6. Generic Programming

I find OOP methodologically wrong. It starts with classes. It is as if mathematicians would start with axioms. You do not start with axioms - you start with proofs. Only when you have found a bunch of related proofs, can you come up with axioms. You end with axioms. The same thing is true in programming: you have to start with interesting algorithms. Only when you understand them well, can you come up with an interface that will let them work. – Alexander Stepanov

6.1. Overview

In this lecture we delve more deeply into the structure of Julia, and in particular into

  • abstract and concrete types

  • the type tree

  • designing and using generic interfaces

  • the role of generic interfaces in Julia performance

Understanding them will help you

  • form a “mental model” of the Julia language

  • design code that matches the “white-board” mathematics

  • create code that can use (and be used by) a variety of other packages

  • write “well organized” Julia code that’s easy to read, modify, maintain and debug

  • improve the speed at which your code runs

(Special thank you to Jeffrey Sarnoff)

6.1.1. Generic Programming is an Attitude

From Mathematics to Generic Programming [SR14]

Generic programming is an approach to programming that focuses on designing algorithms and data structures so that they work in the most general setting without loss of efficiency… Generic programming is more of an attitude toward programming than a particular set of tools.

In that sense, it is important to think of generic programming as an interactive approach to uncover generality without compromising performance rather than as a set of rules.

As we will see, the core approach is to treat data structures and algorithms as loosely coupled, and is in direct contrast to the is-a approach of object-oriented programming.

This lecture has the dual role of giving an introduction into the design of generic algorithms and describing how Julia helps make that possible.

using LinearAlgebra, Statistics
using Distributions, Plots, QuadGK, Polynomials, Interpolations

6.2. Exploring Type Trees

The connection between data structures and the algorithms which operate on them is handled by the type system.

Concrete types (i.e., Float64 or Array{Float64, 2}) are the data structures we apply an algorithm to, and the abstract types (e.g. the corresponding Number and AbstractArray) provide the mapping between a set of related data structures and algorithms.

using Distributions
x = 1
y = Normal()
z = "foo"
@show x, y, z
@show typeof(x), typeof(y), typeof(z)
@show supertype(typeof(x))

# pipe operator, |>, is is equivalent
@show typeof(x) |> supertype
@show supertype(typeof(y))
@show typeof(z) |> supertype
@show typeof(x) <: Any;
(x, y, z) = (1, Normal{Float64}(μ=0.0, σ=1.0), "foo")
(typeof(x), typeof(y), typeof(z)) = (Int64, Normal{Float64}, String)
supertype(typeof(x)) = Signed
typeof(x) |> supertype = Signed
supertype(typeof(y)) = Distribution{Univariate, Continuous}
typeof(z) |> supertype = AbstractString
typeof(x) <: Any = true

Beyond the typeof and supertype functions, a few other useful tools for analyzing the tree of types are discussed in the introduction to types lecture

using Base: show_supertypes # import the function from the `Base` package

Int64 <: Signed <: Integer <: Real <: Number <: Any
3-element Vector{Any}:

Using the subtypes function, we can write an algorithm to traverse the type tree below any time t – with the confidence that all types support subtypes

#  from https://github.com/JuliaLang/julia/issues/24741
function subtypetree(t, level=1, indent=4)
        if level == 1
        for s in subtypes(t)
            println(join(fill(" ", level * indent)) * string(s))  # print type
            subtypetree(s, level+1, indent)  # recursively print the next type, indenting
subtypetree (generic function with 3 methods)

Applying this to Number, we see the tree of types currently loaded

subtypetree(Number) # warning: do not use this function on ``Any``!

For the most part, all of the “leaves” will be concrete types.

6.2.1. Any

At the root of all types is Any

There are a few functions that work in the “most generalized” context: usable with anything that you can construct or access from other packages.

We have already called typeof, show and supertype – which will apply to a custom struct type since MyType <: Any

# custom type
struct MyType

myval = MyType(2.0)
@show myval
@show typeof(myval)
@show supertype(typeof(myval))
@show typeof(myval) <: Any;
myval = MyType(2.0)
typeof(myval) = MyType
supertype(typeof(myval)) = Any
typeof(myval) <: Any = true

Here we see another example of generic programming: every type <: Any supports the @show macro, which in turn, relies on the show function.

The @show macro (1) prints the expression as a string; (2) evaluates the expression; and (3) calls the show function on the returned values.

To see this with built-in types

x = [1, 2]
[1, 2]

The Any type is useful, because it provides a fall-back implementation for a variety of functions.

