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"markdown": "# The SciML suite of packages\n\n\n\nThe `Julia` ecosystem advances rapidly. For much of it, the driving force is the [SciML](https://github.com/SciML) organization (Scientific Machine Learning).\n\n\nIn this section we describe some packages provided by this organization that could be used as alternatives to the ones utilized in these notes. Members of this organization created many packages for solving different types of differential equations, and have branched out from there. Many newer efforts of this organization have been to write uniform interfaces to other packages in the ecosystem, some of which are discussed below. We don't discuss the promise of SCIML: \"Performance is considered a priority, and performance issues are considered bugs,\" as we don't pursue features like in-place modification, sparsity, etc. Interested readers should consult the relevant packages documentation.\n\n\nThe basic structure to use these packages is the \"problem-algorithm-solve\" interface described in [The problem-algorithm-solve interface](../ODEs/solve.html). We also discussed this interface a bit in [ODEs](../ODEs/differential_equations.html).\n\n\n:::{.callout-note}\n## Note\nThese packages are in a process of rapid development and change to them is expected. These notes were written using the following versions:\n\n:::\n\n::: {.cell execution_count=1}\n``` {.julia .cell-code}\npkgs = [\"Symbolics\", \"NonlinearSolve\", \"Optimization\", \"Integrals\"]\nimport Pkg; Pkg.status(pkgs)\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n Project CalculusWithJuliaNotes v0.1.3\n Status `~/julia/CalculusWithJuliaNotes/Project.toml` (empty project)\n```\n:::\n:::\n\n\n## Symbolic math (`Symbolics`)\n\n\nThe `Symbolics`, `SymbolicUtils`, and `ModelingToolkit` packages are provided by this organization. These can be viewed as an alternative to `SymPy`, which is used throughout this set of notes. See the section on [Symbolics](./symbolics.html) for some additional details, the package [documentation](https://symbolics.juliasymbolics.org/stable/), or the documentation for [SymbolicsUtils](https://github.com/JuliaSymbolics/SymbolicUtils.jl).\n\n\n## Solving equations\n\n\nSolving one or more equations (simultaneously) is different in the linear case (where solutions are readily found – though performance can distinguish approaches – and the nonlinear case – where for most situations, numeric approaches are required.\n\n\n### `LinearSolve`\n\n\nThe `LinearSolve` package aims to generalize the solving of linear equations. For many cases these are simply represented as matrix equations of the form `Ax=b`, from which `Julia` (borrowing from MATLAB) offers the interface `A \\ b` to yield `x`. There are scenarios that don't naturally fit this structure and perhaps problems where different tolerances need to be specified, and the `LinearSolve` package aims to provide a common interface to handle these scenarios. As this set of notes doesn't bump into such, this package is not described here. In the symbolic case, the `Symbolics.solve_for` function was described in [Symbolics](./symbolics.html).\n\n\n### `NonlinearSolve`\n\n\nThe `NonlinearSolve` package can be seen as an alternative to the use of the `Roots` package in this set of notes. The package presents itself as \"Fast implementations of root finding algorithms in Julia that satisfy the SciML common interface.\"\n\n\nThe package is loaded through the following command:\n\n``` {.julia .cell-code}\nusing NonlinearSolve\n```\n\n\nUnlike `Roots`, the package handles problems beyond the univariate case, as such the simplest problems have a little extra setup required.\n\n\nFor example, suppose we want to use this package to solve for zeros of $f(x) = x^5 - x - 1$. We could do so a few different ways.\n\n\nFirst, we need to define a `Julia` function representing `f`. We do so with:\n\n::: {.cell execution_count=3}\n``` {.julia .cell-code}\nf(u, p) = @. (u^5 - u - 1)\n```\n\n::: {.cell-output .cell-output-display execution_count=4}\n```\nf (generic function with 1 method)\n```\n:::\n:::\n\n\nThe function definition expects a container for the \"`x`\" variables and allows the passing of a container to hold parameters. We could have used the dotted operations for the power and each subtraction to allow vectorization of these basic math operations, as `u` is a container of values. The `@.` macro makes adding the \"dots\" quite easy, as illustrated above. It converts \"every function call or operator in expr into a `dot call`.\"\n\n\nA problem is set up with this function and an initial guess. The `@SVector` specification for the guess is for performance purposes and is provided by the `StaticArrays` package.\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\nusing StaticArrays\nu0 = @SVector[1.0]\nprob = NonlinearProblem(f, u0)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n\n::: {.ansi-escaped-output}\n```{=html}\n<pre><spaLine truncated
