# API

## Guide

`SVector`

The simplest static array is the type `SVector{N,T}`

, which provides an immutable vector of fixed length `N`

and type `T`

.

`SVector`

defines a series of convenience constructors, so you can just type e.g. `SVector(1,2,3)`

. Alternatively there is an intelligent `@SVector`

macro where you can use native Julia array literals syntax, comprehensions, and the `zeros()`

, `ones()`

, `fill()`

, `rand()`

and `randn()`

functions, such as `@SVector [1,2,3]`

, `@SVector Float64[1,2,3]`

, `@SVector [f(i) for i = 1:10]`

, `@SVector zeros(3)`

, `@SVector randn(Float32, 4)`

, etc (Note: the range of a comprehension is evaluated at global scope by the macro, and must be made of combinations of literal values, functions, or global variables, but is not limited to just simple ranges. Extending this to (hopefully statically known by type-inference) local-scope variables is hoped for the future. The `zeros()`

, `ones()`

, `fill()`

, `rand()`

and `randn()`

functions do not have this limitation.)

`SMatrix`

Statically sized `N×M`

matrices are provided by `SMatrix{N,M,T,L}`

.

Here `L`

is the `length`

of the matrix, such that `N × M = L`

. However, convenience constructors are provided, so that `L`

, `T`

and even `M`

are unnecessary. At minimum, you can type `SMatrix{2}(1,2,3,4)`

to create a 2×2 matrix (the total number of elements must divide evenly into `N`

). A convenience macro `@SMatrix [1 2; 3 4]`

is provided (which also accepts comprehensions and the `zeros()`

, `ones()`

, `fill()`

, `rand()`

, `randn()`

and `eye()`

functions).

`SArray`

A container with arbitrarily many dimensions is defined as `struct SArray{Size,T,N,L} <: StaticArray{Size,T,N}`

, where `Size = Tuple{S1, S2, ...}`

is a tuple of `Int`

s. You can easily construct one with the `@SArray`

macro, supporting all the features of `@SVector`

and `@SMatrix`

(but with arbitrary dimension).

The main reason `SVector`

and `SMatrix`

are defined is to make it easier to define the types without the extra tuple characters (compare `SVector{3}`

to `SArray{Tuple{3}}`

).

`Scalar`

Sometimes you want to broadcast an operation, but not over one of your inputs. A classic example is attempting to displace a collection of vectors by the same vector. We can now do this with the `Scalar`

type:

`[[1,2,3], [4,5,6]] .+ Scalar([1,0,-1]) # [[2,2,2], [5,5,5]]`

`Scalar`

is simply an implementation of an immutable, 0-dimensional `StaticArray`

.

### The `Size`

trait

The size of a statically sized array is a static parameter associated with the type of the array. The `Size`

trait is provided as an abstract representation of the dimensions of a static array. An array `sa::SA`

of size `(dims...)`

is associated with `Size{(dims...)}()`

. The following are equivalent (`@pure`

) constructors:

```
Size{(dims...,)}()
Size(dims...)
Size(sa::StaticArray)
Size(SA) # SA <: StaticArray
```

This is extremely useful for (a) performing dispatch depending on the size of an array, and (b) passing array dimensions that the compiler can reason about.

An example of size-based dispatch for the determinant of a matrix would be:

```
det(x::StaticMatrix) = _det(Size(x), x)
_det(::Size{(1,1)}, x::StaticMatrix) = x[1,1]
_det(::Size{(2,2)}, x::StaticMatrix) = x[1,1]*x[2,2] - x[1,2]*x[2,1]
# and other definitions as necessary
```

Examples of using `Size`

as a compile-time constant include

```
reshape(svector, Size(2,2)) # Convert SVector{4} to SMatrix{2,2}
SizedMatrix{3,3}(rand(3,3)) # Construct a random 3×3 SizedArray (see below)
```

### Indexing

Statically sized indexing can be realized by indexing each dimension by a scalar, a `StaticVector`

or `:`

. Indexing in this way will result a statically sized array (even if the input was dynamically sized, in the case of `StaticVector`

indices) of the closest type (as defined by `similar_type`

).

Conversely, indexing a statically sized array with a dynamically sized index (such as a `Vector{Integer}`

or `UnitRange{Integer}`

) will result in a standard (dynamically sized) `Array`

.

`similar_type()`

Since immutable arrays need to be constructed "all-at-once", we need a way of obtaining an appropriate constructor if the element type or dimensions of the output array differs from the input. To this end, `similar_type`

is introduced, behaving just like `similar`

, except that it returns a type. Relevant methods are:

```
similar_type(::Type{A}) where {A <: StaticArray} # defaults to A
similar_type(::Type{A}, ::Type{ElType}) where {A <: StaticArray, ElType} # Change element type
similar_type(::Type{A}, size::Size) where {A <: AbstractArray} # Change size
similar_type(::Type{A}, ::Type{ElType}, size::Size) where {A <: AbstractArray, ElType} # Change both
```

These setting will affect everything, from indexing, to matrix multiplication and `broadcast`

. Users wanting introduce a new array type should *only* overload the last method in the above.

Use of `similar`

will fall back to a mutable container, such as a `MVector`

(see below), and it requires use of the `Size`

trait if you wish to set a new static size (or else a dynamically sized `Array`

will be generated when specifying the size as plain integers).

### Collecting directly into static arrays

You can collect iterators into static arrays directly with `StaticArrays.sacollect`

. The size needs to be specified, but the element type is optional.