Hence, calling show on our custom type dispatches to the fallback function

myval = MyType(2.0)

The default fallback implementation used by Julia would be roughly equivalent to

function show(io::IO, x)
    str = string(x)
    print(io, str)

To implement a specialized implementation of the show function for our type, rather than using this fallback

import Base.show  # to extend an existing function

function show(io::IO, x::MyType)
    str = "(MyType.a = $(x.a))"  # custom display
    print(io, str)
show(myval)  # it creates an IO value first and then calls the above show
(MyType.a = 2.0)

At that point, we can use the @show macro, which in turn calls show

@show myval;
myval = (MyType.a = 2.0)

Here we see another example of generic programming: any type with a show function works with @show.

Layering of functions (e.g. @show calling show) with a “fallback” implementation makes it possible for new types to be designed and only specialized where necessary.

6.2.2. Unlearning Object Oriented (OO) Programming (Advanced)

See Types for more on OO vs. generic types.

If you have never used programming languages such as C++, Java, and Python, then the type hierarchies above may seem unfamiliar and abstract.

In that case, keep an open mind that this discussion of abstract concepts will have practical consequences, but there is no need to read this section.

Otherwise, if you have used object-oriented programming (OOP) in those languages, then some of the concepts in these lecture notes will appear familiar.

Don’t be fooled!

The superficial similarity can lead to misuse: types are not classes with poor encapsulation, and methods are not the equivalent to member functions with the order of arguments swapped.

In particular, previous OO knowledge often leads people to write Julia code such as

# BAD! Replicating an OO design in Julia
mutable struct MyModel

    MyModel(a, b) = new(a, b, 0.0) # an inner constructor

function myalgorithm!(m::MyModel, x)
    m.algorithmcalculation = m.a + m.b + x # some algorithm

function set_a!(m::MyModel, a)
    m.a = a

m = MyModel(2.0, 3.0)
x = 0.1
set_a!(m, 4.1)
myalgorithm!(m, x)
@show m.algorithmcalculation;
m.algorithmcalculation = 7.199999999999999

You may think to yourself that the above code is similar to OO, except that you

  • reverse the first argument, i.e., myalgorithm!(m, x) instead of the object-oriented m.myalgorithm!(x)

  • cannot control encapsulation of the fields a, b, but you can add getter/setters like set_a

  • do not have concrete inheritance

While this sort of programming is possible, it is (verbosely) missing the point of Julia and the power of generic programming.

When programming in Julia

  • there is no encapsulation and most custom types you create will be immutable.

  • Polymorphism is achieved without anything resembling OOP inheritance.

  • Abstraction is implemented by keeping the data and algorithms that operate on them as orthogonal as possible – in direct contrast to OOP’s association of algorithms and methods directly with a type in a tree.

  • The supertypes in Julia are simply used for selecting which specialized algorithm to use (i.e., part of generic polymorphism) and have nothing to do with OO inheritance.

  • The looseness that accompanies keeping algorithms and data structures as orthogonal as possible makes it easier to discover commonality in the design. Iterative Design of Abstractions

As its essence, the design of generic software is that you will start with creating algorithms which are largely orthogonal to concrete types.

In the process, you will discover commonality which leads to abstract types with informally defined functions operating on them.

Given the abstract types and commonality, you then refine the algorithms as they are more limited or more general than you initially thought.

This approach is in direct contrast to object-oriented design and analysis (OOAD).

With that, where you specify a taxonomies of types, add operations to those types, and then move down to various levels of specialization (where algorithms are embedded at points within the taxonomy, and potentially specialized with inheritance).

In the examples that follow, we will show for exposition the hierarchy of types and the algorithms operating on them, but the reality is that the algorithms are often designed first, and the abstact types came later.

6.3. Distributions

First, consider working with “distributions”.

Algorithms using distributions might (1) draw random numbers for Monte-Carlo methods; and (2) calculate the pdf or cdf – if it is defined.

The process of using concrete distributions in these sorts of applications led to the creation of the Distributions.jl package.

Let’s examine the tree of types for a Normal distribution

using Distributions
d1 = Normal(1.0, 2.0) # an example type to explore
@show d1
d1 = Normal{Float64}(μ=1.0, σ=2.0)
Normal{Float64} <: Distribution{Univariate, Continuous} <: Sampleable{Univariate, Continuous} <: Any

The Sampleable{Univariate,Continuous} type has a limited number of functions, chiefly the ability to draw a random number

@show rand(d1);
rand(d1) = 2.9725381243370244

The purpose of that abstract type is to provide an interface for drawing from a variety of distributions, some of which may not have a well-defined predefined pdf.