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"markdown": "# Calculus plots with Makie\n\n\n\nThe [Makie.jl webpage](https://github.com/JuliaPlots/Makie.jl) says\n\n\n> From the Japanese word Maki-e, which is a technique to sprinkle lacquer with gold and silver powder. Data is basically the gold and silver of our age, so let's spread it out beautifully on the screen!\n\n\n\n`Makie` itself is a metapackage for a rich ecosystem. We show how to use the interface provided by the `GLMakie` backend to produce the familiar graphics of calculus.\n\n\n:::{.callout-note}\n## Examples and tutorials\n`Makie` is a sophisticated plotting package, and capable of an enormous range of plots (cf. [examples](https://makie.juliaplots.org/stable/examples/plotting_functions/).) `Makie` also has numerous [tutorials](https://makie.juliaplots.org/stable/tutorials/) to learn from. These are far more extensive that what is described herein, as this section focuses just on the graphics from calculus.\n\n:::\n\n## Figures\n\n\nMakie draws graphics onto a canvas termed a \"scene\" in the Makie documentation. A scene is an implementation detail, the basic (non-mutating) plotting commands described below return a `FigureAxisPlot` object, a compound object that combines a figure, an axes, and a plot object. The `show` method for these objects display the figure.\n\n\nFor `Makie` there are the `GLMakie`, `WGLMakie`, and `CairoMakie` backends for different types of canvases. In the following, we have used `GLMakie`. `WGLMakie` is useful for incorporating `Makie` plots into web-based technologies.\n\n\nWe begin by loading the main package and the `norm` function from the standard `LinearAlgebra` package:\n\n``` {.julia .cell-code}\nusing GLMakie\nimport LinearAlgebra: norm\n```\n\n\nThe `Makie` developers have workarounds for the delayed time to first plot, but without utilizing these the time to load the package is lengthy.\n\n\n## Points (`scatter`)\n\n\nThe task of plotting the points, say $(1,2)$, $(2,3)$, $(3,2)$ can be done different ways. Most plotting packages, and `Makie` is no exception, allow the following: form vectors of the $x$ and $y$ values then plot those with `scatter`:\n\n::: {.cell execution_count=3}\n``` {.julia .cell-code}\nxs = [1,2,3]\nys = [2,3,2]\nscatter(xs, ys)\n```\n\n::: {.cell-output .cell-output-display execution_count=4}\n{}\n:::\n:::\n\n\nThe `scatter` function creates and returns an object, which when displayed shows the plot.\n\n\n### `Point2`, `Point3`\n\n\nWhen learning about points on the Cartesian plane, a \"`t`\"-chart is often produced:\n\n\n```\nx | y\n-----\n1 | 2\n2 | 3\n3 | 2\n```\n\nThe `scatter` usage above used the columns. The rows are associated with the points, and these too can be used to produce the same graphic. Rather than make vectors of $x$ and $y$ (and optionally $z$) coordinates, it is more idiomatic to create a vector of \"points.\" `Makie` utilizes a `Point` type to store a 2 or 3 dimensional point. The `Point2` and `Point3` constructors will be utilized.\n\n\n`Makie` uses a GPU, when present, to accelerate the graphic rendering. GPUs employ 32-bit numbers. Julia uses an `f0` to indicate 32-bit floating points. Hence the alternate types `Point2f0` to store 2D points as 32-bit numbers and `Points3f0` to store 3D points as 32-bit numbers are seen in the documentation for Makie.\n\n\nWe can plot a vector of points in as direct manner as vectors of their coordinates:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\npts = [Point2(1,2), Point2(2,3), Point2(3,2)]\nscatter(pts)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n{}\n:::\n:::\n\n\nA typical usage is to generate points from some vector-valued function. Say we have a parameterized function `r` taking $R$ into $R^2$ defined by:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nr(t) = [sin(t), cos(t)]\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nr (generic function with 1 method)\n```\n:::\n:::\n\n\nThen broadcasting values gives a vector of vectors, each identified with a point:\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nts = [1,2,3]\nr.(ts)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```\n3-element Vector{Vector{Float64}}:\n [0.8414709848078965, 0.5403023058681398]\n [0.9092974268256817, -0.4161468365471424]\n [0.1411200080598672, -0.9899924966004454]\n```\n:::\n:::\n\n\nWe can broadcast `Point2` over this to create a vector of `Point` objects:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\npts = Point2.(r.(ts))\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```\n3-element Vector{Point2{Float64}}:\n [0.8414709848078965, 0.5403023058681398]\n [0.9092974268256817, -0.4161468365471424]\n [0.1411200080598672, -0.9899924966004454]\n```\n:::\n:::\n\n\nThese then can be plotted directly:\n\n::: {.cell execution_count=8}\n``` {.juliLine truncated
