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diff --git a/notebook/MOO.jl b/notebook/MOO.jl
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+++ b/notebook/MOO.jl
@@ -0,0 +1,1115 @@
+### A Pluto.jl notebook ###
+# v0.17.1
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ 5000b8f3-8a09-4440-80e9-ab56325de90f
+begin
+using DataFrames
+using Plots
+end
+
+# ╔═╡ 266c7f54-ce35-4b97-bdb6-bb5611332811
+md"""
+# Multi-objective optimization in RNA-seq deconvolution.
+
+Multi-objective optimization is used to solve problems that require more than one objective function to be optimized simultaneously. Typically, there is not one single solution that optimizes each objective. 
+"""
+
+# ╔═╡ d6285877-636a-4804-8790-7e4525d870b9
+md"""
+## Easy example
+
+
+Consider two towns, with two paths between them. The objectives that we care about are the distance (km) between the towns, and the travel time (h).
+
+Both paths are 1000 km long, so path A and path B are equally good. Each path has a 50% chance of being choosen.
+
+![]
+(https://github.com/licollier/Multiobjectopt/blob/master/notebook/Figures/1000 km.jpg?raw=true)
+"""
+
+
+# ╔═╡ c6af78f8-d0f2-48ed-b607-7c09dd3d535d
+md"""
+
+Now, let's imagine that the first path has a speed limit of 50 km/h and the second path one of 100 km/h. If we would always drive at this speed limit, it would take us about 20 hours to reach the other city. If we would take the second path, this would take us only 10 hours. Based on the travel time alone, the second path is 2 times as good as the first path. The red path has a 33% chance of being choosen, while the green path has a 66% chance of being choosen.
+
+
+![]
+(https://github.com/licollier/Multiobjectopt/blob/master/notebook/Figures/speedlimits.jpg?raw=true)
+
+"""
+
+# ╔═╡ 7cb56496-5184-44ad-9323-db908a8253c8
+md"""
+
+If we take both the distance and travel time into account, we can clearly see that the green path is better. But how much better? 
+
+If we would take a naive approach, we could just combine the distance and travel time to compute the fitness of each path.
+
+$$red: 1000 + 20 = 1020$$
+$$green: 1000 + 10 = 1010$$
+
+The red path now has a 50,2% chance of being chosen and the green path 49,8%. The travel time has almost no impact on the fitness of the path. We could solve this by multiplying the time distance with some factor, so it becomes similar in scale to the distance:
+
+$$red: 1000 + 20*50 = 2000$$
+$$green: 1000 +10*50 = 1500$$
+
+Now the red path has a 57% chance of being chosen, and the green path a 43% chance.
+This is slightly better, but when we put time on the same scale as the distance, we assume that travel time and distance are equally important. 
+
+"""
+
+# ╔═╡ f398de0e-f8e6-4f0f-931e-aa015d2aeb7e
+md"""
+By scaling one objective and just combining them to calculate the fitness of each path, we also loose information about the solutions. Look at the following example:
+
+![]
+(https://github.com/licollier/Multiobjectopt/blob/master/notebook/Figures/speedlimits2.jpg?raw=true)
+"""
+
+# ╔═╡ 4265b668-3c60-4859-98e6-71e65e8dccd8
+md"""
+
+$$red: 1000 + 10 *50 = 1500$$
+$$green: 500 + 20*50 = 1500$$
+
+Now, the red and green path have the same fitness, even tough they are very different solutions.
