Functions

julia is a compiled language
code is never interpreted
everything is compiled using llvm

Defintion



In [1]:

    
function plus2(x)
    x + 2
end









    Out[1]:





plus2 (generic function with 1 method)



In [11]:

    
function plus2(x)
    return x+2
end









    Out[11]:





plus2 (generic function with 1 method)



In [3]:

    
f(x) = x + 2









    Out[3]:





f (generic function with 1 method)

lambda functions



In [12]:

    
f = function (x) x + 2 end









    Out[12]:





(::#1) (generic function with 1 method)



In [13]:

    
f = x -> x +2









    Out[13]:





(::#3) (generic function with 1 method)

Naming



In [5]:

    
f(x) = 2*x









    Out[5]:





f (generic function with 1 method)



In [7]:

    
ϕ(x) = 2*x









    Out[7]:





ϕ (generic function with 1 method)



In [4]:

    
2f(x) = 2*x









    



syntax: "2" is not a valid function argument name

Chaining



In [17]:

    
π |> sin |> exp









    Out[17]:





1.0000000000000002



In [18]:

    
(exp∘sin)(π)









    Out[18]:





1.0000000000000002

Optional Arguments



In [17]:

    
function logly(x,y=10)
    log(y,x)
end









    Out[17]:





logly (generic function with 2 methods)



In [19]:

    
logly(4,2)









    Out[19]:





2.0



In [20]:

    
function logly(y=10,x)
    log(y,x)
end









    



syntax: optional positional arguments must occur at end

Keyword Arguments



In [1]:

    
function logly(x; base=10)
    log(base,x)
end









    Out[1]:





logly (generic function with 1 method)



In [26]:

    
logly(10)









    Out[26]:





1.0



In [27]:

    
logly(8,base=2)









    Out[27]:





3.0

Types of Arguments



In [2]:

    
logly("breakme",base=10)









    



MethodError: no method matching log(::Int64, ::String)
Closest candidates are:
  log(::Real) at math.jl:421
  log(::T<:Number, ::T<:Number) where T<:Number at math.jl:159
  log(::Number, ::Number) at math.jl:187
  ...

Stacktrace:
 [1] (::#kw##logly)(::Array{Any,1}, ::#logly, ::String) at ./<missing>:0



In [7]:

    
workspace()



In [17]:

    
function logly(x::Real;base::Real=10)
    log(base,x)
end









    Out[17]:





logly (generic function with 1 method)



In [18]:

    
logly("foo",10)









    



MethodError: no method matching logly(::String, ::Int64)



In [19]:

    
function logly(x::Real;base::Real=10)::Real
    log(base,x)
end









    Out[19]:





logly (generic function with 1 method)

Argument Passing

immutable types
- basic types e.g. int, float, bool,
- cannot be changed
- passed by value
mutable types
- arrays, user defined types
- can be changed
- passed by refrence
- function may modify them

Immutable



In [11]:

    
function thumbsUp(x)
    x += 1
end









    Out[11]:





thumbsUp (generic function with 1 method)



In [12]:

    
x = 1;
thumbsUp(x)
x









    Out[12]:





1

Mutable



In [16]:

    
function multiThumbsUp(x)
    x[:] += 1
end









    Out[16]:





multiThumbsUp (generic function with 1 method)



In [18]:

    
multiThumbsUp([0 2 2 6])









    Out[18]:





4-element Array{Int64,1}:
 1
 3
 3
 7



In [ ]:

    
# convention: name of functions modifying the input should end with !

Vectorization



In [38]:

    
f(x) = 2*sin(x)-2;



In [41]:

    
f.([1,2,3])









    Out[41]:





3-element Array{Float64,1}:
 -0.317058
 -0.181405
 -1.71776

Multiple Return Values



In [33]:

    
function duplicate(x)
    (x,x+1)
end









    Out[33]:





duplicate (generic function with 1 method)



In [36]:

    
y,z = duplicate(1);z









    Out[36]:





2

Multiple Dispatch



In [42]:

    
function f(x::Float64)
    1
end
function f(x::Int64)
    0
end









    Out[42]:





f (generic function with 3 methods)



In [44]:

    
f(1)









    Out[44]:





0



In [45]:

    
f(1.0)









    Out[45]:





1



In [46]:

    
function f(x::Float64,y::Int64)
    x*y
end
function f(x::Int64,y::Float64)
    x+y
end









    Out[46]:





f (generic function with 5 methods)



In [49]:

    
f(2.0,2)









    Out[49]:





4.0



In [50]:

    
f(2,3.0)









    Out[50]:





5.0



In [51]:

    
f(2.0,2.0)









    



MethodError: no method matching f(::Float64, ::Float64)
Closest candidates are:
  f(::Int64, ::Float64) at In[46]:5
  f(::Float64, ::Int64) at In[46]:2
  f(::Float64) at In[42]:2
  ...

