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107 μs

Computational Thinking

Math from Computation

Math with Computation

Alan Edelman

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2.1 ms

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@bind prof_size Slider(0 : 0.01 : 2)

278 ms

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81.2 μs

n = 3

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18.5 ms

Any

🌕🌑🌑
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Any[prettylist(combinations(1:n, k) .|> onoff) for k in 1:n]

167 ms

More options

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165 μs

Any

[1]
[2]
[3]

[1, 2]
[1, 3]
[2, 3]

[1, 2, 3]

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Any[prettylist(combinations(1:n, k)) for k in 1:n]

45.8 μs

let

bar(

[length(combinations(1:n, k)) for k in 1:n],

dpi=96, size=(400,200), leg=false

)

end

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4.2 s

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5.7 ms

let

bar(

[length(combinations(1:n_again, k)) for k in 1:n_again],

dpi=96, size=(600,300), leg=false

)

end

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847 μs

k = 1

md"k = $(@bind k Slider(1:n, show_value=true))"

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12.3 ms

[1]
[2]
[3]

combinations(1:n, k) |> prettylist

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4.4 ms

prettylist (generic function with 1 method)

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815 μs

onoff (generic function with 1 method)

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1.0 ms

begin

using Pkg

Pkg.activate(mktempdir())

Pkg.add([

Pkg.PackageSpec(name="PlutoUI", version="0.6.7-0.6"),

Pkg.PackageSpec(name="Plots", version="1.10.1-1"),

Pkg.PackageSpec(name="Combinatorics"),

])

using Plots

gr()

using PlutoUI

using Combinatorics

end

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  Activating new project at `/tmp/jl_AD35XV`
    Updating registry at `~/.julia/registries/General.toml`
   Resolving package versions...
   Installed Combinatorics ─ v1.0.3
    Updating `/tmp/jl_AD35XV/Project.toml`
  [861a8166] + Combinatorics v1.0.3
  [91a5bcdd] + Plots v1.40.14
  [7f904dfe] + PlutoUI v0.6.11
    Updating `/tmp/jl_AD35XV/Manifest.toml`
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  [8bb1440f] + DelimitedFiles
  [f43a241f] + Downloads
  [7b1f6079] + FileWatching
  [b77e0a4c] + InteractiveUtils
  [b27032c2] + LibCURL
  [76f85450] + LibGit2
  [8f399da3] + Libdl
  [37e2e46d] + LinearAlgebra
  [56ddb016] + Logging
  [d6f4376e] + Markdown
  [a63ad114] + Mmap
  [ca575930] + NetworkOptions
  [44cfe95a] + Pkg
  [de0858da] + Printf
  [3fa0cd96] + REPL
  [9a3f8284] + Random
  [ea8e919c] + SHA
  [9e88b42a] + Serialization
  [6462fe0b] + Sockets
  [2f01184e] + SparseArrays
  [10745b16] + Statistics
  [fa267f1f] + TOML
  [a4e569a6] + Tar
  [8dfed614] + Test
  [cf7118a7] + UUIDs
  [4ec0a83e] + Unicode
  [e66e0078] + CompilerSupportLibraries_jll
  [deac9b47] + LibCURL_jll
  [29816b5a] + LibSSH2_jll
  [c8ffd9c3] + MbedTLS_jll
  [14a3606d] + MozillaCACerts_jll
  [4536629a] + OpenBLAS_jll
  [05823500] + OpenLibm_jll
  [efcefdf7] + PCRE2_jll
  [83775a58] + Zlib_jll
  [8e850b90] + libblastrampoline_jll
  [8e850ede] + nghttp2_jll
  [3f19e933] + p7zip_jll
Precompiling project...
  ✓ Combinatorics
  1 dependency successfully precompiled in 1 seconds (150 already precompiled)

8.0 s

First example

asdfasdf

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206 μs

Exercise 1: Calculus without calculus

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251 μs

Before we jump in to simulating the SIR equations, let's experiment with a simple 1D function. In calculus, we learn techniques for differentiating and integrating symbolic equations, e.g. $\frac{d}{d x} x^{n} = n x^{n - 1}$ . But in real applications, it is often impossible to apply these techniques, either because the problem is too complicated to solve symbolically, or because our problem has no symbolic expression, like when working with experimental results.

Instead, we use ✨ computers ✨ to approximate derivatives and integrals. Instead of applying rules to symbolic expressions, we use much simpler strategies that only use the output values of our function.

As a first example, we will approximate the derivative of a function. Our method is inspired by the analytical definition of the derivative!

$f^{'} (a) := lim_{h \to 0} \frac{f (a + h) - f (a)}{h} .$

The finite difference method simply fixes a small value for $h$ , say $h = 10^{- 3}$ , and then approximates the derivative as:

$f^{'} (a) ≃ \frac{f (a + h) - f (a)}{h} .$

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516 μs

Exercise 1.1 - tangent line

👉 Write a function finite_difference_slope that takes a function f and numbers a and h. It returns the slope $f^{'} (a)$ , approximated using the finite difference formula above.