If you were writing a function to simulate a stochastic process with arbitrary iid shocks, where you did not need to assume an existing pdf etc., this is a natural candidate.

For example, to simulate \(x_{t+1} = a x_t + b \epsilon_{t+1}\) where \(\epsilon \sim D\) for some \(D\), which allows drawing random values.

function simulateprocess(x₀; a = 1.0, b = 1.0, N = 5, d::Sampleable{Univariate,Continuous})
    x = zeros(typeof(x₀), N+1) # preallocate vector, careful on the type
    x[1] = x₀
    for t in 2:N+1
        x[t] = a * x[t-1] + b * rand(d) # draw
    return x
@show simulateprocess(0.0, d=Normal(0.2, 2.0));
simulateprocess(0.0, d = Normal(0.2, 2.0)) = [0.0, -1.2542271808161505, -0.6967427991169426, -0.2758021851586304, -3.5515196860936133, -3.2271103839727138]

The Sampleable{Univariate,Continuous} and, especially, the Sampleable{Multivariate,Continuous} abstract types are useful generic interfaces for Monte-Carlo and Bayesian methods.

Moving down the tree, the Distributions{Univariate, Continuous} abstract type has other functions we can use for generic algorithms operating on distributions.

These match the mathematics, such as pdf, cdf, quantile, support, minimum, maximum, etc.

d1 = Normal(1.0, 2.0)
d2 = Exponential(0.1)
@show d1
@show d2
@show supertype(typeof(d1))
@show supertype(typeof(d2))

@show pdf(d1, 0.1)
@show pdf(d2, 0.1)
@show cdf(d1, 0.1)
@show cdf(d2, 0.1)
@show support(d1)
@show support(d2)
@show minimum(d1)
@show minimum(d2)
@show maximum(d1)
@show maximum(d2);
d1 = Normal{Float64}(μ=1.0, σ=2.0)
d2 = Exponential{Float64}(θ=0.1)
supertype(typeof(d1)) = Distribution{Univariate, Continuous}
supertype(typeof(d2)) = Distribution{Univariate, Continuous}
pdf(d1, 0.1) = 0.18026348123082397
pdf(d2, 0.1) = 3.6787944117144233
cdf(d1, 0.1) = 0.32635522028792
cdf(d2, 0.1) = 0.6321205588285577
support(d1) = RealInterval{Float64}(-Inf, Inf)
support(d2) = RealInterval{Float64}(0.0, Inf)
minimum(d1) = -Inf
minimum(d2) = 0.0
maximum(d1) = Inf
maximum(d2) = Inf

You could create your own Distributions{Univariate, Continuous} type by implementing those functions – as is described in the documentation.

If you fulfill all of the conditions of a particular interface, you can use algorithms from the present, past, and future that are written for the abstract Distributions{Univariate, Continuous} type.

As an example, consider the StatsPlots package

using StatsPlots
d = Normal(2.0, 1.0)
plot(d) # note no other arguments!

Calling plot on any subtype of Distributions{Univariate, Continuous} displays the pdf and uses minimum and maximum to determine the range.

Let’s create our own distribution type

struct OurTruncatedExponential <: Distribution{Univariate,Continuous}
Distributions.pdf(d::OurTruncatedExponential, x::Real) = d.α *exp(-d.α * x)/exp(-d.α * d.xmax)
Distributions.minimum(d::OurTruncatedExponential) = 0
Distributions.maximum(d::OurTruncatedExponential) = d.xmax
# ... more to have a complete type

To demonstrate this

d = OurTruncatedExponential(1.0,2.0)
@show minimum(d), maximum(d)
@show support(d) # why does this work?
(minimum(d), maximum(d)) = (0, 2.0)
support(d) = 
RealInterval{Float64}(0.0, 2.0)
RealInterval{Float64}(0.0, 2.0)

Curiously, you will note that the support function works, even though we did not provide one.

This is another example of the power of multiple dispatch and generic programming.

In the background, the Distributions.jl package has something like the following implemented

    Distributions.support(d::Distribution) = RealInterval(minimum(d), maximum(d))

Since OurTruncatedExponential <: Distribution, and we implemented minimum and maximum, calls to support get this implementation as a fallback.

These functions are enough to use the StatsPlots.jl package

plot(d) # uses the generic code!

