From 326439f72543914b236f2f5ba5ef9a80cf980901 Mon Sep 17 00:00:00 2001
From: OlivierHnt <38465572+OlivierHnt@users.noreply.github.com>
Date: Thu, 5 Sep 2024 08:30:27 -0400
Subject: [PATCH] Add Taylor integration

---
 src/mod2/mod2.md                  |    9 -
 src/mod2/taylor_lecture_notes.jl  | 1519 +++++++++++++++++++++++++++++
 src/mod3/fourier_lecture_notes.jl |   96 +-
 3 files changed, 1567 insertions(+), 57 deletions(-)
 delete mode 100644 src/mod2/mod2.md
 create mode 100644 src/mod2/taylor_lecture_notes.jl

diff --git a/src/mod2/mod2.md b/src/mod2/mod2.md
deleted file mode 100644
index 8a81505..0000000
--- a/src/mod2/mod2.md
+++ /dev/null
@@ -1,9 +0,0 @@
----
-title: "Taylor integration"
-order: 1
-chapter: 3
-layout: "md.jlmd"
-tags: ["module2"]
----
-
-##
diff --git a/src/mod2/taylor_lecture_notes.jl b/src/mod2/taylor_lecture_notes.jl
new file mode 100644
index 0000000..29c610a
--- /dev/null
+++ b/src/mod2/taylor_lecture_notes.jl
@@ -0,0 +1,1519 @@
+### A Pluto.jl notebook ###
+# v0.19.46
+
+#> [frontmatter]
+#> chapter = 2
+#> order = 1
+#> title = "Lecture notes: Taylor integration"
+#> tags = ["module2"]
+#> layout = "layout.jlhtml"
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ d0e623ee-b096-4a27-977d-dc32567d6020
+using PlutoTeachingTools, PlutoUI
+
+# ╔═╡ 018ecc45-8638-4a59-b561-efb086bdc751
+using RadiiPolynomial, LinearAlgebra, Plots
+
+# ╔═╡ 7fc40507-eda3-474d-a454-04e9173a7adb
+html"""<style>
+main {
+    max-width: 1000px;
+    margin-left: auto;
+    margin-right: auto;
+    text-align: justify;
+}
+"""
+
+# ╔═╡ d5a510a3-c518-47ed-96bc-7bb22e3b08b5
+md"""
+In this lecture we will learn how to solve initial value problems for ODEs using Taylor series.
+This constitutes the first example of infinite dimensional problems.
+One important aspect of these proofs rely on understanding sequence spaces, and our first illustration is a rigorous square root operation.
+"""
+
+# ╔═╡ f73b88cb-19e2-4d50-a45e-fbe90fb691cd
+md"""
+# Simple example: square root in $\ell^1$
+"""
+
+# ╔═╡ cf717e9b-26d8-449d-9f3b-d1d5d0bd2bb5
+md"""
+## Introduction and setup
+"""
+
+# ╔═╡ b662e57e-ce55-4eb7-8efe-ccbdf7266c44
+md"""
+We consider the simple quadratic equation
+
+$\begin{align}
+a*a = b,
+\end{align}$
+
+where $a, b \in \ell^1$. That is, given $b \in \ell^1$ compute the square root $a \in \ell^1$ relative to the Cauchy product operaton.
+"""
+
+# ╔═╡ 7c9ca144-02f1-4941-bb0b-becb96b4fc8a
+md"""
+### Cauchy product
+"""
+
+# ╔═╡ b5d97ade-bcde-4c7a-ae78-697e73c10a0c
+md"""
+The Cauchy product is given by
+
+$\begin{align}
+(a*b)_n := \sum_{k=0}^n a_k b_{n-k}.
+\end{align}$
+"""
+
+# ╔═╡ f68f02a2-43cb-492a-a97f-0614c5266e03
+Markdown.MD(Markdown.Admonition("tip", "Lemma", [md"""When $u(t) = \sum_{n \ge 0} a_n t^n$
+and $\tilde{u}(t) = \sum_{n \ge 0} \tilde{a}_n t^n$ then for their product we have $u(t)\tilde{u}(t) = \sum_{n \ge 0} (a*\tilde{a})_n t^n$.
+ """]))
+
+# ╔═╡ 10958af8-a261-42d7-aa32-6f412142d2f0
+md"""
+An implementation of the Cauchy product is given below.
+"""
+
+# ╔═╡ 0e917ec7-6f7b-498a-a544-c84f8e17d93f
+function cauchy_product(a, b, nab = length(a)+length(b)-2)
+	# Cauchy product for 1D Taylor; nab is size of output (default is all nonzero coefficients)
+
+    na = length(a)-1
+    nb = length(b)-1
+	
+	c = zeros(eltype(a), nab+1)
+	for n = 0:nab
+		for k = max(n-na, 0):min(n, nb)
+			c[n+1] += a[n-k+1]*b[k+1]
+		end
+	end
+	
+	return c
+end
+
+# ╔═╡ cdccebca-5fd0-4381-b2f7-78cfc7c1e405
+md"""
+### Norm
+"""
+
+# ╔═╡ b33d8335-0abd-4018-adfa-63707eac6a22
+md"""
+We will work in the sequence space
+
+
+$\begin{align}
+X := \{ a \in \mathbb{R}^\mathbb{N} : \|a\|_X:= \sum_{n \ge 0} |a_n| < \infty \}.
+\end{align}$
+"""
+
+# ╔═╡ 4bd0c9e7-ffb2-4c13-b93d-96cfc530bff7
+Foldable("Is X with this norm a Banach space?", md"""**Exercise**: Yes""")
+
+# ╔═╡ 6bb627ea-0b12-4495-8b39-c79bc5a35890
+md"""
+The space $X$ together with $*$ forms a Banach algebra.
+"""
+
+# ╔═╡ e5754113-7f92-4946-a684-99d9bd5d830f
+Markdown.MD(Markdown.Admonition("tip", "Lemma (Banach algebra property)", [md"""For any $a, b \in X$ we have $\|a*b\|_X \le \|a\|_X \|b\|_X$."""]))
+
+# ╔═╡ 2f541fb5-79a8-4755-a611-a14fdf3f532b
+Foldable("""Proof""", md"""**Exercise** (rearrange the series and use the triangle inequality)""")
+
+# ╔═╡ 9f5cf201-871a-4afb-95ed-4bdba6152d23
+md"""
+The multiplication operator $M_a (b) := a * b$ can be represented by an infinite dimensional matrix. For our proof we will only need to construct a finite part of it.
+"""
+