+
+We have realized that just combining the travel time and distance to find the fitness of each solution is not the best way. This way we assume that each objective is equally important and we loose valuable information about the solution. So, what can we do? """
+
+# ╔═╡ 66d7232f-c7b7-4385-95b8-460b0e570105
+md"""
+## Pareto fronts"""
+
+# ╔═╡ 4bc3066e-a0f5-4829-9c72-fa9cb6046427
+md""" 
+
+Continuing our problem of travelling between two towns. We now have 4 different solutions. We can visualise each solution on a simple plot. """
+
+# ╔═╡ af094e89-71e5-4468-93a9-941a1c32f8ff
+DataFrame(Distance =[1000, 750, 800, 500], Time = [10, 14, 14.5, 20])
+
+# ╔═╡ 8147f6ce-16ac-4085-a5d5-ef3c863d1018
+begin
+
+	Distance =[1000 750 800 500 ]
+	Time = [10 14 14.5 20]
+	markercolor = [:red :blue :purple :green]
+
+plot(Time,Distance,seriestype = :scatter, color = markercolor, markersize = 10,xlabel="Time (h)",ylabel="Distance (km)",legend = false)
+end
+
+# ╔═╡ 4c7e1677-bd51-475d-b996-1a3bf2d31ae2
+md"""
+When we compare the blue solution with the red solution, we see that blue is better in distance, but not in time. When we compare blue with green, we see that blue is better in time, but not in distance. Blue is another equally good solution as green and red, but with other trade offs between our objectives.
+
+When we look at the purple solution, we see that blue is better both in distance and in time. We can say that purple is **dominated** by blue.
+
+We have now found a way to order our solutions by fitness, when we have more than one objective. By identifying the dominating solutions, we can find the best solutions in our population. But what about the remainder of the solutions? Let us look at a few more solutions."""
+
+# ╔═╡ 24785674-8024-4d9e-9bb3-8b5e4589e7b5
+begin
+	Distance1 =[800 700 650 600 570 525 1000 900 750 1000 850 700 850]
+	Time1 = [10 11 13 15 15.5 19 11 12 14 16 13.5 16 17]
+	
+
+
+plot(Time1,Distance1,seriestype = :scatter, markersize =8,xlabel="Time (h)", ylabel="Distance (km)",legend=false)
+end
+
+# ╔═╡ 52aaf73e-7d97-43e6-810d-f3a8eea2e081
+md"""
+
+We can immediately identify the non-dominated solutions (in red), for these solutions we can not necessarily find a better solution, while for the blue ones we can.
+"""
+
+# ╔═╡ d84017c8-3e45-495d-8748-e758cca32932
+begin
+markercolor2=[:red :red :red :red :red :red :blue :blue :blue :blue :blue :blue :blue]
+plot(Time1,Distance1,seriestype = :scatter, markersize =8,xlabel="Time (h)", ylabel="Distance (km)",legend=false,color=markercolor2)
+end
+
+# ╔═╡ aca14538-cc05-49bd-9371-f4358e70150e
+md"""
+The line that is formed between these non-dominated points, is called a Pareto front. Every solution on the pareto front is equally fit. Once we have the first front we can identify a second, third,fourth,... front:""" 
+
+# ╔═╡ 78bb0c77-8722-4854-aa74-d502525469c5
+begin
+markercolor3=[:red :red :red :red :red :red :green :green :green :yellow :green :green :yellow]
+plot(Time1,Distance1,seriestype = :scatter, markersize =8,xlabel="Time (h)", ylabel="Distance (km)",legend=false,color=markercolor3)
+end
+
+# ╔═╡ 92652f0b-8112-4ed9-9412-ed714e66b2f8
+md"""
+
+We have found the red solutions to be the best, we can choose one solution on this Pareto front based on our own preferences. For example, I would rather drive more kilometers on a highway at a higher speed, than less kilometers on a "boerewegel" at low speed. So I would probably choose one of the most left solutions. """
+
+# ╔═╡ 9350e66f-bfa0-4772-a8c9-8d42df4c76ed
+md"""
+
+Now that we understand the basic concept of multi-object optimization, how can this be used to deconvolute bulk RNA-seq data? """
+
+# ╔═╡ ad3efd53-7051-4007-a276-b1b91aa76339
+md"""
+## Deconvolution"""
+
+# ╔═╡ f8b45b20-1a32-49c1-ad84-26e165328614
+md"""
+
+Deconvolution is typically used to obtain cell fractions (composition deconvolution) from heterogenous bulk gene expression data such as blood samples or tumor samples. 