### Mutable arrays: `MVector`

, `MMatrix`

and `MArray`

These statically sized arrays are identical to the above, but are defined as `mutable struct`

s, instead of immutable `struct`

s. Because they are mutable, they allow `setindex!`

to be defined (achieved through pointer manipulation, into a tuple).

As a consequence of Julia's internal implementation, these mutable containers live on the heap, not the stack. Their memory must be allocated and tracked by the garbage collector. Nevertheless, there is opportunity for speed improvements relative to `Base.Array`

because (a) there may be one less pointer indirection, (b) their (typically small) static size allows for additional loop unrolling and inlining, and consequentially (c) their mutating methods like `map!`

are extremely fast. Benchmarking shows that operations such as addition and matrix multiplication are faster for `MMatrix`

than `Matrix`

, at least for sizes up to 14 × 14, though keep in mind that optimal speed will be obtained by using mutating functions (like `map!`

or `A_mul_B!`

) where possible, rather than reallocating new memory.

Mutable static arrays also happen to be very useful containers that can be constructed on the heap (with the ability to use `setindex!`

, etc), and later copied as e.g. an immutable `SVector`

to the stack for use, or into e.g. an `Array{SVector}`

for storage.

Convenience macros `@MVector`

, `@MMatrix`

and `@MArray`

are provided.

`SizedArray`

: a decorate size wrapper for `Array`

Another convenient mutable type is the `SizedArray`

, which is just a wrapper-type about a standard Julia `Array`

which declares its known size. For example, if we knew that `a`

was a 2×2 `Matrix`

, then we can type `sa = SizedArray{Tuple{2,2}}(a)`

to construct a new object which knows the type (the size will be verified automatically). For one and two dimensions, a more convenient syntax for obtaining a `SizedArray`

is by using the `SizedMatrix`

and `SizedVector`

aliases, e.g. `sa = SizedMatrix{2,2}(a)`

.

Then, methods on `sa`

will use the specialized code provided by the *StaticArrays* package, which in many cases will be much, much faster. For example, calling `eigen(sa)`

will be signficantly faster than `eigen(a)`

since it will perform a specialized 2×2 matrix diagonalization rather than a general algorithm provided by Julia and *LAPACK*.

In some cases it will make more sense to use a `SizedArray`

, and in other cases an `MArray`

might be preferable.

`FieldVector`

Sometimes it is useful to give your own struct types the properties of a vector. *StaticArrays* can take care of this for you by allowing you to inherit from `FieldVector{N, T}`

. For example, consider:

```
struct Point3D <: FieldVector{3, Float64}
x::Float64
y::Float64
z::Float64
end
```

With this type, users can easily access fields to `p = Point3D(x,y,z)`

using `p.x`

, `p.y`

or `p.z`

, or alternatively via `p[1]`

, `p[2]`

, or `p[3]`

. You may even permute the coordinates with `p[SVector(3,2,1)]`

). Furthermore, `Point3D`

is a complete `AbstractVector`

implementation where you can add, subtract or scale vectors, multiply them by matrices, etc.

*Note*: the three components of an ordinary `v::SVector{3}`

can also be accessed as `v.x`

, `v.y`

, and `v.z`

, so there is no need for a `FieldVector`

to use this convention.

It is also worth noting that `FieldVector`

s may be mutable or immutable, and that `setindex!`

is defined for use on mutable types. For immutable containers, you may want to define a method for `similar_type`

so that operations leave the type constant (otherwise they may fall back to `SVector`

). For mutable containers, you may want to define a default constructor (no inputs) and an appropriate method for `similar`

,

### Implementing your own types

You can easily create your own `StaticArray`

type, by defining linear `getindex`

(and optionally `setindex!`

for mutable types –- see `setindex!(::MArray, val, i)`

in *MArray.jl* for an example of how to achieve this through pointer manipulation). Your type should define a constructor that takes a tuple of the data (and mutable containers may want to define a default constructor).

Other useful functions to overload may be `similar_type`

(and `similar`

for mutable containers).

### Conversions from `Array`

In order to convert from a dynamically sized `AbstractArray`

to one of the statically sized array types, you must specify the size explicitly. For example,

```
v = [1,2]
m = [1 2;
3 4]
# ... a lot of intervening code
sv = SVector{2}(v)
sm = SMatrix{2,2}(m)
sa = SArray{Tuple{2,2}}(m)
sized_v = SizedVector{2}(v)
sized_m = SizedMatrix{2,2}(m)
```

We have avoided adding `SVector(v::AbstractVector)`

as a valid constructor to help users avoid the type instability (and potential performance disaster, if used without care) of this innocuous looking expression.

### Arrays of static arrays

Storing a large number of static arrays is convenient as an array of static arrays. For example, a collection of positions (3D coordinates –- `SVector{3,Float64}`

) could be represented as a `Vector{SVector{3,Float64}}`

.

Another common way of storing the same data is as a 3×`N`

`Matrix{Float64}`

. Rather conveniently, such types have *exactly* the same binary layout in memory, and therefore we can use `reinterpret`

to convert between the two formats

```
function svectors(x::Matrix{T}, ::Val{N}) where {T,N}
size(x,1) == N || error("sizes mismatch")
isbitstype(T) || error("use for bitstypes only")
reinterpret(SVector{N,T}, vec(x))
end
```

Such a conversion does not copy the data, rather it refers to the *same* memory. Arguably, a `Vector`

of `SVector`

s is often preferable to a `Matrix`

because it provides a better abstraction of the objects contained in the array and it allows the fast *StaticArrays* methods to act on elements.