+# ╔═╡ 27de2c2a-f7b6-46c5-97e7-e15341706cd5
+function cauchy_matrix(a, N)
+    Da = zeros(eltype(a), N+1, N+1)
+
+    for n = 0:N
+        for k = n:N
+            Da[k+1, n+1] = a[k-n+1]
+        end
+    end
+	
+	return Da
+end
+
+# ╔═╡ 1be891e6-f624-4c9e-b031-b7a4012b3362
+md"""
+## Zero finding problem
+"""
+
+# ╔═╡ 90fe50ee-89e8-4047-a115-533c90082957
+md"""
+We define the zero finding problem $F(a)=0$ on $X$ by ($n \ge 0$)
+
+$\begin{align}
+F_n(a) := (a*a)_n - b_n.
+\end{align}$
+"""
+
+# ╔═╡ fed8bd7d-fd50-4f48-9f0d-c28790294f4c
+md"""
+### Finite dimensional projection
+"""
+
+# ╔═╡ e97b6b16-82b8-482b-8aa8-8c75d8ddf23d
+md"""
+We introduce a projection operator on a finite number of modes ($N+1$ coefficients) and its complement:
+
+$\begin{align}
+(\pi^{\leq N} a)_n := \begin{cases}
+a_n, & n \le N, \\
+0, & n > N,
+\end{cases}
+& \qquad\qquad
+(\pi^{>N} a)_n := \begin{cases}
+0, & n \le N, \\
+a_n, & n > N,
+\end{cases}
+\end{align}$
+
+so that $a=\pi^{\leq N}a+\pi^{>N}a$ and $\pi^{\leq N}\pi^{>N}=0=\pi^{>N}\pi^{\leq N}$.
+"""
+
+# ╔═╡ 2c076e0d-5f35-40e3-9612-d5351b019559
+F(a, b, Nf = length(a)-1) = cauchy_product(a, a, Nf) - [b ; zeros(eltype(b), Nf - length(b)+1)]
+
+# ╔═╡ d3221c19-2720-43d5-9573-958ff69c5730
+md"""
+The derivative of $F(a)=a*a-b$ can now be written compactly as 
+
+$\begin{align}
+DF(a)c = 2a*c.
+\end{align}$
+"""
+
+# ╔═╡ f688e26c-dc4f-4faa-bd69-82c2366bd696
+DF(a, Ndf = length(a)-1) = 2 * cauchy_matrix(a, Ndf)
+
+# ╔═╡ 606533bd-a81a-4f33-b94f-fc4afb7fa5d6
+md"""
+## The Newton-Kantorovich proof
+"""
+
+# ╔═╡ af117c60-f433-4751-92ff-6d5a3744fbb1
+md"""
+We choose $b_n = 1/3^n$.
+"""
+
+# ╔═╡ 4fa97d29-4415-45e0-ac95-34cb1e74e55f
+begin
+	N = 35
+	
+	tauBar = interval(1)/interval(3)
+	
+	bbar = zeros(Interval{Float64}, N+1)
+	for n = 0:N
+		bbar[n+1] = tauBar^interval(n)
+	end
+	
+	b = zeros(Float64, N+1)
+	for n = 0:N
+		b[n+1] = mid(tauBar)^n
+	end
+	
+	epsilonN = tauBar^interval(N+1) / (interval(1) - tauBar)
+end
+
+# ╔═╡ db4c62ae-bef8-4c90-8e54-e38e7ec357b0
+begin
+	a = zeros(Float64, N+1)
+	a[1] = sqrt(b[1])
+	a, _ = newton(a -> (F(a, b, N), DF(a, N)), a)
+	
+	abar = interval(a)
+end
+
+# ╔═╡ cde7982c-8cd1-4369-be74-1c543704f933
+md"""
+We construct a numerical approximation of $\big(\pi^{\le N} DF(\bar{a}) \pi^{\le N}\big)^{-1}$.
+"""
+
+# ╔═╡ f3d4ab9f-24f6-4d65-93fc-e8155fe4f055
+begin
+	A = interval(inv(DF(a, N)))
+	normA = maximum(interval(ones(1, N+1)) * abs.(A)) # matrix 1-norm
+end
+
+# ╔═╡ 55580d08-5668-4a98-9082-20792e7a58c8
+begin	
+	YN = sum(abs.(A*F(abar, bbar, N)))
+	Yinf = sum(abs.(F(abar, bbar, 2N)[N+2:end]))
+	Y_bound = YN + Yinf + epsilonN
+end
+
+# ╔═╡ 8bd6e02f-2b26-4ba4-affc-19038ea60e31
+function Z1Bound(a, N)
+	sum = zero(eltype(a))
+	for n = 1:N
+		sum = sum + abs(a[n+1]) 
+	end
+	return interval(1)/interval(2)*(interval(1) + sum/a[1])
+end
+
+# ╔═╡ e782bccc-71fe-4170-9ac1-0027bbf1ad8a
+Z1_bound = Z1Bound(abar, N)
+
+# ╔═╡ 310397bd-ff77-44a3-b70c-7954ed944f37
+Z2_bound = interval(2)*normA
+
+# ╔═╡ c187aff6-087d-43f5-8042-d3ea1d09ba7a
+r = interval_of_existence(Y_bound, Z1_bound, Z2_bound, Inf)
+
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+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
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+git-tree-sha1 = "bcd466676fef0878338c61e655629fa7bbc69d8e"
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+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
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+[[deps.Xorg_xcb_util_wm_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
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+[[deps.Xorg_xkbcomp_jll]]
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+[[deps.Xorg_xkeyboard_config_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
+git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
+uuid = "33bec58e-1273-512f-9401-5d533626f822"
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+[[deps.Xorg_xtrans_jll]]
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+[[deps.Zlib_jll]]
+deps = ["Libdl"]
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+[[deps.Zstd_jll]]
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+version = "1.5.6+0"
+
+[[deps.eudev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
+git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
+uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
+version = "3.2.9+0"
+
+[[deps.fzf_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
+uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
+version = "0.43.0+0"