+
+A general pipeline for deconvolution can be seen below. Tumor or blood samples can be sequenced to generate a bulk RNA gene expression profile. To determine the abundance of each cell type that is present in this cell mixture, deconvolution is used. For each cell type, the gene expression profile is established, preferably by single cell RNA sequencing (scRNA-seq). These gene expression profiles can then be used to generate a signature matrix. By combining these two inputs, the cell proportions can be determined.
+
+![]
+(https://github.com/licollier/Multiobjectopt/blob/master/notebook/Figures/Deconvolution_pipeline.jpg?raw=true)"""
+
+# ╔═╡ a7c46cef-e37d-41ee-acfa-234c527906cf
+md"""
+
+The signature matrix contains a gene expression profile for each cell type that is present in the bulk mixture. These expression profiles contain marker genes and there expression values. However, some cell types are closely related, so finding a signature profile that is unique enough to seperate them can be difficult. This is where multi-objective optimization can be used. We want to select a set of genes with two objectives: simultaneously minimize the correlation and maximize the distance between the cell types. First the Pareto optimals are found, and then it is again up to the researchers to choose what they find most important. Aliee and Theis (2021) found that using the solution with the least correlation will be the best. However, other solutions will still work well. """
+
+# ╔═╡ 23a2aa94-aee7-44d3-9658-d41724081b0a
+md"""
+
+## References
+
+Multiobjective Optimization | NEOS. (z.d.). Neos-Guide.Org. Geraadpleegd op 30 januari 2022, van https://neos-guide.org/content/multiobjective-optimizationTutorials with Gary. (2020, 13 juni). 
+
+Multi-Objective Optimisation - Writing your own Genetic Algorithm Part 6 [Video]. YouTube. https://www.youtube.com/watch?v=3JrpyuSHEWQ&t=569sWikipedia contributors. (2021, 26 november). 
+
+Pareto front. Wikipedia. Geraadpleegd op 30 januari 2022, van https://en.wikipedia.org/wiki/Pareto_front"""
+
+
+
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+[[Tables]]
+deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
+git-tree-sha1 = "bb1064c9a84c52e277f1096cf41434b675cd368b"
+uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
+version = "1.6.1"
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+[[Tar]]
+deps = ["ArgTools", "SHA"]
+uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
+
+[[Test]]
+deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
+uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+
+[[URIs]]
+git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
+uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
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+[[UUIDs]]
+deps = ["Random", "SHA"]
+uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"
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+[[Unicode]]
+uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
+
+[[UnicodeFun]]
+deps = ["REPL"]
+git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
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+[[Unzip]]
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+[[Wayland_protocols_jll]]
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+[[XSLT_jll]]
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+git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
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+[[Xorg_libX11_jll]]
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+git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
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+
+[[Xorg_libXdmcp_jll]]
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+git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
+uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
+version = "1.1.3+4"
+
+[[Xorg_libXext_jll]]
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+git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
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+[[Xorg_libXfixes_jll]]
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+git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
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+[[Xorg_libXi_jll]]
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+git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
+uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
+version = "1.7.10+4"
+
+[[Xorg_libXinerama_jll]]
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+git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
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+version = "1.1.4+4"
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+[[Xorg_libXrandr_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
+git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
+uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
+version = "1.5.2+4"
+
+[[Xorg_libXrender_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
+git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
+uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
+version = "0.9.10+4"
+
+[[Xorg_libpthread_stubs_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
+uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