However, the resulting object is a Base.ReinterpretArray, not an Array, which carries some runtime penalty on every single access. If you can afford the memory for a copy and can live with the non-shared mutation semantics, then it is better to pull a copy by e.g.

```
function svectorscopy(x::Matrix{T}, ::Val{N}) where {T,N}
size(x,1) == N || error("sizes mismatch")
isbitstype(T) || error("use for bitstypes only")
copy(reinterpret(SVector{N,T}, vec(x)))
end
```

For example:

```
julia> M=reshape(collect(1:6), (2,3))
2×3 Array{Int64,2}:
1 3 5
2 4 6
julia> svectors(M, Val{2}())
3-element reinterpret(SArray{Tuple{2},Int64,1,2}, ::Array{Int64,1}):
[1, 2]
[3, 4]
[5, 6]
julia> svectorscopy(M, Val{2}())
3-element Array{SArray{Tuple{2},Int64,1,2},1}:
[1, 2]
[3, 4]
[5, 6]
```

### Working with mutable and immutable arrays

Generally, it is performant to rebind an *immutable* array, such as

```
function average_position(positions::Vector{SVector{3,Float64}})
x = zeros(SVector{3,Float64})
for pos ∈ positions
x = x + pos
end
return x / length(positions)
end
```

so long as the `Type`

of the rebound variable (`x`

, above) does not change.

On the other hand, the above code for mutable containers like `Array`

, `MArray`

or `SizedArray`

is *not* very efficient. Mutable containers in Julia 0.5 must be *allocated* and later *garbage collected*, and for small, fixed-size arrays this can be a leading contribution to the cost. In the above code, a new array will be instantiated and allocated on each iteration of the loop. In order to avoid unnecessary allocations, it is best to allocate an array only once and apply mutating functions to it:

```
function average_position(positions::Vector{SVector{3,Float64}})
x = zeros(MVector{3,Float64})
for pos ∈ positions
# Take advantage of Julia 0.5 broadcast fusion
x .= (+).(x, pos) # same as broadcast!(+, x, x, positions[i])
end
x .= (/).(x, length(positions))
return x
end
```

Keep in mind that Julia 0.5 does not fuse calls to `.+`

, etc (or `.+=`

etc), however the `.=`

and `(+).()`

syntaxes are fused into a single, efficient call to `broadcast!`

. The simpler syntax `x .+= pos`

is expected to be non-allocating (and therefore faster) in Julia 0.6.

The functions `setindex`

, `push`

, `pop`

, `pushfirst`

, `popfirst`

, `insert`

and `deleteat`

are provided for performing certain specific operations on static arrays, in analogy with the standard functions `setindex!`

, `push!`

, `pop!`

, etc. (Note that if the size of the static array changes, the type of the output will differ from the input.)

When building static arrays iteratively, it is usually efficient to build up an `MArray`

first and then convert. The allocation will be elided by recent Julia compilers, resulting in very efficient code:

```
function standard_basis_vector(T, ::Val{I}, ::Val{N}) where {I,N}
v = zero(MVector{N,T})
v[I] = one(T)
SVector(v)
end
```

### SIMD optimizations

It seems Julia and LLVM are smart enough to use processor vectorization extensions like SSE and AVX - however they are currently partially disabled by default. Run Julia with `julia -O`

or `julia -O3`

to enable these optimizations, and many of your (immutable) `StaticArray`

methods *should* become significantly faster!

## Docstrings

`StaticArrays.Dynamic`

`StaticArrays.FieldArray`

`StaticArrays.FieldMatrix`

`StaticArrays.FieldVector`

`StaticArrays.MArray`

`StaticArrays.MMatrix`

`StaticArrays.MVector`

`StaticArrays.SA`

`StaticArrays.SArray`

`StaticArrays.SHermitianCompact`

`StaticArrays.SMatrix`

`StaticArrays.SOneTo`

`StaticArrays.SVector`

`StaticArrays.Scalar`

`StaticArrays.Size`

`StaticArrays.SizedArray`

`StaticArrays.StaticArray`

`StaticArrays.StaticMatMulLike`

`StaticArrays.TSize`

`StaticArrays._InitialValue`

`Base.setindex`

`Base.similar`

`LinearAlgebra.qr`

`StaticArrays._construct_similar`

`StaticArrays._lind`

`StaticArrays._muladd_expr`

`StaticArrays._size`

`StaticArrays.arithmetic_closure`

`StaticArrays.check_dims`

`StaticArrays.deleteat`

`StaticArrays.dimmatch`

`StaticArrays.gen_by_access`

`StaticArrays.gen_by_access`

`StaticArrays.insert`

`StaticArrays.mul_result_structure`

`StaticArrays.multiplied_dimension`

`StaticArrays.pop`

`StaticArrays.popfirst`

`StaticArrays.push`

`StaticArrays.pushfirst`

`StaticArrays.sacollect`

`StaticArrays.same_size`

`StaticArrays.similar_type`

`StaticArrays.size_to_tuple`

`StaticArrays.sizematch`

`StaticArrays.sizematch`

`StaticArrays.uplo_access`

`StaticArrays.@MArray`

`StaticArrays.@MMatrix`

`StaticArrays.@MVector`

`StaticArrays.@SArray`

`StaticArrays.@SMatrix`

`StaticArrays.@SVector`

`StaticArrays.StaticMatMulLike`

— Type`StaticMatMulLike`

Static wrappers used for multiplication dispatch.