+
+[[deps.gperf_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
+uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
+version = "3.1.1+0"
+
+[[deps.libaom_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
+uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
+version = "3.9.0+0"
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+[[deps.libass_jll]]
+deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
+git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
+uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
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+
+[[deps.libblastrampoline_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
+version = "5.11.0+0"
+
+[[deps.libdecor_jll]]
+deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
+git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
+uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
+version = "0.2.2+0"
+
+[[deps.libevdev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
+uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
+version = "1.11.0+0"
+
+[[deps.libfdk_aac_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
+uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
+version = "2.0.2+0"
+
+[[deps.libinput_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
+git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
+uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
+version = "1.18.0+0"
+
+[[deps.libpng_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4"
+uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
+version = "1.6.43+1"
+
+[[deps.libvorbis_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
+git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
+uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
+version = "1.3.7+1"
+
+[[deps.mtdev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
+uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
+version = "1.1.6+0"
+
+[[deps.nghttp2_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
+version = "1.52.0+1"
+
+[[deps.p7zip_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
+version = "17.4.0+2"
+
+[[deps.x264_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
+uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
+version = "2021.5.5+0"
+
+[[deps.x265_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
+uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
+version = "3.5.0+0"
+
+[[deps.xkbcommon_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
+git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
+uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
+version = "1.4.1+1"
+"""
+
+# ╔═╡ Cell order:
+# ╟─7fc40507-eda3-474d-a454-04e9173a7adb
+# ╠═d0e623ee-b096-4a27-977d-dc32567d6020
+# ╠═018ecc45-8638-4a59-b561-efb086bdc751
+# ╟─d5a510a3-c518-47ed-96bc-7bb22e3b08b5
+# ╟─f73b88cb-19e2-4d50-a45e-fbe90fb691cd
+# ╟─cf717e9b-26d8-449d-9f3b-d1d5d0bd2bb5
+# ╟─b662e57e-ce55-4eb7-8efe-ccbdf7266c44
+# ╟─7c9ca144-02f1-4941-bb0b-becb96b4fc8a
+# ╟─b5d97ade-bcde-4c7a-ae78-697e73c10a0c
+# ╟─f68f02a2-43cb-492a-a97f-0614c5266e03
+# ╟─10958af8-a261-42d7-aa32-6f412142d2f0
+# ╠═0e917ec7-6f7b-498a-a544-c84f8e17d93f
+# ╟─cdccebca-5fd0-4381-b2f7-78cfc7c1e405
+# ╟─b33d8335-0abd-4018-adfa-63707eac6a22
+# ╟─4bd0c9e7-ffb2-4c13-b93d-96cfc530bff7
+# ╟─6bb627ea-0b12-4495-8b39-c79bc5a35890
+# ╟─e5754113-7f92-4946-a684-99d9bd5d830f
+# ╟─2f541fb5-79a8-4755-a611-a14fdf3f532b
+# ╟─9f5cf201-871a-4afb-95ed-4bdba6152d23
+# ╠═27de2c2a-f7b6-46c5-97e7-e15341706cd5
+# ╟─1be891e6-f624-4c9e-b031-b7a4012b3362
+# ╟─90fe50ee-89e8-4047-a115-533c90082957
+# ╟─fed8bd7d-fd50-4f48-9f0d-c28790294f4c
+# ╟─e97b6b16-82b8-482b-8aa8-8c75d8ddf23d
+# ╠═2c076e0d-5f35-40e3-9612-d5351b019559
+# ╟─d3221c19-2720-43d5-9573-958ff69c5730
+# ╠═f688e26c-dc4f-4faa-bd69-82c2366bd696
+# ╟─606533bd-a81a-4f33-b94f-fc4afb7fa5d6
+# ╟─af117c60-f433-4751-92ff-6d5a3744fbb1
+# ╠═4fa97d29-4415-45e0-ac95-34cb1e74e55f
+# ╠═db4c62ae-bef8-4c90-8e54-e38e7ec357b0
+# ╟─cde7982c-8cd1-4369-be74-1c543704f933
+# ╠═f3d4ab9f-24f6-4d65-93fc-e8155fe4f055
+# ╠═55580d08-5668-4a98-9082-20792e7a58c8
+# ╠═8bd6e02f-2b26-4ba4-affc-19038ea60e31
+# ╠═e782bccc-71fe-4170-9ac1-0027bbf1ad8a
+# ╠═310397bd-ff77-44a3-b70c-7954ed944f37
+# ╠═c187aff6-087d-43f5-8042-d3ea1d09ba7a
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002
diff --git a/src/mod3/fourier_lecture_notes.jl b/src/mod3/fourier_lecture_notes.jl
index d045883..b838e23 100644
--- a/src/mod3/fourier_lecture_notes.jl
+++ b/src/mod3/fourier_lecture_notes.jl
@@ -76,7 +76,7 @@ md"""
 