+version = "0.1.0+3"
+
+[[Xorg_libxcb_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
+git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
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+
+[[Xorg_libxkbfile_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
+git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
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+[[Xorg_xcb_util_image_jll]]
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+git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
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+[[Xorg_xcb_util_jll]]
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+git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
+uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
+version = "0.4.0+1"
+
+[[Xorg_xcb_util_keysyms_jll]]
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+git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
+uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
+version = "0.4.0+1"
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+[[Xorg_xcb_util_renderutil_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
+uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
+version = "0.3.9+1"
+
+[[Xorg_xcb_util_wm_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
+uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
+version = "0.4.1+1"
+
+[[Xorg_xkbcomp_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
+git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
+uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
+version = "1.4.2+4"
+
+[[Xorg_xkeyboard_config_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
+git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
+uuid = "33bec58e-1273-512f-9401-5d533626f822"
+version = "2.27.0+4"
+
+[[Xorg_xtrans_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
+uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
+version = "1.4.0+3"
+
+[[Zlib_jll]]
+deps = ["Libdl"]
+uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
+
+[[Zstd_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
+uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
+version = "1.5.0+0"
+
+[[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"
+version = "0.15.1+0"
+
+[[libfdk_aac_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
+uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
+version = "2.0.2+0"
+
+[[libpng_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
+git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
+uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
+version = "1.6.38+0"
+
+[[libvorbis_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
+git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
+uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
+version = "1.3.7+1"
+
+[[nghttp2_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
+
+[[p7zip_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
+
+[[x264_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
+uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
+version = "2021.5.5+0"
+
+[[x265_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
+uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
+version = "3.5.0+0"
+
+[[xkbcommon_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
+git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
+uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
+version = "0.9.1+5"
+"""
+
+# ╔═╡ Cell order:
+# ╟─266c7f54-ce35-4b97-bdb6-bb5611332811
+# ╟─5000b8f3-8a09-4440-80e9-ab56325de90f
+# ╟─d6285877-636a-4804-8790-7e4525d870b9
+# ╟─c6af78f8-d0f2-48ed-b607-7c09dd3d535d
+# ╟─7cb56496-5184-44ad-9323-db908a8253c8
+# ╟─f398de0e-f8e6-4f0f-931e-aa015d2aeb7e
+# ╠═4265b668-3c60-4859-98e6-71e65e8dccd8
+# ╟─66d7232f-c7b7-4385-95b8-460b0e570105
+# ╟─4bc3066e-a0f5-4829-9c72-fa9cb6046427
+# ╟─af094e89-71e5-4468-93a9-941a1c32f8ff
+# ╟─8147f6ce-16ac-4085-a5d5-ef3c863d1018
+# ╟─4c7e1677-bd51-475d-b996-1a3bf2d31ae2
+# ╟─24785674-8024-4d9e-9bb3-8b5e4589e7b5
+# ╟─52aaf73e-7d97-43e6-810d-f3a8eea2e081
+# ╟─d84017c8-3e45-495d-8748-e758cca32932
+# ╟─aca14538-cc05-49bd-9371-f4358e70150e
+# ╟─78bb0c77-8722-4854-aa74-d502525469c5
+# ╟─92652f0b-8112-4ed9-9412-ed714e66b2f8
+# ╟─9350e66f-bfa0-4772-a8c9-8d42df4c76ed
+# ╟─ad3efd53-7051-4007-a276-b1b91aa76339
+# ╟─f8b45b20-1a32-49c1-ad84-26e165328614
+# ╟─a7c46cef-e37d-41ee-acfa-234c527906cf
+# ╠═23a2aa94-aee7-44d3-9658-d41724081b0a
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002
diff --git a/notebook/example.jl b/notebook/example.jl
deleted file mode 100644
index 950417f0..00000000
--- a/notebook/example.jl
+++ /dev/null
@@ -1,159 +0,0 @@
-### A Pluto.jl notebook ###
-# v0.14.8
-
-using Markdown
-using InteractiveUtils
-
-# ╔═╡ 5ad5c202-20f8-11eb-23f1-4f38b687c285
-using STMOZOO.Example
-
-# ╔═╡ d2d007b8-20f8-11eb-0ddd-1181d4565a85
-using Plots
-
-# ╔═╡ 45189a82-20fa-11eb-0423-05ce1b84639d
-using Zygote
-
-# ╔═╡ 171fee18-20f6-11eb-37e5-2d04caea8c35
-md"""
-# Example: solving quadratic systems
-
-By Michiel Stock
-
-## Introduction
-
-In this notebook, I have chosen to write code so minimize *quadratic functions* of the form:
-
-$$f(\mathbf{x}) =  \frac{1}{2} \mathbf{x}^\intercal P\mathbf{x} + \mathbf{q} \cdot \mathbf{x} + r\,,$$
-
-where $P \succ 0$.