`StaticArrays.Dynamic`

— Type`Dynamic()`

Used to signify that a dimension of an array is not known statically.

`StaticArrays.FieldArray`

— Type`abstract FieldArray{N, T, D} <: StaticArray{N, T, D}`

Inheriting from this type will make it easy to create your own rank-D tensor types. A `FieldArray`

will automatically define `getindex`

and `setindex!`

appropriately. An immutable `FieldArray`

will be as performant as an `SArray`

of similar length and element type, while a mutable `FieldArray`

will behave similarly to an `MArray`

.

Note that you must define the fields of any `FieldArray`

subtype in column major order. If you want to use an alternative ordering you will need to pay special attention in providing your own definitions of `getindex`

, `setindex!`

and tuple conversion.

If you define a `FieldArray`

which is parametric on the element type you should consider defining `similar_type`

as in the `FieldVector`

example.

**Example**

```
struct Stiffness <: FieldArray{Tuple{2,2,2,2}, Float64, 4}
xxxx::Float64
yxxx::Float64
xyxx::Float64
yyxx::Float64
xxyx::Float64
yxyx::Float64
xyyx::Float64
yyyx::Float64
xxxy::Float64
yxxy::Float64
xyxy::Float64
yyxy::Float64
xxyy::Float64
yxyy::Float64
xyyy::Float64
yyyy::Float64
end
```

`StaticArrays.FieldMatrix`

— Type`abstract FieldMatrix{N1, N2, T} <: FieldArray{Tuple{N1, N2}, 2}`

Inheriting from this type will make it easy to create your own rank-two tensor types. A `FieldMatrix`

will automatically define `getindex`

and `setindex!`

appropriately. An immutable `FieldMatrix`

will be as performant as an `SMatrix`

of similar length and element type, while a mutable `FieldMatrix`

will behave similarly to an `MMatrix`

.

Note that the fields of any subtype of `FieldMatrix`

must be defined in column major order unless you are willing to implement your own `getindex`

.

If you define a `FieldMatrix`

which is parametric on the element type you should consider defining `similar_type`

as in the `FieldVector`

example.

**Example**

```
struct Stress <: FieldMatrix{3, 3, Float64}
xx::Float64
yx::Float64
zx::Float64
xy::Float64
yy::Float64
zy::Float64
xz::Float64
yz::Float64
zz::Float64
end
```

Note that the fields of any subtype of `FieldMatrix`

must be defined in column major order. This means that formatting of constructors for literal `FieldMatrix`

can be confusing. For example

```
sigma = Stress(1.0, 2.0, 3.0,
4.0, 5.0, 6.0,
7.0, 8.0, 9.0)
3×3 Stress:
1.0 4.0 7.0
2.0 5.0 8.0
3.0 6.0 9.0
```

will give you the transpose of what the multi-argument formatting suggests. For clarity, you may consider using the alternative

```
sigma = Stress(SA[1.0 2.0 3.0;
4.0 5.0 6.0;
7.0 8.0 9.0])
```

`StaticArrays.FieldVector`

— Type`abstract FieldVector{N, T} <: FieldArray{Tuple{N}, 1}`

Inheriting from this type will make it easy to create your own vector types. A `FieldVector`

will automatically define `getindex`

and `setindex!`

appropriately. An immutable `FieldVector`

will be as performant as an `SVector`

of similar length and element type, while a mutable `FieldVector`

will behave similarly to an `MVector`

.

If you define a `FieldVector`

which is parametric on the element type you should consider defining `similar_type`

to preserve your array type through array operations as in the example below.

**Example**

```
struct Vec3D{T} <: FieldVector{3, T}
x::T
y::T
z::T
end
StaticArrays.similar_type(::Type{<:Vec3D}, ::Type{T}, s::Size{(3,)}) where {T} = Vec3D{T}
```

`StaticArrays.MArray`

— Type```
MArray{S, T, N, L}(undef)
MArray{S, T, N, L}(x::NTuple{L})
MArray{S, T, N, L}(x1, x2, x3, ...)
```

Construct a statically-sized, mutable array `MArray`

. The data may optionally be provided upon construction and can be mutated later. The `S`

parameter is a Tuple-type specifying the dimensions, or size, of the array - such as `Tuple{3,4,5}`

for a 3×4×5-sized array. The `N`

parameter is the dimension of the array; the `L`

parameter is the `length`

of the array and is always equal to `prod(S)`

. Constructors may drop the `L`

, `N`

and `T`

parameters if they are inferrable from the input (e.g. `L`

is always inferrable from `S`

).

`MArray{S}(a::Array)`

Construct a statically-sized, mutable array of dimensions `S`

(expressed as a `Tuple{...}`

) using the data from `a`

. The `S`

parameter is mandatory since the size of `a`

is unknown to the compiler (the element type may optionally also be specified).

`StaticArrays.MMatrix`

— Type```
MMatrix{S1, S2, T, L}(undef)
MMatrix{S1, S2, T, L}(x::NTuple{L, T})
MMatrix{S1, S2, T, L}(x1, x2, x3, ...)
```

Construct a statically-sized, mutable matrix `MMatrix`

. The data may optionally be provided upon construction and can be mutated later. The `L`

parameter is the `length`

of the array and is always equal to `S1 * S2`

. Constructors may drop the `L`

, `T`

and even `S2`

parameters if they are inferrable from the input (e.g. `L`

is always inferrable from `S1`

and `S2`

).