 # ╔═╡ 8f3e96dc-0cdc-411e-8f94-e9e47488abf3
 md"""
-We substitute the Fourier series 
+We substitute the Fourier series
 
 $\begin{align}
 u(t) = \sum_{n\in\mathbb{Z}} a_n e^{int},
@@ -102,7 +102,7 @@ md"""
 
 # ╔═╡ b5d97ade-bcde-4c7a-ae78-697e73c10a0c
 md"""
-The convolution product is given by 
+The convolution product is given by
 
 $\begin{align}
 (a*b)_n := \sum_{k\in\mathbb{Z}} a_k b_{n-k}.
@@ -116,7 +116,7 @@ and $\tilde{u}(t) = \sum_{n\in\mathbb{Z}} \tilde{a}_n e^{int}$ then for their pr
 
 # ╔═╡ ac3e3cf5-e5c8-4863-8c7b-ccd5f58a3073
 md"""
-When $a_n=0$ for $|n|>N_a$ and $b_n=0$ for $|n|>N_b$ then the convolution can be computed. 
+When $a_n=0$ for $|n|>N_a$ and $b_n=0$ for $|n|>N_b$ then the convolution can be computed.
 An implementation is below.
 """
 
@@ -129,7 +129,7 @@ function convolution(a, b, nab = (length(a)-1)÷2 + (length(b)-1)÷2)
 
     # initialize the result array with zeros
     result = zeros(eltype(a), 2*nab+1)
-    
+
     # perform the convolution
     for ia = -na:na
 		minib = max(-nab-ia, -nb)
@@ -149,7 +149,7 @@ md"""
 
 # ╔═╡ c4d9c4f0-fd89-4a2a-96a7-17c4ba6dd747
 md"""
-We will work in the sequence space 
+We will work in the sequence space
 
 $\begin{align}
 X = X_\nu := \{ a \in \mathbb{C}^\mathbb{Z} : \|a\|_X:= \sum_{n\in\mathbb{Z}} |a_n| \nu^{|n|} < \infty \}.
@@ -182,14 +182,14 @@ The operator norm (recall: $\|\Gamma\|_{B(X)}$ is the smallest number such that
 