-
-Hence, solving
-
-$$\min_\mathbf{x}\, f(\mathbf{x})\,.$$
-"""
-
-# ╔═╡ fb4aeb8c-20f7-11eb-0444-259de7b76883
-md"""
-
-## Derivation
-
-The gradient of the quadratic function is
-
-$$\nabla f(\mathbf{x})=P\mathbf{x} +\mathbf{q}\,.$$
-
-Setting this to zero gives
-
-$$\mathbf{x}^\star=-P^{-1}\mathbf{q}\,.$$
-
-We also know that $\nabla^2f(\mathbf{x}) = P \succ 0$, so $\mathbf{x}^\star$ is a minimizer.
-"""
-
-# ╔═╡ 52e87238-20f8-11eb-2ea0-27ee1208d3c3
-md"""
-## Illustration
-
-We will use a simple 2D example to illustrate this code.
-"""
-
-# ╔═╡ e9a9d69e-20f8-11eb-0d9d-330ee5e9cf25
-P = [7 -1; -1 2]
-
-# ╔═╡ fdd4e550-20f8-11eb-227b-25f36708484d
-q = [-14.0, -2.0]
-
-# ╔═╡ 025fd6e8-20f9-11eb-3e7d-3519f3c4b58f
-r = 1
-
-# ╔═╡ 096eff98-20f9-11eb-1e61-99d5714895ba
-md"We have a function to generate the quadratic function."
-
-# ╔═╡ 165509ca-20f9-11eb-107c-550cbba0f0e9
-f_quadr = quadratic_function(P, q, r)
-
-# ╔═╡ 1fffc82a-20f9-11eb-198c-c160d7dac87d
-f_quadr([2, 1])
-
-# ╔═╡ 26ab6ce2-20f9-11eb-1836-1756b290e5e3
-md"No more need to remember the formulla for the minimizer! Just use `solve_quadratic_system`!"
-
-# ╔═╡ 49832a8e-20f9-11eb-0841-19a40a12db18
-x_star = solve_quadratic_system(P, q, r)
-
-# ╔═╡ 55e0e274-20f9-11eb-36c0-753f228f7e9b
-begin
-	contourf(-20:0.1:20, -20:0.1:20, (x, y) -> f_quadr([x,y]), color=:speed)
-	scatter!([x_star[1]], [x_star[2]], label="minimizer")
-end
-
-# ╔═╡ b1551758-20f9-11eb-3e8f-ff9a7127d7f8
-md"""
-## Approximating non-quadratic functions
-
-We can approximate non-quadratic functions by a quadratic function: The second order Taylor approximation $\hat{f}$ of a function $f$ at $\mathbf{x}$ is
-$$f(\mathbf{x}+\mathbf{v})\approx\hat{f}(\mathbf{x}+\mathbf{v}) = f(\mathbf{x}) + \nabla f(\mathbf{x})^\top \mathbf{v} + \frac{1}{2} \mathbf{v}^\top \nabla^2 f(\mathbf{x}) \mathbf{v}\,.$$
-
-Let us use this idea for the Rosenbrock function.
