Methods

So far, we defined all functions (with some exceptions) without annotating the types of input arguments. When the type annotation is omitted, the default behaviour in Julia is to allow values to be of any type. One can write many useful functions without stating the types. When additional expressiveness is needed, it is easy to introduce type annotations into previously untyped code.

In Julia, one function consists of multiple methods. A prime example is the convert function. When a user calls a function, the process of choosing which method to execute is called dispatch. The dispatch system in Julia decides which method to execute based on:

• the number of function arguments;
• the types of function arguments.

Using all function arguments to choose which method should be invoked is known as multiple dispatch.

As an example of multiple dispatch, we define the product function that computes the product of two numbers.

julia> product(x, y) = x * y
product (generic function with 1 method)

In the REPL, we can see that the product function has only one method. In this case, we defined the method for any two input arguments without type specification.

julia> product(1, 4.5)
4.5

julia> product(2.4, 3.1)
7.4399999999999995

The methods function lists all methods for a function.

julia> methods(product)
# 1 method for generic function "product" from Main:
 [1] product(x, y)
     @ none:1

Because we did not specify types of input arguments, the product function accepts arguments of all types. For some inputs, such as symbols, the * operator will not work.

julia> product(:a, :b)
ERROR: MethodError: no method matching *(::Symbol, ::Symbol)
[...]

We can avoid such errors by specifying types of input arguments. Since we want to create a function that computes the product of two numbers, it makes sense to allow input arguments to be only numbers.

product(x::Number, y::Number) = x * y
product(x, y) = throw(ArgumentError("product is defined for numbers only."))

The second line redefined the original definition of the product function. It now throws an error if product is called with non-numeric inputs.

julia> methods(product)
# 2 methods for generic function "product" from Main:
 [1] product(x::Number, y::Number)
     @ none:1
 [2] product(x, y)
     @ none:1

Now, we have a function with two methods, that returns a product if the input arguments are numbers, and throws an error otherwise.

julia> product(1, 4.5)
4.5

julia> product(:a, :b)
ERROR: ArgumentError: product is defined for numbers only.

julia> product("a", "b")
ERROR: ArgumentError: product is defined for numbers only.

Type hierarchy

It is always better to use abstract types like Number or Real instead of concrete types like Float64, Float32, or Int64. The reason is that if we use an abstract type, the function will work for all its subtypes. To find a supertype for a specific type, we can use the supertype function from the InteractiveUtils package.

julia> using InteractiveUtils: supertype

julia> supertype(Float64)
AbstractFloat

The problem with the supertype function is that it does not return the whole supertype hierarchy, but only the closest larger supertype. For Float64 the closest larger supertype is AbstractFloat. However, as in the example above, we do not want to use this supertype, since then the function will only work for floating point numbers.

function supertypes_tree(T::Type, level::Int = 0)
    # code
end

function supertypes_tree(T::Type, level::Int = 0)
    isequal(T, Any) && return
    println(repeat("   ", level), T)
    supertypes_tree(supertype(T), level + 1)
    return
end

Now we can use the supertypes_tree function to get the whole supertype hierarchy for Float64.

julia> supertypes_tree(Float64)
Float64
   AbstractFloat
      Real
         Number

We can check the type hierarchy by the <: operator for comparing types: If T1 <: T2 is true, then T1 is a subtype (or the same type) of T2.

julia> Float64 <: AbstractFloat <: Real <: Number
true

Similarly to the supertype function, there is the subtypes function that returns all subtypes for the given type.

julia> using InteractiveUtils: subtypes

julia> subtypes(Number)
2-element Vector{Any}:
 Complex
 Real

This function suffers from a similar disadvantage as the supertype function: It is impossible to get the whole hierarchy of all subtypes using only this function.

function subtypes_tree(T::Type, level::Int = 0)
    # code
end

function subtypes_tree(T::Type, level::Int = 0)
    println(repeat("   ", level), T)
    subtypes_tree.(subtypes(T), level + 1)
    return
end

Now we can use the subtypes_tree function to get the whole subtypes hierarchy for the Number type.

julia> subtypes_tree(Number)
Number
   Complex
   Real
      AbstractFloat
         BigFloat
         Float16
         Float32
         Float64
      AbstractIrrational
         Irrational
      Integer
         Bool
         Signed
            BigInt
            Int128
            Int16
            Int32
            Int64
            Int8
         Unsigned
            UInt128
            UInt16
            UInt32
            UInt64
            UInt8
      Rational

This tree shows the whole structure of Julia numerical types. If we want to define a function that accepts all numeric types, we should use inputs of type Number. However, many operations are restricted to only real numbers. In such a case, we want to use the Real type instead of Number.