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"markdown": "# JavaScript based plotting libraries\n\n\n\n:::{.callout-note}\n## Not working with quarto\nCurrently, the plots generated here are not rendering within quarto.\n\n:::\n\nThis section uses this add-on package:\n\n``` {.julia .cell-code}\nusing PlotlyLight\n```\n\n\nTo avoid a dependence on the `CalculusWithJulia` package, we load two utility packages:\n\n``` {.julia .cell-code}\nusing PlotUtils\nusing SplitApplyCombine\n```\n\n\n---\n\n\n`Julia` has different interfaces to a few JavaScript plotting libraries, notably the [vega](https://vega.github.io/vega/) and [vega-lite](https://vega.github.io/vega-lite/) through the [VegaLite.jl](https://github.com/queryverse/VegaLite.jl) package, and [plotly](https://plotly.com/javascript/) through several interfaces: `Plots.jl`, `PlotlyJS.jl`, and `PlotlyLight.jl`. These all make web-based graphics, for display through a web browser.\n\n\nThe `Plots.jl` interface is a backend for the familiar `Plots` package, making the calling syntax familiar, as is used throughout these notes. The `plotly()` command, from `Plots`, switches to this backend.\n\n\nThe `PlotlyJS.jl` interface offers direct translation from `Julia` structures to the underlying `JSON` structures needed by plotly, and has mechanisms to call back into `Julia` from `JavaScript`. This allows complicated interfaces to be produced.\n\n\nHere we discuss `PlotlyLight` which conveniently provides the translation from `Julia` structures to the `JSON` structures needed in a light-weight package, which plots quickly, without the delays due to compilation of the more complicated interfaces. Minor modifications would be needed to adjust the examples to work with `PlotlyJS` or `PlotlyBase`. The documentation for the `JavaScript` [library](https://plotly.com/javascript/) provides numerous examples which can easily be translated. The [one-page-reference](https://plotly.com/javascript/reference/) gives specific details, and is quoted from below, at times.\n\n\nThis discussion covers the basic of graphing for calculus purposes. It does not cover, for example, the faceting common in statistical usages, or the chart types common in business and statistics uses. The `plotly` library is much more extensive than what is reviewed below.\n\n\n## Julia dictionaries to JSON\n\n\n`PlotlyLight` uses the `JavaScript` interface for the `plotly` libraries. Unlike more developed interfaces, like the one for `Python`, `PlotlyLight` only manages the translation from `Julia` structures to `JavaScript` structures and the display of the results.\n\n\nThe key to translation is the mapping for `Julia`'s dictionaries to the nested `JSON` structures needed by the `JavaScript` library.\n\n\nFor example, an introductory [example](https://plotly.com/javascript/line-and-scatter/) for a scatter plot includes this `JSON` structure:\n\n``` {.julia .cell-code}\nvar trace1 = {\n x: [1, 2, 3, 4],\n y: [10, 15, 13, 17],\n mode: 'markers',\n type: 'scatter'\n};\n```\n\n\nThe `{}` create a list, the `[]` an Array (or vector, as it does with `Julia`), the `name:` are keys. The above is simply translated via:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nConfig(x = [1,2,3,4],\n y = [10, 15, 13, 17],\n mode = \"markers\",\n type = \"scatter\"\n )\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\nConfig with 4 entries:\n :x => [1, 2, 3, 4]\n :y => [10, 15, 13, 17]\n :mode => \"markers\"\n :type => \"scatter\"\n```\n:::\n:::\n\n\nThe `Config` constructor (from the `EasyConfig` package loaded with `PlotlyLight`) is an interface for a dictionary whose keys are symbols, which are produced by the named arguments passed to `Config`. By nesting `Config` statements, nested `JavaScript` structures can be built up. As well, these can be built on the fly using `.` notation, as in:\n\n::: {.cell hold='true' execution_count=6}\n``` {.julia .cell-code}\ncfg = Config()\ncfg.key1.key2.key3 = \"value\"\ncfg\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nConfig with 1 entry:\n :key1 => Config(:key2=>Config(:key3=>\"value\"))\n```\n:::\n:::\n\n\nTo produce a figure with `PlotlyLight` then is fairly straightforward: data and, optionally, a layout are created using `Config`, then passed along to the `Plot` command producing a `Plot` object which has `display` methods defined for it. This will be illustrated through the examples.\n\n\n## Scatter plot\n\n\nA basic scatter plot of points $(x,y)$ is created as follows:\n\n::: {.cell hold='true' execution_count=7}\n``` {.julia .cell-code}\nxs = 1:5\nys = rand(5)\ndata = Config(x = xs,\n y = ys,\n type=\"scatter\",\n mode=\"markers\"\n )\nPlot(data)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```{=html}\n<div class=\"\" style=\"height: 100vh;\" id=\"parent-of-rSE4HwRQR9\">\n <div class=\"\" style=\"height: 100%;\" id=\"rSE4HwRQR9\"></div>\n</div>\nLine truncated