A few things to point out

  • Even if it worked for StatsPlots, our implementation is incomplete, as we haven’t fulfilled all of the requirements of a Distribution.

  • We also did not implement the rand function, which means we are breaking the implicit contract of the Sampleable abstract type.

  • It turns out that there is a better way to do this precise thing already built into Distributions.

d = Truncated(Exponential(0.1), 0.0, 2.0)
@show typeof(d)
typeof(d) = Truncated{Exponential{Float64}, Continuous, Float64}

This is the power of generic programming in general, and Julia in particular: you can combine and compose completely separate packages and code, as long as there is an agreement on abstract types and functions.

6.4. Numbers and Algebraic Structures

Define two binary functions, \(+\) and \(\cdot\), called addition and multiplication – although the operators can be applied to data structures much more abstract than a Real.

In mathematics, a ring is a set with associated additive and multiplicative operators where

  • the additive operator is associative and commutative

  • the multiplicative operator is associative and distributive with respect to the additive operator

  • there is an additive identity element, denoted \(0\), such that \(a + 0 = a\) for any \(a\) in the set

  • there is an additive inverse of each element, denoted \(-a\), such that \(a + (-a) = 0\)

  • there is a multiplicative identity element, denoted \(1\), such that \(a \cdot 1 = a = 1 \cdot a\)

  • a total or partial ordering is not required (i.e., there does not need to be any meaningful \(<\) operator defined)

  • a multiplicative inverse is not required

While this skips over some parts of the mathematical definition, this algebraic structure provides motivation for the abstract Number type in Julia

  • Remark: We use the term “motivation” because they are not formally connected and the mapping is imperfect.

  • The main difficulty when dealing with numbers that can be concretely created on a computer is that the requirement that the operators are closed in the set are difficult to ensure (e.g. floating points have finite numbers of bits of information).

Let typeof(a) = typeof(b) = T <: Number, then under an informal definition of the generic interface for Number, the following must be defined

  • the additive operator: a + b

  • the multiplicative operator: a * b

  • an additive inverse operator: -a

  • an inverse operation for addition a - b = a + (-b)

  • an additive identity: zero(T) or zero(a) for convenience

  • a multiplicative identity: one(T) or one(a) for convenience

The core of generic programming is that, given the knowledge that a value is of type Number, we can design algorithms using any of these functions and not concern ourselves with the particular concrete type.

Furthermore, that generality in designing algorithms comes with no compromises on performance compared to carefully designed algorithms written for that particular type.

To demonstrate this for a complex number, where Complex{Float64} <: Number

a = 1.0 + 1.0im
b = 0.0 + 2.0im
@show typeof(a)
@show typeof(a) <: Number
@show a + b
@show a * b
@show -a
@show a - b
@show zero(a)
@show one(a);
typeof(a) = ComplexF64
typeof(a) <: Number = true
a + b = 1.0 + 3.0im
a * b = -2.0 + 2.0im
-a = -1.0 - 1.0im
a - b = 1.0 - 1.0im
zero(a) = 0.0 + 0.0im
one(a) = 1.0 + 0.0im

And for an arbitrary precision integer where BigInt <: Number (i.e., a different type than the Int64 you have worked with, but nevertheless a Number)

a = BigInt(10)
b = BigInt(4)
@show typeof(a)
@show typeof(a) <: Number
@show a + b
@show a * b
@show -a
@show a - b
@show zero(a)
@show one(a);
typeof(a) = BigInt
typeof(a) <: Number = true
a + b = 14
a * b = 40
-a = -10
a - b = 6
zero(a) = 0
one(a) = 1

6.4.1. Complex Numbers and Composition of Generic Functions

This allows us to showcase further how different generic packages compose – even if they are only loosely coupled through agreement on common generic interfaces.

The Complex numbers require some sort of storage for their underlying real and imaginary parts, which is itself left generic.

This data structure is defined to work with any type <: Number, and is parameterized (e.g. Complex{Float64} is a complex number storing the imaginary and real parts in Float64)

x = 4.0 + 1.0im
@show x, typeof(x)

xbig = BigFloat(4.0) + 1.0im
@show xbig, typeof(xbig);
(x, typeof(x)) = (4.0 + 1.0im, ComplexF64)
(xbig, typeof(xbig)) = 
(4.0 + 1.0im, Complex{BigFloat})

The implementation of the Complex numbers use the underlying operations of storage type, so as long as +, * etc. are defined – as they should be for any Number – the complex operation can be defined

@which +(x,x)
+(z::Complex, w::Complex) in Base at complex.jl:288

Following that link, the implementation of + for complex numbers is

+(z::Complex, w::Complex) = Complex(real(z) + real(w), imag(z) + imag(w))

real(z) and imag(z) returns the associated components of the complex number in the underlying storage type (e.g. Float64 or BigFloat).