Note

no need to write a function for all possible types
in general types may be omitted
optimized machine code is generate for each combination of input types whenever it is encountered



In [2]:

    
# example
h(x,y) = x*y+2









    Out[2]:





h (generic function with 1 method)



In [3]:

    
@code_native h(1,2)









    



	.section	__TEXT,__text,regular,pure_instructions
Filename: In[1]
	pushq	%rbp
	movq	%rsp, %rbp
Source line: 1
	imulq	%rsi, %rdi
	leaq	2(%rdi), %rax
	popq	%rbp
	retq
	nop



In [4]:

    
@code_native h(1.0,2.0)









    



	.section	__TEXT,__text,regular,pure_instructions
Filename: In[1]
	pushq	%rbp
	movq	%rsp, %rbp
Source line: 1
	mulsd	%xmm1, %xmm0
	movabsq	$4739742496, %rax       ## imm = 0x11A82BB20
	addsd	(%rax), %xmm0
	popq	%rbp
	retq
	nopl	(%rax,%rax)



In [5]:

    
@code_native h(1.0,2)









    



	.section	__TEXT,__text,regular,pure_instructions
Filename: In[1]
	pushq	%rbp
	movq	%rsp, %rbp
Source line: 1
	xorps	%xmm1, %xmm1
	cvtsi2sdq	%rdi, %xmm1
	mulsd	%xmm1, %xmm0
	movabsq	$4739743136, %rax       ## imm = 0x11A82BDA0
	addsd	(%rax), %xmm0
	popq	%rbp
	retq

Important note: The compiled code does NOT perform any type checking!



In [3]:

    
# view type information extracted by compiler
@code_typed h(1.0,2)









    Out[3]:





CodeInfo(:(begin 
        return (Base.add_float)((Base.mul_float)(x, (Base.sitofp)(Float64, y)::Float64)::Float64, (Base.sitofp)(Float64, 2)::Float64)::Float64
    end))=>Float64



In [6]:

    
code_native(h,(Float64,Int32))









    



	.section	__TEXT,__text,regular,pure_instructions
Filename: In[2]
	pushq	%rbp
	movq	%rsp, %rbp
Source line: 2
	xorps	%xmm1, %xmm1
	cvtsi2sdl	%edi, %xmm1
	mulsd	%xmm1, %xmm0
	movabsq	$4911194656, %rax       ## imm = 0x124BAE220
	addsd	(%rax), %xmm0
	popq	%rbp
	retq
	nop



In [10]:

    
code_native(h,(Int32,Int64))









    



	.section	__TEXT,__text,regular,pure_instructions
Filename: In[2]
	pushq	%rbp
	movq	%rsp, %rbp
Source line: 2
	movslq	%edi, %rax
	imulq	%rsi, %rax
	addq	$2, %rax
	popq	%rbp
	retq
	nopw	%cs:(%rax,%rax)



In [11]:

    
code_llvm(h,(Int32,Int64))









    



define i64 @julia_h_61157(i32, i64) #0 !dbg !5 {
top:
  %2 = sext i32 %0 to i64
  %3 = mul i64 %2, %1
  %4 = add i64 %3, 2
  ret i64 %4
}

Dispatch on Values



In [70]:

    
fv(x,::Val{true})  = x+2
fv(x,::Val{false}) = x-5









    Out[70]:





fv (generic function with 2 methods)



In [74]:

    
fv(1,Val(true))









    Out[74]:





3



In [76]:

    
fv(1,Val(false))









    Out[76]:





-4

Exercises

Task 1

Write a function implementing the following map $$ \begin{align} f \colon &\mathbb{R} \to \mathbb{R}\\ & x \mapsto x^2 + \frac{\sin(x+13)}{x} \end{align} $$

Task 2

Create a function $g$ mapping a function $f: \mathbb{R} \to \mathbb{R}$ to the function $h(x) = |f(x)|$.

With f being the function of task 1, you should get g(f)(-1) = 0.46342708199956506

Task 3

One options to calculate the inverse of a real number $x$ is given by the recurrence relation.

$$ y_{k+1} = 2y_k - y_k^2 x $$

with sattisfies $y_{k+1} \to \frac{1}{x}$ for $k \to \infty$.

Create a function newtonInv calculating the inverse of a given value $x$ using this iterations. The arguments of the function should be

x (real number to be inverted)
y0 (starting value for iteration, optional, default: 0.1)
iterations (number of iterations to performed, optional, default: 10, keyword option)

Bonus Task: Modify the function such that it has an option "relTol" specifying a relative tolerance (default: 1e-8). Instead of performing a fixed amount of iterations, the modified function should iterate until the solution satisfies the given tolernace.

Task 4

Write a function crossSwap!(x,y) which exchanges x[1] with y[end] and x[end] with y[1].

Task 5

Create a function f(x,+/-1) mapping the interval [-1,1] to right(+1) respectively left part of the unit circle in the complex plane.

$$ \begin{align*} f(x, 1) = e^{ \pi i \frac{x}{2}}\\ f(x,-1) = e^{ \pi i (\frac{x}{2} + 1)} \end{align*} $$