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"markdown": "# Symbolics.jl\n\n\n\n\nThere are a few options in `Julia` for symbolic math, for example, the `SymPy` package which wraps a Python library. This section describes a collection of native `Julia` packages providing many features of symbolic math.\n\n\n## About\n\n\nThe `Symbolics` package bills itself as a \"fast and modern Computer Algebra System (CAS) for a fast and modern programming language.\" This package relies on the `SymbolicUtils` package and is built upon by the `ModelingToolkit` package, which we don't describe here.\n\n\nWe begin by loading the `Symbolics` package which when loaded re-exports the `SymbolicUtils` package.\n\n``` {.julia .cell-code}\nusing Symbolics\n```\n\n\n## Symbolic variables\n\n\nSymbolic math at its core involves symbolic variables, which essentially defer evaluation until requested. The creation of symbolic variables differs between the two packages discussed here.\n\n\n`SymbolicUtils` creates variables which carry `Julia` type information (e.g. `Int`, `Float64`, ...). This type information carries through operations involving these variables. Symbolic variables can be created with the `@syms` macro. For example\n\n::: {.cell execution_count=2}\n``` {.julia .cell-code}\n@syms x y::Int f(x::Real)::Real\n```\n\n::: {.cell-output .cell-output-display execution_count=3}\n```\n(x, y, f)\n```\n:::\n:::\n\n\nThis creates `x` a symbolic value with symbolic type `Number`, `y` a symbolic variable holding integer values, and `f` a symbolic function of a single real variable outputting a real variable.\n\n\nThe non-exported `symtype` function reveals the underlying type:\n\n::: {.cell execution_count=3}\n``` {.julia .cell-code}\nimport Symbolics.SymbolicUtils: symtype\n\nsymtype(x), symtype(y)\n```\n\n::: {.cell-output .cell-output-display execution_count=4}\n```\n(Number, Int64)\n```\n:::\n:::\n\n\nFor `y`, the symbolic type being real does not imply the type of `y` is a subtype of `Real`:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\nisa(y, Real)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\nfalse\n```\n:::\n:::\n\n\nWe see that the function `f` when called with `y` would return a value of (symbolic) type `Real`:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nf(y) |> symtype\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nReal\n```\n:::\n:::\n\n\nAs the symbolic type of `x` is `Number` – which is not a subtype of `Real` – the following will error:\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nf(x)\n```\n\n::: {.cell-output .cell-output-error}\n```\nLoadError: Tuple{Number} is not a subtype of Tuple{Real}.\n```\n:::\n:::\n\n\n:::{.callout-note}\n## Note\nThe `SymPy` package also has an `@syms` macro to create variables. Though their names agree, they do different things. Using both packages together would require qualifying many shared method names. For `SymbolicUtils`, the `@syms` macro uses `Julia` types to parameterize the variables. In `SymPy` it is possible to specify *assumptions* on the variables, but that is different and not useful for dispatch without some extra effort.\n\n:::\n\nFor `Symbolics`, symbolic variables are created using a wrapper around an underlying `SymbolicUtils` object. This wrapper, `Num`, is a subtype of `Real` (the underlying `SymbolicUtils` object may have symbolic type `Real`, but it won't be a subtype of `Real`.)\n\n\nSymbolic values are created with the `@variables` macro. For example:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\n@variables x y::Int z[1:3]::Int f(..)::Int\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```\n4-element Vector{Any}:\n x\n y\n z[1:3]\n f⋆\n```\n:::\n:::\n\n\nThis creates\n\n\n * a symbolic value `x` of `symtype` `Real`\n * a symbolic value `y` of `symtype` `Int`\n * a vector of symbolic values each of `symtype` `Int`\n * a symbolic function `f` returning an object of `symtype` `Int`\n\n\nThe symbolic type reflects that of the underlying object behind the `Num` wrapper:\n\n::: {.cell execution_count=8}\n``` {.julia .cell-code}\ntypeof(x), symtype(x), typeof(Symbolics.value(x))\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n```\n(Num, Real, SymbolicUtils.Sym{Real, Base.ImmutableDict{DataType, Any}})\n```\n:::\n:::\n\n\n(The `value` method unwraps the `Num` wrapper.)\n\n\n## Symbolic expressions\n\n\nSymbolic expressions are built up from symbolic variables through natural `Julia` idioms. `SymbolicUtils` privileges a few key operations: `Add`, `Mul`, `Pow`, and `Div`. For examples:\n\n::: {.cell execution_count=9}\n``` {.julia .cell-code}\n@syms x y\ntypeof(x + y) # `Add`\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n```\nSymbolicUtils.Add{Number, Int64, Dict{Any, Number}, Nothing}\n```\n:::\n:::\n\n\n::: {.cell execution_count=10}\n``` {.julia .cell-code}\ntypeof(x * y) # `Mul`\n```\n\n::: {.cell-output .cell-output-display executioLine truncated
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Reference in new issue
Block a user