 # ╔═╡ 7e78028c-f70b-4474-8bee-783ad7d99d56
 Markdown.MD(Markdown.Admonition("tip", "Lemma (characterization of operator norm)",[md"""
-Let $\Gamma$ be a bounded operator on $X$, represented by $(\Gamma a)_n=\sum_{k\in\mathbb{Z}} \Gamma_{nk} a_k$. 
-Then 
+Let $\Gamma$ be a bounded operator on $X$, represented by $(\Gamma a)_n=\sum_{k\in\mathbb{Z}} \Gamma_{nk} a_k$.
+Then
 
 $\begin{align}
   \|\Gamma\|_{B(X)} = \sup_{k\in\mathbb{Z}} \frac{1}{\nu^{|k|}} \sum_{n\in\mathbb{Z}} |\Gamma_{nk}| \nu^{|n|}.
 \end{align}$
 
-Let $e_k$ denote the basis vector given by $(e_k)_n=\delta_{kn}$ for any $k,n \in \mathbb{Z}$. Then 
+Let $e_k$ denote the basis vector given by $(e_k)_n=\delta_{kn}$ for any $k,n \in \mathbb{Z}$. Then
 
 $\begin{align}
   \|\Gamma\|_{B(X)} = \sup_{k\in\mathbb{Z}} \frac{\|\Gamma e_k\|_X}{\|e_k\|_X}.
@@ -290,13 +290,13 @@ The projections provide a sleek way to do the algebra that essentially splits th
 
 # ╔═╡ 740167dc-5bcd-48c0-b1f0-a7bf5e1b6cb7
 md"""
-The range $\pi^{\leq N}X$ is finite dimensional, and 
+The range $\pi^{\leq N}X$ is finite dimensional, and
 the restriction of $\pi^{\leq N} F$ to $\pi^{\leq N} X$, which we denote by $F^{\leq N}$, is what we will work with in the computer. An implementation is given below."""
 
 # ╔═╡ 026aec24-4ed0-4189-af4f-28f56e6964ef
 function F(a, beta, Nf = (length(a)-1)÷2)
 	# default length of output is same as input
-	
+
 	Na = (length(a)-1)÷2
 
 	# deal with sizes
@@ -314,23 +314,23 @@ function F(a, beta, Nf = (length(a)-1)÷2)
 
 	# linear terms
     lambdainv = [inv(n^2 - im * beta[1] * n - beta[2]) for n = -Nf:Nf]
-	
+
     return a1 + lambdainv .* nonlinear
 end
 
 # ╔═╡ 06f3d467-af63-440d-b486-8da7f67c314f
-md"""It is convenient to introduce the diagonal linear operator 
+md"""It is convenient to introduce the diagonal linear operator
 
 $\begin{align}
 (\Lambda a)_n := \lambda_n a_n.
 \end{align}$
 
-When $\beta_1 \neq 0$ then the inverse $\Lambda^{-1}$ is well-defined as a bounded operator on $X$. 
+When $\beta_1 \neq 0$ then the inverse $\Lambda^{-1}$ is well-defined as a bounded operator on $X$.
 """
 
 # ╔═╡ f5cc03ca-09eb-4419-9d51-78c956ee73d9
 md"""
-The derivative of $F(a)=a+\Lambda^{-1}[a*a+c]$ can now be written compactly as 
+The derivative of $F(a)=a+\Lambda^{-1}[a*a+c]$ can now be written compactly as
 
 $\begin{align}
 DF(a)b = b + 2 \Lambda^{-1} [a*b].
@@ -345,7 +345,7 @@ An implementation of the Jacobian $DF^{\leq N}$ of the truncated problem is give
 # ╔═╡ e5b2628d-2e64-47ca-b594-a0e4cb8ea64a
 function DF(a, beta, Ndf = (length(a)-1)÷2)
 	# Jacobian; default size of output is same as input
-	
+
     Na = (length(a)-1)÷2
 
 	# deal with nonlinear term in terms of a Toeplitz matrix
@@ -359,7 +359,7 @@ function DF(a, beta, Ndf = (length(a)-1)÷2)
 	# linear terms
     lambdainv = [inv(n^2 - im * beta[1] * n - beta[2]) for n = -Ndf:Ndf]
 
-    return I + 2*Diagonal(lambdainv)*Ta	
+    return I + 2*Diagonal(lambdainv)*Ta
 end
 
 # ╔═╡ c4363fdf-6500-4830-bc42-9777d616ac34
@@ -392,7 +392,7 @@ md"""
 
 # ╔═╡ 4a9daa68-86ed-4327-98fc-03d436885d2e
 md"""
-We consider the finite truncation $F^{\leq N}$ and use a Newton algorithm to find a point $\bar{a} \in \pi^{\leq N} X$ such that $\pi^{\leq N} F (\bar{a}) \approx 0$. If the truncation dimensions $N$ is large enough we may hope that also 
+We consider the finite truncation $F^{\leq N}$ and use a Newton algorithm to find a point $\bar{a} \in \pi^{\leq N} X$ such that $\pi^{\leq N} F (\bar{a}) \approx 0$. If the truncation dimensions $N$ is large enough we may hope that also
 $\pi^{> N} F (\bar{a}) \approx 0$, since we expect the coefficients of $\bar{a}$ to decrease for moderately large $n$.
 """
 