-"""
-
-# ╔═╡ 41d8f1dc-20fa-11eb-3586-a989427c1fd6
-f_nonquadr((x1, x2); a=1, b=5) = (a-x1)^2 + b * (x2 - x1^2)^2
-
-# ╔═╡ 4ed4215e-20fa-11eb-11ee-f7741591163c
-x = [0.0, 0.0]
-
-# ╔═╡ 56af99ee-20fa-11eb-0240-69c675efb78c
-fx = f_nonquadr(x)
-
-# ╔═╡ 6c5473b4-20fa-11eb-327b-51ac560530eb
-∇fx = f_nonquadr'(x)
-
-# ╔═╡ 7518c2c0-20fa-11eb-32c0-a9db2a91cbc5
-∇²fx = Zygote.hessian(f_nonquadr, x)
-
-# ╔═╡ 34027942-20fb-11eb-261e-3b991ce4c9f8
-v = solve_quadratic_system(∇²fx, ∇fx, fx)
-
-# ╔═╡ 3bbeb85c-20fc-11eb-04d0-fb12d8ace50a
-f̂(x′) = quadratic_function(∇²fx, ∇fx, fx)(x′ .- x)
-
-# ╔═╡ 8623ac1a-20fa-11eb-2d45-49cce0fdac86
-begin
-	plot_nonquadr = contourf(-2:0.01:2, -2:0.01:2, (x, y) -> f_nonquadr([x,y]), color=:speed, title="non-quadratic function")
-	scatter!(plot_nonquadr, [x[1]], [x[2]], label="x")
-	scatter!(plot_nonquadr, [x[1]+v[1]], [x[2]+v[2]], label="x + v")
-	
-	plot_approx = contourf(-2:0.01:2, -2:0.01:2, (x, y) -> f̂([x,y]), color=:speed,
-		title="quadratic approximation")
-	scatter!(plot_approx, [x[1]], [x[2]], label="x")
-	scatter!(plot_approx, [x[1]+v[1]], [x[2]+v[2]], label="x + v")
-	
-	plot(plot_nonquadr, plot_approx, layout=(2,1), size=(600, 800))
-	
-end
-
-
-# ╔═╡ Cell order:
-# ╟─171fee18-20f6-11eb-37e5-2d04caea8c35
-# ╠═5ad5c202-20f8-11eb-23f1-4f38b687c285
-# ╠═d2d007b8-20f8-11eb-0ddd-1181d4565a85
-# ╟─fb4aeb8c-20f7-11eb-0444-259de7b76883
-# ╟─52e87238-20f8-11eb-2ea0-27ee1208d3c3
-# ╠═e9a9d69e-20f8-11eb-0d9d-330ee5e9cf25
-# ╠═fdd4e550-20f8-11eb-227b-25f36708484d
-# ╠═025fd6e8-20f9-11eb-3e7d-3519f3c4b58f
-# ╟─096eff98-20f9-11eb-1e61-99d5714895ba
-# ╠═165509ca-20f9-11eb-107c-550cbba0f0e9
-# ╠═1fffc82a-20f9-11eb-198c-c160d7dac87d
-# ╟─26ab6ce2-20f9-11eb-1836-1756b290e5e3
-# ╠═49832a8e-20f9-11eb-0841-19a40a12db18
-# ╠═55e0e274-20f9-11eb-36c0-753f228f7e9b
-# ╟─b1551758-20f9-11eb-3e8f-ff9a7127d7f8
-# ╠═41d8f1dc-20fa-11eb-3586-a989427c1fd6
-# ╠═45189a82-20fa-11eb-0423-05ce1b84639d
-# ╠═4ed4215e-20fa-11eb-11ee-f7741591163c
-# ╠═56af99ee-20fa-11eb-0240-69c675efb78c
-# ╠═6c5473b4-20fa-11eb-327b-51ac560530eb
-# ╠═7518c2c0-20fa-11eb-32c0-a9db2a91cbc5
-# ╠═34027942-20fb-11eb-261e-3b991ce4c9f8
-# ╠═3bbeb85c-20fc-11eb-04d0-fb12d8ace50a
-# ╟─8623ac1a-20fa-11eb-2d45-49cce0fdac86