Multiple dispatch

Now we can go back to our example with the product function. The problem with this function is that it is too restrictive because the product of two strings is a legitimate operation that should return their concatenation. We should define a method for strings. To use the proper type, we can use the supertypes_tree function for the String type.

julia> supertypes_tree(String)
String
   AbstractString

We see that the largest supertype for String is AbstractString. This leads to

product(x::AbstractString, y::AbstractString) = x * y
product(x, y) = throw(ArgumentError("product is defined for numbers and strings only."))

We also redefined the original definition of the product function to throw an appropriate error.

julia> product(1, 4.5)
4.5

julia> product("a", "b")
"ab"

julia> product(:a, :b)
ERROR: ArgumentError: product is defined for numbers and strings only.

Sometimes, it may be complicated to guess which method is used for concrete inputs. In such a case, there is a useful macro @which that returns the method that is called for given arguments.

julia> using InteractiveUtils: @which

julia> @which product(1, 4.5)
product(x::Number, y::Number)
     @ Main none:1

julia> @which product("a", :a)
product(x, y)
     @ Main none:1

julia> @which product("a", "b")
product(x::AbstractString, y::AbstractString)
     @ Main none:1

The previous example with the product function shows how methods in Julia works. However, it is a good practice to use type annotation only if we want to have a specialized function or if we want to define a function, which does different things for different types of input arguments.

g(x::Real) = x + 1
g(x::String) = repeat(x, 4)

For example, the g function returns x + 1 if the input x is a real number or repeats four times the input argument if it is a string. Otherwise, it will throw a method error.

julia> g(1.2)
2.2

julia> g("a")
"aaaa"

julia> g(:a)
ERROR: MethodError: no method matching g(::Symbol)

Closest candidates are:
  g(!Matched::Real)
   @ Main none:1
  g(!Matched::String)
   @ Main none:1
[...]

product_new(x, y) = x * y

julia> product(1, 4.5)
4.5

julia> product_new(1, 4.5)
4.5

julia> product("a", "b")
"ab"

julia> product_new("a", "b")
"ab"

julia> product("a", :a)
ERROR: ArgumentError: product is defined for numbers and strings only.

julia> product_new("a", :a)
ERROR: MethodError: no method matching *(::String, ::Symbol)
[...]

We show a simple example when the multiple dispatch is useful.

abstract type Student end

struct Master <: Student
    salary
end

struct Doctoral <: Student
    salary
    exam_mid::Bool
    exam_english::Bool
end

julia> subtypes_tree(Student)
Student
   Doctoral
   Master

s1 = Master(5000)
s2 = Doctoral(30000, 1, 0)

salary_yearly(s::Student) = 12*salary_monthly(s)

salary_monthly(s::Master) = s.salary
salary_monthly(s::Doctoral) = s.salary + s.exam_mid*2000 + s.exam_english*1000

println("The yearly salary is $(salary_yearly(s1)).")
println("The yearly salary is $(salary_yearly(s2)).")

The yearly salary is 60000.
The yearly salary is 384000.

Method ambiguities

It is possible to define a set of function methods with no most specific method applicable to some combinations of arguments.

f(x::Float64, y) = x * y
f(x, y::Float64) = x + y

Here, f has two methods. The first method applies if the first argument is of type Float64, and the second method applies if the second argument is of type Float64.

julia> f(2.0, 3)
6.0

julia> f(2, 3.0)
5.0

Both methods can be used if both arguments are of type Float64. The problem is that neither method is more specific than the other. This results in MethodError.

julia> f(2.0, 3.0)
ERROR: MethodError: f(::Float64, ::Float64) is ambiguous.

Candidates:
  f(x::Float64, y)
    @ Main none:1
  f(x, y::Float64)
    @ Main none:1

Possible fix, define
  f(::Float64, ::Float64)
[...]

We can avoid method ambiguities by specifying an appropriate method for the intersection case.

julia> f(x::Float64, y::Float64) = x - y
f (generic function with 3 methods)

Now f has three methods.

julia> f(2.0, 3.0)
-1.0