@@ -421,7 +421,7 @@ To simplify some of the expressions appearing in the bounds, we will for conveni
 """
 
 # ╔═╡ a89606ee-b46f-4656-9cb9-d89850646593
-println(N >= sup(sqrt(beta[3])))
+N >= sup(sqrt(beta[3]))
 
 # ╔═╡ 192507bd-8285-4581-8a6d-f9ff1b2c4793
 md"""
@@ -435,7 +435,7 @@ begin
 	initialdata = (initialdata + conj(reverse(initialdata))) / 2
 
 	Fbeta = mid.(beta) # floats
-	
+
 	a0, success = newton(a -> (F(a, Fbeta), DF(a, Fbeta)), initialdata)
 	a0 = (a0 + conj(reverse(a0))) / 2 # symmetry
 end
@@ -447,14 +447,14 @@ We plot the approximate periodic solution
 
 # ╔═╡ d5800e45-8809-46d6-9f41-7565e5b1cbfd
 begin
-	Na= (length(a0)-1) ÷ 2
-	Nplot=201;
-	t=LinRange(0, 2*pi, Nplot)
-	u=zeros(typeof(0.0im),Nplot)
-	for n=-Na:Na
+	Na = (length(a0)-1) ÷ 2
+	Nplot = 201
+	t = LinRange(0, 2*pi, Nplot)
+	u = zeros(typeof(0.0im),Nplot)
+	for n = -Na:Na
 		u += a0[Na+1+n]*exp.(im*n*t)
 	end
-	u=real(u)
+	u  =real(u)
 	plot(t, u, legend=false)
 	xlabel!("t")
 	ylabel!("u")
@@ -493,7 +493,7 @@ nu = interval(1.1)
 
 # ╔═╡ 83f772ea-aeec-49a3-baa2-d2fd8771c0a7
 md"""
-Since we want to prove something we need to resort to interval arithmetic. Hence we will need to convert $\bar{a}$ and $A$ to interval-valued variables. 
+Since we want to prove something we need to resort to interval arithmetic. Hence we will need to convert $\bar{a}$ and $A$ to interval-valued variables.
 """
 
 # ╔═╡ 1ace9001-93ae-46f2-98e6-7b371221beb3
@@ -502,7 +502,7 @@ begin
 	Ibeta = beta
 	IAN = interval(AN)
 	Inu = interval(nu)
-	# for testing without intervals:	
+	# for testing without intervals:
     # Ia0 = a0
     # Ibeta = Fbeta
 	# IA = AN
@@ -511,7 +511,7 @@ end
 
 # ╔═╡ 9433b6f1-3460-48c4-b587-9009823327c0
 md"""
-We are now in the Newton-Kantorovich context and thus need to derive the appropriate bounds. 
+We are now in the Newton-Kantorovich context and thus need to derive the appropriate bounds.
 """
 
 # ╔═╡ 9fe5120a-824a-4395-afe8-325a36ab9a07
@@ -550,7 +550,7 @@ md"""
 
 # ╔═╡ 002c7c0a-2b38-4a31-9e08-a424fa5679b8
 md"""
-This concerns the bound on the operator $I-A DF(\bar{a})$. In the analysis we will split the bound in a part which is computable by "brute force" and an estimate of the tail. The triangle inequality is often helpful for such arguments, but in our case the characterization of the operator norm gives an additional tool. In particular we will use that for any $M \in \mathbb{N}$ we have that 
+This concerns the bound on the operator $I-A DF(\bar{a})$. In the analysis we will split the bound in a part which is computable by "brute force" and an estimate of the tail. The triangle inequality is often helpful for such arguments, but in our case the characterization of the operator norm gives an additional tool. In particular we will use that for any $M \in \mathbb{N}$ we have that
 
 $\begin{align}
 \|\Gamma\|_{B(X)} = \max \Big( \|\Gamma \pi^{\le M}\|_{B(X)}, \|\Gamma \pi^{> M}\|_{B(X)} \Big).
@@ -564,7 +564,7 @@ Markdown.MD(Markdown.Admonition("note", "Finite bandwidth",[md"""
 It follows from the expression for $DF(a)$ that $DF(\bar{a})$ is an finite bandwidth operator of width $N+1$.
 In particular, for any $M \in \mathbb{N}$ we have $\pi^{> M+N} DF(\bar{a}) \pi^{\leq M} = 0$ and for any $M \ge N$ we have that $\pi^{\le M-N} DF(\bar{a}) \pi^{> M} =0$.
 