`MMatrix{S1, S2}(mat::Matrix)`

Construct a statically-sized, mutable matrix of dimensions `S1 × S2`

using the data from `mat`

. The parameters `S1`

and `S2`

are mandatory since the size of `mat`

is unknown to the compiler (the element type may optionally also be specified).

`StaticArrays.MVector`

— Type```
MVector{S,T}(undef)
MVector{S,T}(x::NTuple{S, T})
MVector{S,T}(x1, x2, x3, ...)
```

Construct a statically-sized, mutable vector `MVector`

. Data may optionally be provided upon construction, and can be mutated later. Constructors may drop the `T`

and `S`

parameters if they are inferrable from the input (e.g. `MVector(1,2,3)`

constructs an `MVector{3, Int}`

).

`MVector{S}(vec::Vector)`

Construct a statically-sized, mutable vector of length `S`

using the data from `vec`

. The parameter `S`

is mandatory since the length of `vec`

is unknown to the compiler (the element type may optionally also be specified).

`StaticArrays.SA`

— Type```
SA[ elements ]
SA{T}[ elements ]
```

Create `SArray`

literals using array construction syntax. The element type is inferred by promoting `elements`

to a common type or set to `T`

when `T`

is provided explicitly.

**Examples:**

`SA[1.0, 2.0]`

creates a length-2`SVector`

of`Float64`

elements.`SA[1 2; 3 4]`

creates a 2×2 SMatrix of`Int`

s.`SA[1 2]`

creates a 1×2 SMatrix of`Int`

s.`SA{Float32}[1, 2]`

creates a length-2`SVector`

of`Float32`

elements.

A couple of helpful type aliases are also provided:

`SA_F64[1, 2]`

creates a length-2`SVector`

of`Float64`

elements`SA_F32[1, 2]`

creates a length-2`SVector`

of`Float32`

elements

`StaticArrays.SArray`

— Type```
SArray{S, T, N, L}(x::NTuple{L})
SArray{S, T, N, L}(x1, x2, x3, ...)
```

Construct a statically-sized array `SArray`

. Since this type is immutable, the data must be provided upon construction and cannot be mutated later. The `S`

parameter is a Tuple-type specifying the dimensions, or size, of the array - such as `Tuple{3,4,5}`

for a 3×4×5-sized array. The `N`

parameter is the dimension of the array; the `L`

parameter is the `length`

of the array and is always equal to `prod(S)`

. Constructors may drop the `L`

, `N`

and `T`

parameters if they are inferrable from the input (e.g. `L`

is always inferrable from `S`

).

`SArray{S}(a::Array)`

Construct a statically-sized array of dimensions `S`

(expressed as a `Tuple{...}`

) using the data from `a`

. The `S`

parameter is mandatory since the size of `a`

is unknown to the compiler (the element type may optionally also be specified).

`StaticArrays.SHermitianCompact`

— Type`SHermitianCompact{N, T, L} <: StaticMatrix{N, N, T}`

A `StaticArray`

subtype that represents a Hermitian matrix. Unlike `LinearAlgebra.Hermitian`

, `SHermitianCompact`

stores only the lower triangle of the matrix (as an `SVector`

). The lower triangle is stored in column-major order. For example, for an `SHermitianCompact{3}`

, the indices of the stored elements can be visualized as follows:

```
┌ 1 ⋅ ⋅ ┐
| 2 4 ⋅ |
└ 3 5 6 ┘
```

Type parameters:

`N`

: matrix dimension;`T`

: element type for lower triangle;`L`

: length of the`SVector`

storing the lower triangular elements.

Note that `L`

is always the `N`

th triangular number.

An `SHermitianCompact`

may be constructed either:

- from an
`AbstractVector`

containing the lower triangular elements; or - from a
`Tuple`

containing both upper and lower triangular elements in column major order; or - from another
`StaticMatrix`

.

For the latter two cases, only the lower triangular elements are used; the upper triangular elements are ignored.

`StaticArrays.SMatrix`

— Type```
SMatrix{S1, S2, T, L}(x::NTuple{L, T})
SMatrix{S1, S2, T, L}(x1, x2, x3, ...)
```

Construct a statically-sized matrix `SMatrix`

. Since this type is immutable, the data must be provided upon construction and cannot be mutated later. The `L`

parameter is the `length`

of the array and is always equal to `S1 * S2`

. Constructors may drop the `L`

, `T`

and even `S2`

parameters if they are inferrable from the input (e.g. `L`

is always inferrable from `S1`

and `S2`

).

`SMatrix{S1, S2}(mat::Matrix)`

Construct a statically-sized matrix of dimensions `S1 × S2`

using the data from `mat`

. The parameters `S1`

and `S2`

are mandatory since the size of `mat`

is unknown to the compiler (the element type may optionally also be specified).

`StaticArrays.SOneTo`

— Type`SOneTo(n)`

Return a statically-sized `AbstractUnitRange`

starting at `1`

, functioning as the `axes`

of a `StaticArray`

.

