[, base]]) → FloatingPoint¶

Returns the nearest integer not greater than x. digits and base work as above.

trunc(x[, digits[, base]]) → FloatingPoint¶

Returns the nearest integer not greater in magnitude than x. digits and base work as above.

iround(x) → Integer¶

Returns the nearest integer to x.

iceil(x) → Integer¶

Returns the nearest integer not less than x.

ifloor(x) → Integer¶

Returns the nearest integer not greater than x.

itrunc(x) → Integer¶

Returns the nearest integer not greater in magnitude than x.

signif(x, digits[, base]) → FloatingPoint¶

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 x and y

max(x, y)

Return the maximum of x and y

clamp(x, lo, hi)¶

Return x if lo <= x <= y. If x < lo, return lo. If x > hi, return hi.

abs(x)¶

Absolute value of x

abs2(x)¶

Squared absolute value of x

copysign(x, y)¶

Return x such that it has the same sign as y

sign(x)¶

Return +1 if x is positive, 0 if x == 0, and -1 if x is negative.

signbit(x)¶

Returns 1 if the value of the sign of x is negative, otherwise 0.

flipsign(x, y)¶

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

sqrt(x)¶

Return \(\sqrt{x}\)

cbrt(x)¶

Return \(x^{1/3}\)

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 \(\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)\).

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 of a complex number z

cis(z)¶

Return cos(z) + i*sin(z) if z is real. Return (cos(real(z)) + i*sin(real(z)))/exp(imag(z)) if z is complex

binomial(n, k)¶

Number of ways to choose k out of n items

factorial(n)¶

Factorial of n

factorial(n, k)

Compute factorial(n)/factorial(k)

factor(n)¶

Compute the prime factorization of an integer n. Returns a dictionary. The keys of the dictionary correspond to the factors, and hence are of the same type as n. The value associated with each key indicates the number of times the factor appears in the factorization.

Example: \(100=2*2*5*5\); then, factor(100) -> [5=>2,2=>2]

gcd(x, y)¶

Greatest common divisor

lcm(x, y)¶

Least common multiple

gcdx(x, y)¶

Greatest common divisor, also returning integer coefficients u and v that solve ux+vy == gcd(x,y)

ispow2(n)¶

Test whether n is a power of two

nextpow2(n)¶

Next power of two not less than n

prevpow2(n)¶

Previous power of two not greater than n

nextpow(a, n)¶

Next power of a not less than n

prevpow(a, n)¶

Previous power of a not greater than n

nextprod([a, b, c], n)¶

Next integer not less than n that can be written a^i1 * b^i2 * c^i3 for integers i1, i2, i3.

prevprod([a, b, c], n)¶

Previous integer not greater than n that can be written a^i1 * b^i2 * c^i3 for integers i1, i2, i3.

invmod(x, m)¶

Inverse of x, modulo m

powermod(x, p, m)¶

Compute mod(x^p, m)

gamma(x)¶

Compute the gamma function of x

lgamma(x)¶

Compute 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))

airy(k, x)¶

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 \(\operatorname{Bi}'(x)\).

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, \(J_\nu(x)\).

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, \(Y_\nu(x)\).

hankelh1(nu, x)¶

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

hankelh2(nu, x)¶

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

besseli(nu, x)¶

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

besselk(nu, x)¶

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

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 beta function \(\log(\operatorname{B}(x,y))\).

eta(x)¶

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

zeta(x)¶

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

bitmix(x, y)¶

Hash two integers into a single integer. Useful for constructing hash functions.

ndigits(n, b)¶

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

Data Formats¶

bin(n[, pad])¶

Convert an integer to a binary string, optionally specifying a number of digits to pad to.

hex(n[, pad])¶

Convert an integer to a hexadecimal string, optionally specifying a number of digits to pad to.

dec(n[, pad])¶

Convert an integer to a decimal string, optionally specifying a number of digits to pad to.