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285 μs

finite_difference_slope (generic function with 2 methods)

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730 μs

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8.5 μs

0.2

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3.9 ms

Got it!

Good job!

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2.4 ms

👉 Write a function tangent_line that takes the same arguments f, a and g, but it returns a function. This function ( $R \to R$ ) is the tangent line with slope $f^{'} (a)$ (computed using finite_difference_slope) that passes through $(a, f (a))$ .

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322 μs

Hint

Remember that functions are objects! For example, here is a function that returns the square root function:

function the_square_root_function()
	f = x -> sqrt(x)
	return f
end

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42.2 ms

tangent_line (generic function with 1 method)

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1.0 ms

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8.4 μs

Got it!

Awesome!

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3.2 ms

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12.1 ms

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912 ms

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573 μs

The slider below controls $h$ using a log scale. In the (mathematical) definition of the derivative, we take $lim_{h \to 0}$ . This corresponds to moving the slider to the left.

Notice that, as you decrease $h$ , the tangent line gets more accurate, but what happens if you make $h$ too small?

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315 μs

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15.7 ms

0.31622776601683794

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14.6 μs

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14.8 μs

Exercise 1.2 - antiderivative

In the finite differences method, we approximated the derivative of a function:

$f^{'} (a) ≃ \frac{f (a + h) - f (a)}{h}$

We can do something very similar to approximate the 'antiderivate' of a function. Finding the antiderivative means that we use the slope $f^{'}$ to compute $f$ numerically!

This antiderivative problem is illustrated below. The only information that we have is the slope at any point $a \in R$ , and we have one initial value, $f (1)$ .

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651 μs

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618 μs

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164 ms

Using only this information, we want to reconstruct $f$ .

By rearranging the equation above, we get the Euler method:

$f (a + h) ≃ h f^{'} (a) + f (a)$

Using this formula, we only need to know the value $f (a)$ and the slope $f^{'} (a)$ of a function at $a$ to get the value at $a + h$ . Doing this repeatedly can give us the value at $a + 2 h$ , at $a + 3 h$ , etc., all from one initial value $f (a)$ .

👉 Write a function euler_integrate_step that applies this formula to a known function $f^{'}$ at $a$ , with step size $h$ and the initial value $f (a)$ . It returns the next value, $f (a + h)$ .

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613 μs

euler_integrate_step (generic function with 1 method)

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494 μs

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8.8 μs

Got it!

Keep it up!

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27.7 ms

👉 Write a function euler_integrate that takes takes a known function $f^{'}$ , the initial value $f (a)$ and a range T with a == first(T) and h == step(T). It applies the function euler_integrate_step repeatedly, once per entry in T, to produce the sequence of values $f (a + h)$ , $f (a + 2 h)$ , etc.

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248 μs

euler_integrate (generic function with 1 method)

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1.2 ms

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8.9 μs

Let's try it out on $f^{'} (x) = 3 x^{2}$ and T ranging from $0$ to $10$ .

We already know the analytical solution $f (x) = x^{3}$ , so the result should be an array going from (approximately) 0.0 to 1000.0.

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265 μs

Float64

0.003

0.015

0.042

0.09

0.165

0.273

0.42

0.612

0.855

1.155

1.518

1.95

2.457

3.045

3.72

4.488

5.355

6.327

7.41

8.61

791.43

817.377

843.885

870.96

898.608

926.835

955.647

985.05

100

1015.05

101

1045.65

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84.5 ms

Got it!

Good job!

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10.7 ms

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625 μs

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426 ms

You see that our numerical antiderivate is not very accurate, but we can get a smaller error by choosing a smaller step size. Try it out!

There are also alternative integration methods that are more accurate with the same step size. Some methods also use the second derivative, other methods use multiple steps at once, etc.! This is the study of Numerical Methods.

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237 μs

Some cool things

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201 μs

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66.0 μs

asdfasdf

md"asdfasdf"

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168 μs

Function library

Just some helper functions used in the notebook.

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210 μs

hint (generic function with 1 method)

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464 μs

almost (generic function with 1 method)

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450 μs

still_missing (generic function with 2 methods)

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801 μs

keep_working (generic function with 2 methods)

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1.2 ms

Markdown.MD

Fantastic!

Splendid!

Great!

Yay ❤

Great! 🎉

Well done!

Keep it up!

Good job!

Awesome!

You got the right answer!

Let's move on to the next section.

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9.8 ms

correct (generic function with 2 methods)

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679 μs

not_defined (generic function with 1 method)

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755 μs

In the cloud (experimental)

On your computer

Frontmatter

Preview

Computational Thinking

Math from Computation

Math with Computation

More options

First example

Exercise 1: Calculus without calculus

Exercise 1.1 - tangent line

Exercise 1.2 - antiderivative

Some cool things

Function library