`StaticArrays.SVector`

— Type```
SVector{S, T}(x::NTuple{S, T})
SVector{S, T}(x1, x2, x3, ...)
```

Construct a statically-sized vector `SVector`

. Since this type is immutable, the data must be provided upon construction and cannot be mutated later. Constructors may drop the `T`

and `S`

parameters if they are inferrable from the input (e.g. `SVector(1,2,3)`

constructs an `SVector{3, Int}`

).

`SVector{S}(vec::Vector)`

Construct a statically-sized vector of length `S`

using the data from `vec`

. The parameter `S`

is mandatory since the length of `vec`

is unknown to the compiler (the element type may optionally also be specified).

`StaticArrays.Scalar`

— Type`Scalar{T}(x::T)`

Construct a statically-sized 0-dimensional array that contains a single element, `x`

. This type is particularly useful for influencing broadcasting operations.

`StaticArrays.Size`

— Type`Size(dims::Int...)`

`Size`

is used extensively throughout the `StaticArrays`

API to describe *compile-time* knowledge of the size of an array. The dimensions are stored as a type parameter and are statically propagated by the compiler, resulting in efficient, type-inferrable code. For example, to create a static matrix of zeros, use `A = zeros(SMatrix{3,3})`

. The static size of `A`

can be obtained by `Size(A)`

. (rather than `size(zeros(3,3))`

, which returns `Base.Tuple{2,Int}`

).

Note that if dimensions are not known statically (e.g., for standard `Array`

s), `Dynamic()`

should be used instead of an `Int`

.

```
Size(a::AbstractArray)
Size(::Type{T<:AbstractArray})
```

The `Size`

constructor can be used to extract static dimension information from a given array. For example:

```
julia> Size(zeros(SMatrix{3, 4}))
Size(3, 4)
julia> Size(zeros(3, 4))
Size(StaticArrays.Dynamic(), StaticArrays.Dynamic())
```

This has multiple uses, including "trait"-based dispatch on the size of a statically-sized array. For example:

```
det(x::StaticMatrix) = _det(Size(x), x)
_det(::Size{(1,1)}, x::StaticMatrix) = x[1,1]
_det(::Size{(2,2)}, x::StaticMatrix) = x[1,1]*x[2,2] - x[1,2]*x[2,1]
# and other definitions as necessary
```

`StaticArrays.SizedArray`

— Type`SizedArray{Tuple{dims...}}(array)`

Wraps an `AbstractArray`

with a static size, so to take advantage of the (faster) methods defined by the static array package. The size is checked once upon construction to determine if the number of elements (`length`

) match, but the array may be reshaped.

The aliases `SizedVector{N}`

and `SizedMatrix{N,M}`

are provided as more convenient names for one and two dimensional `SizedArray`

s. For example, to wrap a 2x3 array `a`

in a `SizedArray`

, use `SizedMatrix{2,3}(a)`

.

`StaticArrays.StaticArray`

— Type```
abstract type StaticArray{S, T, N} <: AbstractArray{T, N} end
StaticScalar{T} = StaticArray{Tuple{}, T, 0}
StaticVector{N,T} = StaticArray{Tuple{N}, T, 1}
StaticMatrix{N,M,T} = StaticArray{Tuple{N,M}, T, 2}
```

`StaticArray`

s are Julia arrays with fixed, known size.

**Dev docs**

They must define the following methods:

- Constructors that accept a flat tuple of data.
`getindex()`

with an integer (linear indexing) (preferably`@inline`

with`@boundscheck`

).`Tuple()`

, returning the data in a flat Tuple.

It may be useful to implement:

`similar_type(::Type{MyStaticArray}, ::Type{NewElType}, ::Size{NewSize})`

, returning a type (or type constructor) that accepts a flat tuple of data.

For mutable containers you may also need to define the following:

`setindex!`

for a single element (linear indexing).`similar(::Type{MyStaticArray}, ::Type{NewElType}, ::Size{NewSize})`

.- In some cases, a zero-parameter constructor,
`MyStaticArray{...}()`

for unintialized data is assumed to exist.

(see also `SVector`

, `SMatrix`

, `SArray`

, `MVector`

, `MMatrix`

, `MArray`

, `SizedArray`

, `FieldVector`

, `FieldMatrix`

and `FieldArray`

)

`StaticArrays.TSize`

— TypeSize that stores whether a Matrix is a Transpose Useful when selecting multiplication methods, and avoiding allocations when dealing with the `Transpose`

type by passing around the original matrix. Should pair with `parent`

.

`StaticArrays._InitialValue`

— Type`_InitialValue`

A singleton type for representing "universal" initial value (identity element).

The idea is that, given `op`

for `mapfoldl`

, virtually, we define an "extended" version of it by

```
op′(::_InitialValue, x) = x
op′(acc, x) = op(acc, x)
```

This is just a conceptually useful model to have in mind and we don't actually define `op′`

here (yet?). But see `Base.BottomRF`

for how it might work in action.

(It is related to that you can always turn a semigroup without an identity into a monoid by "adjoining" an element that acts as the identity.)

`Base.setindex`

— Method`setindex(vec::StaticArray, x, index::Int)`

Return a new array with the item at `index`

replaced by `x`

.