oct(n[, pad])¶

Convert an integer to an octal string, optionally specifying a number of digits to pad to.

base(base, n[, pad])¶

Convert an integer to a string in the given base, optionally specifying a number of digits to pad to. The base can be specified as either an integer, or as a Uint8 array of character values to use as digit symbols.

bits(n)¶

A string giving the literal bit representation of a number.

parseint([type], str[, base])¶

Parse a string as an integer in the given base (default 10), yielding a number of the specified type (default Int).

parsefloat([type], str)¶

Parse a string as a decimal floating point number, yielding a number of the specified type.

bool(x)¶

Convert a number or numeric array to boolean

isbool(x)¶

Test whether number or array is boolean

int(x)¶

Convert a number or array to the default integer type on your platform. Alternatively, x can be a string, which is parsed as an integer.

uint(x)¶

Convert a number or array to the default unsigned integer type on your platform. Alternatively, x can be a string, which is parsed as an unsigned integer.

integer(x)¶

Convert a number or array to integer type. If x is already of integer type it is unchanged, otherwise it converts it to the default integer type on your platform.

isinteger(x)¶

Test whether a number or array is of integer type

signed(x)¶

Convert a number to a signed integer

unsigned(x)¶

Convert a number to an unsigned integer

int8(x)¶

Convert a number or array to Int8 data type

int16(x)¶

Convert a number or array to Int16 data type

int32(x)¶

Convert a number or array to Int32 data type

int64(x)¶

Convert a number or array to Int64 data type

int128(x)¶

Convert a number or array to Int128 data type

uint8(x)¶

Convert a number or array to Uint8 data type

uint16(x)¶

Convert a number or array to Uint16 data type

uint32(x)¶

Convert a number or array to Uint32 data type

uint64(x)¶

Convert a number or array to Uint64 data type

uint128(x)¶

Convert a number or array to Uint128 data type

float32(x)¶

Convert a number or array to Float32 data type

float64(x)¶

Convert a number or array to Float64 data type

float(x)¶

Convert a number, array, or string to a FloatingPoint data type. For numeric data, the smallest suitable FloatingPoint type is used. For strings, it converts to Float64.

significand(x)¶

Extract the significand(s) (a.k.a. mantissa), in binary representation, of a floating-point number or array.

For example, significand(15.2)/15.2 == 0.125, and significand(15.2)*8 == 15.2

exponent(x) → Int¶

Get the exponent of a normalized floating-point number.

float64_valued(x::Rational)¶

True if x can be losslessly represented as a Float64 data type

complex64(r, i)¶

Convert to r+i*im represented as a Complex64 data type

complex128(r, i)¶

Convert to r+i*im represented as a Complex128 data type

char(x)¶

Convert a number or array to Char data type

complex(r, i)¶

Convert real numbers or arrays to complex

iscomplex(x) → Bool¶

Test whether a number or array is of a complex type

isreal(x) → Bool¶

Test whether a number or array is of a real type

bswap(n)¶

Byte-swap an integer

num2hex(f)¶

Get a hexadecimal string of the binary representation of a floating point number

hex2num(str)¶

Convert a hexadecimal string to the floating point number it represents

Numbers¶

one(x)¶

Get the multiplicative identity element for the type of x (x can also specify the type itself). For matrices, returns an identity matrix of the appropriate size and type.

zero(x)¶

Get the additive identity element for the type of x (x can also specify the type itself).