The rest of the function has been carefully written to use functions defined for any Number (e.g. + but not <, since it is not part of the generic number interface).

To follow another example , look at the implementation of abs specialized for complex numbers

@which abs(x)
abs(z::Complex) in Base at complex.jl:277

The source is

abs(z::Complex)  = hypot(real(z), imag(z))

In this case, if you look at the generic function to get the hypotenuse, hypot, you will see that it has the function signature hypot(x::T, y::T) where T<:Number, and hence works for any Number.

That function, in turn, relies on the underlying abs for the type of real(z).

This would dispatch to the appropriate abs for the type

@which abs(1.0)
abs(x::Union{Float16, Float32, Float64}) in Base at float.jl:486
@which abs(BigFloat(1.0))
abs(x::Real) in Base at number.jl:164

With implementations

abs(x::Real) = ifelse(signbit(x), -x, x)
abs(x::Float64) = abs_float(x)

For a Real number (which we will discuss in the next section) the fallback implementation calls a function signbit to determine if it should flip the sign of the number.

The specialized version for Float64 <: Real calls a function called abs_float – which turns out to be a specialized implementation at the compiler level.

While we have not completely dissected the tree of function calls, at the bottom of the tree you will end at the most optimized version of the function for the underlying datatype.

Hopefully this showcases the power of generic programming: with a well-designed set of abstract types and functions, the code can both be highly general and composable and still use the most efficient implementation possible.

6.5. Reals and Algebraic Structures

Thinking back to the mathematical motivation, a field is a ring with a few additional properties, among them

  • a multiplicative inverse: \(a^{-1}\)

  • an inverse operation for multiplication: \(a / b = a \cdot b^{-1}\)

Furthermore, we will make it a total ordered field with

  • a total ordering binary operator: \(a < b\)

This type gives some motivation for the operations and properties of the Real type.

Of course, Complex{Float64} <: Number but not Real – since the ordering is not defined for complex numbers in mathematics.

These operations are implemented in any subtype of Real through

  • the multiplicative inverse: inv(a)

  • the multiplicative inverse operation: a / b = a * inv(b)

  • an ordering a < b

We have already shown these with the Float64 and BigFloat.

To show this for the Rational number type, where a // b constructs a rational number \(\frac{a}{b}\)

a = 1 // 10
b = 4 // 6
@show typeof(a)
@show typeof(a) <: Number
@show typeof(a) <: Real
@show inv(a)
@show a / b
@show a < b;
typeof(a) = Rational{Int64}
typeof(a) <: Number = true
typeof(a) <: Real = true
inv(a) = 10//1
a / b = 3//20
a < b = true

Remark: Here we see where and how the precise connection to the mathematics for number types breaks down for practical reasons, in particular

  • Integer types (i.e., Int64 <: Integer) do not have a a multiplicative inverse with closure in the set.

  • However, it is necessary in practice for integer division to be defined, and return back a member of the Real’s.

  • This is called type promotion, where a type can be converted to another to ensure an operation is possible by direct conversion between types (i.e., it can be independent of the type hierarchy).

Do not think of the break in the connection between the underlying algebraic structures and the code as a failure of the language or design.

Rather, the underlying algorithms for use on a computer do not perfectly fit the algebraic structures in this instance.

Moving further down the tree of types provides more operations more directly tied to the computational implementation than abstract algebra.

For example, floating point numbers have a machine precision, below which numbers become indistinguishable due to lack of sufficient “bits” of information

@show Float64 <: AbstractFloat
@show BigFloat <: AbstractFloat
@show eps(Float64)
@show eps(BigFloat);
Float64 <: AbstractFloat = true
BigFloat <: AbstractFloat = true
eps(Float64) = 2.220446049250313e-16
eps(BigFloat) = 1.727233711018888925077270372560079914223200072887256277004740694033718360632485e-77

The isless function also has multiple methods.

First let’s try with integers

@which isless(1, 2)
isless(x::Real, y::Real) in Base at operators.jl:434

As we saw previously, the Real data type is an abstract type, and encompasses both floats and integers.

If we go to the provided link in the source, we see the entirety of the function is

isless(x::Real, y::Real) = x<y

That is, for any values where typeof(x) <: Real and typeof(y) <: Real, the definition relies on <.