**Examples**

```
julia> setindex(@SVector[1,2,3], 4, 2)
3-element SArray{Tuple{3},Int64,1,3} with indices SOneTo(3):
1
4
3
julia> setindex(@SMatrix[2 4; 6 8], 1, 2)
2×2 SArray{Tuple{2,2},Int64,2,4} with indices SOneTo(2)×SOneTo(2):
2 4
1 8
```

`Base.similar`

— Method```
similar(static_array)
similar(static_array, T)
similar(array, ::Size)
similar(array, T, ::Size)
```

Constructs and returns a mutable but statically-sized array (i.e. a `StaticArray`

). If the input `array`

is not a `StaticArray`

, then the `Size`

is required to determine the output size (or else a dynamically sized array will be returned).

`LinearAlgebra.qr`

— Method```
qr(A::StaticMatrix,
pivot::Union{Val{true}, Val{false}, LinearAlgebra.PivotingStrategy} = Val(false))
```

Compute the QR factorization of `A`

. The factors can be obtained by iteration:

```
julia> A = @SMatrix rand(3,4);
julia> Q, R = qr(A);
julia> Q * R ≈ A
true
```

or by using `getfield`

:

```
julia> F = qr(A);
julia> F.Q * F.R ≈ A
true
```

`StaticArrays._construct_similar`

— Method`_construct_similar(a, ::Size, elements::NTuple)`

Construct a static array of similar type to `a`

with the given `elements`

.

When `a`

is an instance or a concrete type the element type `eltype(a)`

is used. However, when `a`

is a `UnionAll`

type such as `SMatrix{2,2}`

, the promoted type of `elements`

is used instead.

`StaticArrays._lind`

— MethodObtain an expression for the linear index of var[k,j], taking transposes into account

`StaticArrays._muladd_expr`

— MethodCombine left and right sides of an assignment expression, short-cutting lhs = α * rhs + β * lhs, element-wise. If α = 1, the multiplication by α is removed. If β = 0, the second rhs term is removed.

`StaticArrays._size`

— MethodReturn either the statically known Size() or runtime size()

`StaticArrays.arithmetic_closure`

— Method`arithmetic_closure(T)`

Return the type which values of type `T`

will promote to under a combination of the arithmetic operations `+, -, *`

and `/`

.

```
julia> import StaticArrays.arithmetic_closure
julia> arithmetic_closure(Bool)
Float64
julia> arithmetic_closure(Int32)
Float64
julia> arithmetic_closure(BigFloat)
BigFloat
julia> arithmetic_closure(BigInt)
BigFloat
```

`StaticArrays.check_dims`

— MethodValidate the dimensions of a matrix multiplication, including matrix-vector products

`StaticArrays.deleteat`

— Method`deleteat(vec::StaticVector, index::Integer)`

Return a new vector with the item at the given `index`

removed.

**Examples**

```
julia> deleteat(@SVector[6, 5, 4, 3, 2, 1], 2)
5-element SArray{Tuple{5},Int64,1,5} with indices SOneTo(5):
6
4
3
2
1
```

`StaticArrays.dimmatch`

— Function`dimmatch(x::StaticDimension, y::StaticDimension)`

Return whether dimensions `x`

and `y`

match at compile time, that is:

- if
`x`

and`y`

are both`Int`

s, check that they are equal - if
`x`

or`y`

are`Dynamic()`

, return true

`StaticArrays.gen_by_access`

— Function`gen_by_access(expr_gen, a::Type{<:AbstractArray}, asym = :wrapped_a)`

Statically generate outer code for fully unrolled multiplication loops. Returned code does wrapper-specific tests (for example if a symmetric matrix view is `U`

or `L`

) and the body of the if expression is then generated by function `expr_gen`

. The function `expr_gen`

receives access pattern description symbol as its argument and this symbol is then consumed by uplo_access to generate the right code for matrix element access.

The name of the matrix to test is indicated by `asym`

.

`StaticArrays.gen_by_access`

— Method`gen_by_access(expr_gen, a::Type{<:AbstractArray}, b::Type{<:AbstractArray})`

Simiar to gen*by*access with only one type argument. The difference is that tests for both arrays of type `a`

and `b`

are generated and `expr_gen`

receives two access arguments, first for matrix `a`

and the second for matrix `b`

.

`StaticArrays.insert`

— Method`insert(vec::StaticVector, index::Integer, item)`

Return a new vector with `item`

inserted into `vec`

at the given `index`

.

**Examples**

```
julia> insert(@SVector[6, 5, 4, 2, 1], 4, 3)
6-element SArray{Tuple{6},Int64,1,6} with indices SOneTo(6):
6
5
4
3
2
1
```

`StaticArrays.mul_result_structure`

— Method`mul_result_structure(a::Type, b::Type)`

Get a structure wrapper that should be applied to the result of multiplication of matrices of given types (a*b).

`StaticArrays.multiplied_dimension`

— MethodCalculate the product of the dimensions being multiplied. Useful as a heuristic for unrolling.

`StaticArrays.pop`

— Method`pop(vec::StaticVector)`

Return a new vector with the last item in `vec`

removed.

**Examples**

```
julia> pop(@SVector[1,2,3])
2-element SArray{Tuple{2},Int64,1,2} with indices SOneTo(2):
1
2
```

`StaticArrays.popfirst`

— Method`popfirst(vec::StaticVector)`

Return a new vector with the first item in `vec`

removed.

**Examples**

```
julia> popfirst(@SVector[1,2,3])
2-element SArray{Tuple{2},Int64,1,2} with indices SOneTo(2):
2
3
```

`StaticArrays.push`

— Method`push(vec::StaticVector, item)`

Return a new `StaticVector`

with `item`

inserted on the end of `vec`

.