pi¶

The constant pi

im¶

The imaginary unit

e¶

The constant e

Inf¶

Positive infinity of type Float64

Inf32¶

Positive infinity of type Float32

NaN¶

A not-a-number value of type Float64

NaN32¶

A not-a-number value of type Float32

isdenormal(f) → Bool¶

Test whether a floating point number is denormal

isfinite(f) → Bool¶

Test whether a number is finite

isinf(f)¶

Test whether a number is infinite

isnan(f)¶

Test whether a floating point number is not a number (NaN)

inf(f)¶

Returns infinity in the same floating point type as f (or f can by the type itself)

nan(f)¶

Returns NaN in the same floating point type as f (or f can by the type itself)

nextfloat(f)¶

Get the next floating point number in lexicographic order

prevfloat(f) → Float¶

Get the previous floating point number in lexicographic order

integer_valued(x)¶

Test whether x is numerically equal to some integer

real_valued(x)¶

Test whether x is numerically equal to some real number

BigInt(x)¶

Create an arbitrary precision integer. x may be an Int (or anything that can be converted to an Int) or a String. The usual mathematical operators are defined for this type, and results are promoted to a BigInt.

BigFloat(x)¶

Create an arbitrary precision floating point number. x may be an Integer, a Float64, a String or a BigInt. The usual mathematical operators are defined for this type, and results are promoted to a BigFloat.

Random Numbers¶

Random number generateion in Julia uses the Mersenne Twister library. Julia has a global RNG, which is used by default. Multiple RNGs can be plugged in using the AbstractRNG object, which can then be used to have multiple streams of random numbers. Currently, only MersenneTwister is supported.

srand([rng], seed)¶

Seed the RNG with a seed, which may be an unsigned integer or a vector of unsigned integers. seed can even be a filename, in which case the seed is read from a file. If the argument rng is not provided, the default global RNG is seeded.

MersenneTwister([seed])¶

Create a MersenneTwister RNG object. Different RNG objects can have their own seeds, which may be useful for generating different streams of random numbers.

rand()¶

Generate a Float64 random number uniformly in [0,1)

rand!([rng], A)¶

Populate the array A with random number generated from the specified RNG.

rand(rng::AbstractRNG[, dims...])

Generate a random Float64 number or array of the size specified by dims, using the specified RNG object. Currently, MersenneTwister is the only available Random Number Generator (RNG), which may be seeded using srand.

rand(dims or [dims...])

Generate a random Float64 array of the size specified by dims

rand(Int32|Uint32|Int64|Uint64|Int128|Uint128[, dims...])

Generate a random integer of the given type. Optionally, generate an array of random integers of the given type by specifying dims.

rand(r[, dims...])

Generate a random integer from the inclusive interval specified by Range1 r (for example, 1:n). Optionally, generate a random integer array.

randbool([dims...])¶

Generate a random boolean value. Optionally, generate an array of random boolean values.

randbool!(A)¶

Fill an array with random boolean values. A may be an Array or a BitArray.

randn(dims or [dims...])¶

Generate a normally-distributed random number with mean 0 and standard deviation 1. Optionally generate an array of normally-distributed random numbers.

Arrays¶

Combinatorics¶

nthperm(v, k)¶

Compute the kth lexicographic permutation of a vector.

nthperm!(v, k)¶

In-place version of nthperm().

randperm(n)¶

Construct a random permutation of the given length.

invperm(v)¶

Return the inverse permutation of v.

isperm(v) → Bool¶

Returns true if v is a valid permutation.

permute!(v, p)¶

Permute vector v in-place, according to permutation p. No checking is done to verify that p is a permutation.

To return a new permutation, use v[p]. Note that this is generally faster than permute!(v,p) for large vectors.

ipermute!(v, p)¶

Like permute!, but the inverse of the given permutation is applied.

randcycle(n)¶

Construct a random cyclic permutation of the given length.

shuffle(v)¶

Randomly rearrange the elements of a vector.

shuffle!(v)¶

In-place version of shuffle().

reverse(v)¶

Reverse vector v.

reverse!(v) → v¶

In-place version of reverse().

combinations(array, n)¶

Generate all combinations of n elements from a given array. Because the number of combinations can be very large, this function runs inside a Task to produce values on demand. Write c = @task combinations(a,n), then iterate c or call consume on it.

integer_partitions(n, m)¶

Generate all arrays of m integers that sum to n. Because the number of partitions can be very large, this function runs inside a Task to produce values on demand. Write c = @task integer_partitions(n,m), then iterate c or call consume on it.

partitions(array)¶

Generate all set partitions of the elements of an array, represented as arrays of arrays. Because the number of partitions can be very large, this function runs inside a Task to produce values on demand. Write c = @task partitions(a), then iterate c or call consume on it.