-It follows from the block-diagonal structure of $A$ that this implies that 
+It follows from the block-diagonal structure of $A$ that this implies that
 
 $\begin{align}
 A DF(\bar{a}) \pi^{> 2N} =  (A^{\le N}\pi^{\le N} + I^{>N} \pi^{>N}) DF(\bar{a}) \pi^{> 2N} = \pi^{>N} DF(\bar{a}) \pi^{> 2N},
@@ -572,7 +572,7 @@ A DF(\bar{a}) \pi^{> 2N} =  (A^{\le N}\pi^{\le N} + I^{>N} \pi^{>N}) DF(\bar{a})
 
 and we see that this expression does not involve $A^{\le N}$.
 
-Furthermore, it also follows from the block-diagonal structure of $A$ and the finite bandwidth of $DF(\bar{a})$ that 
+Furthermore, it also follows from the block-diagonal structure of $A$ and the finite bandwidth of $DF(\bar{a})$ that
 
 $\begin{align}
 A DF(\bar{a}) \pi^{\le 2N} = A \pi^{\le 3N} DF(\bar{a}) \pi^{\le 2N} = \pi^{\le 3N} A DF(\bar{a}) \pi^{\le 2N},
@@ -597,12 +597,12 @@ The latter term is computable thanks to the following lemma.
 
 # ╔═╡ d71cba6e-4f62-48d0-aa17-061e1f6bd744
 Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""
-Assume that $N \ge \sqrt{\beta_3}$. 
+Assume that $N \ge \sqrt{\beta_3}$.
 Then
 
 $\begin{align}
-\| [I-ADF(\bar{a})]\pi^{>2N} \|_{B(X)} = 
-\max \left( \frac{\|[I-ADF(\bar{a})]e_{2N+1}\|_X}{\|e_{2N+1}\|_X} ,  
+\| [I-ADF(\bar{a})]\pi^{>2N} \|_{B(X)} =
+\max \left( \frac{\|[I-ADF(\bar{a})]e_{2N+1}\|_X}{\|e_{2N+1}\|_X} ,
 \frac{\|[I-ADF(\bar{a})]e_{-(2N+1)}\|_X}{\|e_{-(2N+1)}\|_X} \right).
 \end{align}$
 """]))
@@ -613,7 +613,7 @@ we have $[I-ADF(\bar{a})]e_k= -2\Lambda^{-1} (\bar{a} * e_k)$ and it follows fro
 
 # ╔═╡ b6ae8c03-57aa-4a45-9d6a-9886421ee934
 md"""
-Collecting and summarizing the above results, we conclude that 
+Collecting and summarizing the above results, we conclude that
 
 $\begin{align}
 \|I-ADF(\bar{a})\|_{B(X)} = \|\pi^{\le 3N+1}[I-ADF(\bar{a})]\pi^{\le 2N+1}\|_{B(X)}.
@@ -627,13 +627,13 @@ Foldable("""Do we need to determine the norm of I-ADF(a̅) exactly?""", md"""No,
 
 We are often willing to spend quite some computer power on getting a rather sharp bound.
 For the current problem we could also get away with a somewhat rougher bound,
-such as (provided $N\geq \sqrt{\beta_3}$) 
+such as (provided $N\geq \sqrt{\beta_3}$)
 
 $\begin{align}
 Z_1=\max\left\{ \|\pi^{\leq 3N}[I- ADF(\bar{a})]\pi^{\leq 2N}\|_{B(X)} , \frac{2  \|\bar{a}\|_X}{|\lambda_{N+1}|} \right\} .
 \end{align}$
 
-We note that sharpness of the $Z_2$-bound is less of an issue, as it plays a considerably less critical role in the radii polynomial inequality. 
+We note that sharpness of the $Z_2$-bound is less of an issue, as it plays a considerably less critical role in the radii polynomial inequality.
 """)
 
 # ╔═╡ 4a57ce24-2872-4ef1-ada1-73a64ab4f17a
@@ -642,7 +642,7 @@ function boundZ1(a, beta, A, nu)
     NA = (size(A, 1)-1)÷2
 
 	# deal with sizes
-	Ncol = max(NA,ceil(Integer,sup(sqrt(beta[2])))) + Na + 1
+	Ncol = max(NA,ceil(Integer, sup(sqrt(beta[2])))) + Na + 1
 	Nrow = Ncol + Na
 
 	# compute A*DF(a)
@@ -651,7 +651,7 @@ function boundZ1(a, beta, A, nu)
 
 	B = I(2*Nrow+1)-Adfa
 	B = B[:,Nrow+1-Ncol:Nrow+1+Ncol]
-	
+
 	return nunorm(B, nu)
 end
 
@@ -662,7 +662,7 @@ IZ1 = boundZ1(Ia0, Ibeta, IAN, Inu)
 Foldable("""Injectivity of A revisited""", md"""**Exercise**: Prove that $\|I-A DF(\bar{a})\|_{B(X)}<1$ implies that $A$ is surjective on $X$.
 