**Examples**

```
julia> push(@SVector[1, 2, 3], 4)
4-element SArray{Tuple{4},Int64,1,4} with indices SOneTo(4):
1
2
3
4
```

`StaticArrays.pushfirst`

— Method`pushfirst(vec::StaticVector, item)`

Return a new `StaticVector`

with `item`

inserted at the beginning of `vec`

.

**Examples**

```
julia> pushfirst(@SVector[1, 2, 3, 4], 5)
5-element SArray{Tuple{5},Int64,1,5} with indices SOneTo(5):
5
1
2
3
4
```

`StaticArrays.sacollect`

— Functionsacollect(SA, gen)

Construct a statically-sized vector of type `SA`

.from a generator `gen`

. `SA`

needs to have a size parameter since the length of `vec`

is unknown to the compiler. `SA`

can optionally specify the element type as well.

Example:

```
sacollect(SVector{3, Int}, 2i+1 for i in 1:3)
sacollect(SMatrix{2, 3}, i+j for i in 1:2, j in 1:3)
sacollect(SArray{2, 3}, i+j for i in 1:2, j in 1:3)
```

This creates the same statically-sized vector as if the generator were collected in an array, but is more efficient since no array is allocated.

Equivalent:

`SVector{3, Int}([2i+1 for i in 1:3])`

`StaticArrays.same_size`

— MethodReturns the common Size of the inputs (or else throws a DimensionMismatch)

`StaticArrays.similar_type`

— Function```
similar_type(static_array)
similar_type(static_array, T)
similar_type(array, ::Size)
similar_type(array, T, ::Size)
```

Returns a constructor for a statically-sized array similar to the input array (or type) `static_array`

/`array`

, optionally with different element type `T`

or size `Size`

. If the input `array`

is not a `StaticArray`

then the `Size`

is mandatory.

This differs from `similar()`

in that the resulting array type may not be mutable (or define `setindex!()`

), and therefore the returned type may need to be *constructed* with its data.

Note that the (optional) size *must* be specified as a static `Size`

object (so the compiler can infer the result statically).

New types should define the signature `similar_type(::Type{A},::Type{T},::Size{S}) where {A<:MyType,T,S}`

if they wish to overload the default behavior.

`StaticArrays.size_to_tuple`

— Method`size_to_tuple(::Type{S}) where S<:Tuple`

Converts a size given by `Tuple{N, M, ...}`

into a tuple `(N, M, ...)`

.

`StaticArrays.sizematch`

— Method```
sizematch(::Size, ::Size)
sizematch(::Tuple, ::Tuple)
```

Determine whether two sizes match, in the sense that they have the same number of dimensions, and their dimensions match as determined by `dimmatch`

.

`StaticArrays.sizematch`

— Method`sizematch(::Size, A::AbstractArray)`

Determine whether array `A`

matches the given size. If `A`

is a `StaticArray`

, the check is performed at compile time, otherwise, the check is performed at runtime.

`StaticArrays.uplo_access`

— Method`uplo_access(sa, asym, k, j, uplo)`

Generate code for matrix element access, for a matrix of size `sa`

locally referred to as `asym`

in the context where the result will be used. Both indices `k`

and `j`

need to be statically known for this function to work. `uplo`

is the access pattern mode generated by the `gen_by_access`

function.

`StaticArrays.@MArray`

— Macro```
@MArray [a b; c d]
@MArray [[a, b];[c, d]]
@MArray [i+j for i in 1:2, j in 1:2]
@MArray ones(2, 2, 2)
```

A convenience macro to construct `MArray`

with arbitrary dimension. See `@SArray`

for detailed features.

`StaticArrays.@MMatrix`

— Macro```
@MMatrix [a b c d]
@MMatrix [[a, b];[c, d]]
@MMatrix [i+j for i in 1:2, j in 1:2]
@MMatrix ones(2, 2, 2)
```

A convenience macro to construct `MMatrix`

. See `@SArray`

for detailed features.

`StaticArrays.@MVector`

— Macro```
@MVector [a, b, c, d]
@MVector [i for i in 1:2]
@MVector ones(2)
```

A convenience macro to construct `MVector`

. See `@SArray`

for detailed features.

`StaticArrays.@SArray`

— Macro```
@SArray [a b; c d]
@SArray [[a, b];[c, d]]
@SArray [i+j for i in 1:2, j in 1:2]
@SArray ones(2, 2, 2)
```

A convenience macro to construct `SArray`

with arbitrary dimension. It supports:

- (typed) array literals.

Every argument inside the square brackets is treated as a scalar during expansion. Thus `@SArray[a; b]`

always forms a `SVector{2}`

and `@SArray [a b; c]`

always throws an error.

- comprehensions

The range of a comprehension is evaluated at global scope by the macro, and must be made of combinations of literal values, functions, or global variables.

- initialization functions

Only support `zeros()`

, `ones()`

, `fill()`

, `rand()`

, `randn()`

, and `randexp()`

`StaticArrays.@SMatrix`

— Macro```
@SMatrix [a b c d]
@SMatrix [[a, b];[c, d]]
@SMatrix [i+j for i in 1:2, j in 1:2]
@SMatrix ones(2, 2, 2)
```

A convenience macro to construct `SMatrix`

. See `@SArray`

for detailed features.

`StaticArrays.@SVector`

— Macro```
@SVector [a, b, c, d]
@SVector [i for i in 1:2]
@SVector ones(2)
```

A convenience macro to construct `SVector`

. See `@SArray`

for detailed features.