Statistics¶

mean(v[, region])¶

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

std(v[, region])¶

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 draw from that generative distribution. This computation is equivalent to calculating sqrt(sum((v - mean(v)).^2) / (length(v) - 1)).

stdm(v, m)¶

Compute the sample standard deviation of a vector v with known mean m.

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 draw from that generative distribution. This computation is equivalent to calculating sum((v - mean(v)).^2) / (length(v) - 1).

varm(v, m)¶

Compute the sample variance of a vector v with known mean m.

median(v)¶

Compute the median of a vector v.

hist(v[, n]) → e, counts¶

Compute the histogram of v, optionally using approximately n bins. The return values are a range e, which correspond to the edges of the bins, and counts containing the number of elements of v in each bin.

hist(v, e) → e, counts

Compute the histogram of v using a vector/range e as the edges for the bins. The result will be a vector of length length(e) - 1, with the i``th element being ``sum(e[i] .< v .<= e[i+1]).

histrange(v, n)¶

Compute nice bin ranges for the edges of a histogram of v, using approximately n bins. The resulting step sizes will be 1, 2 or 5 multiplied by a power of 10.

midpoints(e)¶

Compute the midpoints of the bins with edges e. The result is a vector/range of length length(e) - 1.

quantile(v, p)¶

Compute the quantiles of a vector v at a specified set of probability values p.

quantile(v)

Compute the quantiles of a vector v at the probability values [.0, .2, .4, .6, .8, 1.0].

cov(v1[, v2])¶

Compute the Pearson covariance between two vectors v1 and v2. If called with a single element v, then computes covariance of columns of v.

cor(v1[, v2])¶

Compute the Pearson correlation between two vectors v1 and v2. If called with a single element v, then computes correlation of columns of v.

Signal Processing¶

FFT functions in Julia are largely 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 \(\operatorname{DFT}[k] = \sum_{n=1}^{\operatorname{length}(A)} \exp\left(-i\frac{2\pi (n-1)(k-1)}{\operatorname{length}(A)} \right) A[n]\). A multidimensional FFT simply performs this operation along each transformed dimension of A.

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 backward FFT computes \(\operatorname{BDFT}[k] = \sum_{n=1}^{\operatorname{length}(A)} \exp\left(+i\frac{2\pi (n-1)(k-1)}{\operatorname{length}(A)} \right) A[n]\). A multidimensional backward FFT simply performs this operation along each transformed dimension of A. The inverse FFT computes the same thing divided by the product of the transformed dimensions.

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.)

bfft!(A[, dims])¶

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

plan_fft(A[, dims[, flags[, timelimit]]])¶

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 a function plan(A) that computes fft(A, dims) quickly.

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[, timelimit]]])¶

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

plan_bfft(A[, dims[, flags[, timelimit]]])¶

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

plan_fft!(A[, dims[, flags[, timelimit]]])¶

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

plan_ifft!(A[, dims[, flags[, timelimit]]])¶

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

plan_bfft!(A[, dims[, flags[, timelimit]]])¶

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 (floor(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 d == floor(size(A,dims[1])/2)+1. (This parameter cannot be inferred from size(A) due to the possibility of rounding by the floor function here.)

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[, timelimit]]])¶

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_irfft(A, d[, dims[, flags[, timelimit]]])¶

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.

FFTW.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 FFTW.plan_r2r() to pre-plan optimized r2r transforms.

FFTW.r2r!(A, kind[, dims])¶

FFTW.r2r!() is the same as FFTW.r2r(), but operates in-place on A, which must be an array of real or complex floating-point numbers.

FFTW.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 FFTW.r2r() and FFTW.r2r!(), respectively.

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

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

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.