 **Corollary**: The block-diagonal structure of $A$ implies that it follows from surjectivity of $A$ onto $X$ that $A^{\leq N}$ is surjective as an operator on $X^{\leq N}$. Since $X^{\leq N}$ is finite dimensional, this implies that $A^{\leq N}$  is injective on $X^{\leq N}$. We conclude that $A$ is also injective on $X$.
-""") 
+""")
 
 # ╔═╡ c0ce2c85-eaae-4b08-8ef7-2a9297b10e05
 md"""
@@ -671,7 +671,7 @@ md"""
 
 # ╔═╡ 6f346c67-879a-4427-afd4-6acdeb17af21
 md"""
-This concerns the bound on the operator $A[DF(a)-DF(\bar{a})]$. Since this is a Lipschitz bound on the first derivative, one approach is to bound the second derivative. However, here we just do a direct computation to see that 
+This concerns the bound on the operator $A[DF(a)-DF(\bar{a})]$. Since this is a Lipschitz bound on the first derivative, one approach is to bound the second derivative. However, here we just do a direct computation to see that
 $A[DF(a)-DF(\bar{a})]b =2A\Lambda^{-1}[(a-\bar{a})*b]$, hence by the Banach algebra property and the definition of the operator norm we arrive at
 
 $\begin{align}
@@ -687,7 +687,7 @@ Since the operator $\Lambda^{-1}$ is diagonal, it has a natural restriction to $
 """
 
 # ╔═╡ 62cec2b5-e408-46a4-947a-ebc58745c1ec
-Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""Assume $N\geq\sqrt{\beta_3}$. Then 
+Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""Assume $N\geq\sqrt{\beta_3}$. Then
 $\|A\Lambda^{-1}\|_{B(X)} = \max\{ \|A^{\leq N} \Lambda^{-1}  \|_{B(\pi^{\leq N} X)} , |\lambda_{N+1}|^{-1}  \}$
 """]))
 
@@ -745,7 +745,7 @@ We made sure that the numerical approximation was conjugate symmetric: $\bar{a}^
 """
 
 # ╔═╡ 6df91aa3-ad36-4b9e-ba4a-4dfb46eb1d27
-	println(conj(reverse(a0))==a0)
+conj(reverse(a0))==a0
 
 # ╔═╡ 49eae38b-786f-45fc-b331-3b01c32363a1
 md"""
@@ -760,11 +760,11 @@ Assume that $\bar{a}^\dagger=\bar{a}$ and that the assumptions of the Newton-Kan
 
 # ╔═╡ bae570fe-8219-4cf2-b763-fcd9f0f02735
 md"""
-*Proof*: First, based on the symmetry $F(a^\dagger) = F(a)^\dagger$ of the zero-finding problem, we see that $F(\tilde{a}) = 0$ implies that $F(\tilde{a}^\dagger) = F(\tilde{a})^\dagger = 0$, hence $\tilde{a}^\dagger$ is a zero of $F$. Second, the norm on $X$ is such that $\|a^\dagger\|_X = \|a\|_X$ for all $a \in X$. We find that 
+*Proof*: First, based on the symmetry $F(a^\dagger) = F(a)^\dagger$ of the zero-finding problem, we see that $F(\tilde{a}) = 0$ implies that $F(\tilde{a}^\dagger) = F(\tilde{a})^\dagger = 0$, hence $\tilde{a}^\dagger$ is a zero of $F$. Second, the norm on $X$ is such that $\|a^\dagger\|_X = \|a\|_X$ for all $a \in X$. We find that
 
-$\begin{align} 
+$\begin{align}
 \|\tilde{a}^\dagger-\bar{a}\|_X
-= \|\tilde{a}^\dagger-\bar{a}^\dagger\|_X 
+= \|\tilde{a}^\dagger-\bar{a}^\dagger\|_X
 = \|(\tilde{a}-\bar{a})^\dagger\|_X
 = \|\tilde{a}-\bar{a}\|_X \le r_0.
 \end{align}$
@@ -779,7 +779,7 @@ md"""
 
 # ╔═╡ eef1afcd-4da9-4042-a7c9-0124070740f8
 md"""
-We choose 
+We choose
 """
 
 # ╔═╡ 238c9111-7dda-40e8-8297-9776fe77f7d9
@@ -826,7 +826,7 @@ md"""
 
 # ╔═╡ 9dc65809-74c2-4341-baa8-63900de08df4
 md"""
-In this worksheet we discussed the setup of a CAP for a periodic solution in a simplified problem, including explicit estimates and code. These arguments can be generalized to the more general problem of finding a periodic orbit in a system of ODEs. As mentioned before, the main additional difficulties are the unknown period of the orbit, the phase condition, and the bookkeeping of the different components of the system. 
+In this worksheet we discussed the setup of a CAP for a periodic solution in a simplified problem, including explicit estimates and code. These arguments can be generalized to the more general problem of finding a periodic orbit in a system of ODEs. As mentioned before, the main additional difficulties are the unknown period of the orbit, the phase condition, and the bookkeeping of the different components of the system.
 """
 
 # ╔═╡ 194e7df2-a234-45c7-8760-8c49a0d6e651