We know that < is defined for the types because it is part of the informal interface for the Real abstract type.

Note that this is not defined for Number because not all Number types have the < ordering operator defined (e.g. Complex).

In order to generate fast code, the implementation details may define specialized versions of these operations.

isless(1.0, 2.0)  # applied to two floats
@which isless(1.0, 2.0)
isless(a::T, b::T) where T<:Union{Float16, Float32, Float64} in Base at float.jl:424

Note that the reason Float64 <: Real calls this implementation rather than the one given above, is that Float64 <: Real, and Julia chooses the most specialized implementation for each function.

The specialized implementations are often more subtle than you may realize due to floating point arithmetic, underflow, etc.

6.6. Functions, and Function-Like Types

Another common example of the separation between data structures and algorithms is the use of functions.

Syntactically, a univariate “function” is any f that can call an argument x as f(x).

For example, we can use a standard function

using QuadGK
f(x) = x^2
@show quadgk(f, 0.0, 1.0)  # integral

function plotfunctions(f)
    intf(x) = quadgk(f, 0.0, x)[1]  # int_0^x f(x) dx

    x = 0:0.1:1.0
    f_x = f.(x)
    plot(x, f_x, label="f")
    plot!(x, intf.(x), label="int_f")
plotfunctions(f)  # call with our f
quadgk(f, 0.0, 1.0) = (0.3333333333333333, 5.551115123125783e-17)

Of course, univariate polynomials are another type of univariate function

using Polynomials
p = Polynomial([2, -5, 2], :x)  # :x just gives a symbol for display
@show p
@show p(1.0) # call like a function

plotfunctions(p)  # same generic function
p = Polynomial(2 - 5*x + 2*x^2)
p(1.0) = -1.0

Similarly, the result of interpolating data is also a function

using Interpolations
x = 0.0:0.2:1.0
f(x) = x^2
f_int = LinearInterpolation(x, f.(x))  # interpolates the coarse grid
@show f_int(1.0)  # call like a function

plotfunctions(f_int)  # same generic function
f_int(1.0) = 1.0

Note that the same generic plotfunctions could use any variable passed to it that “looks” like a function, i.e., can call f(x).

This approach to design with types – generic, but without any specific type declarations – is called duck typing.

If you need to make an existing type callable, see Function Like Objects.

6.7. Limitations of Dispatching on Abstract Types

You will notice that types in Julia represent a tree with Any at the root.

The tree structure has worked well for the above examples, but it doesn’t allow us to associate multiple categorizations of types.

For example, a semi-group type would be useful for a writing generic code (e.g. continuous-time solutions for ODEs and matrix-free methods), but cannot be implemented rigorously since the Matrix type is a semi-group as well as an AbstractArray, but not all semi-groups are AbstractArray s.

The main way to implement this in a generic language is with a design approach called “traits”.

6.8. Exercises

6.8.1. Exercise 1a

In a previous exercise, we discussed the trapezoidal rule for numerical integration.

To summarize, the vector

\[ \int_{\underline{x}}^{\bar{x}} f(x) \, dx \approx \omega \cdot \vec{f} \]

where \(\vec{f} \equiv \begin{bmatrix} f(x_1) & \ldots & f(x_N) \end{bmatrix}\in R^N\) and, for a uniform grid spacing of \(\Delta\),

\[ \omega \equiv \Delta \begin{bmatrix} \frac{1}{2} & 1 & \ldots & 1 & \frac{1}{2}\end{bmatrix} \in R^N \]

The quadrature rule can be implemented easily as

using LinearAlgebra
function trap_weights(x)
    return step(x) * [0.5; ones(length(x) - 2); 0.5]
x = range(0.0, 1.0, length = 100)
ω = trap_weights(x)
f(x) = x^2
dot(f.(x), ω)

However, in this case the creation of the ω temporary is inefficient as there are no reasons to allocate an entire vector just to iterate through it with the dot. Instead, create an iterable by following the interface definition for Iteration, and implement the modified trap_weights and integration.

Hint: create a type such as

struct UniformTrapezoidal

and then implement the function Base.iterate(S::UniformTrapezoidal, state=1).

6.8.2. Exercise 1b (Advanced)

Make the UniformTrapezoidal type operate as an array with interface definition for AbstractArray. With this, you should be able it go ω[2] or length(ω) to access the quadrature weights.

6.8.3. Exercise 2 (Advanced)

Implement the same features as Exercise 1a and 1b, but for the non-uniform trapezoidal rule.