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quarto/_freeze/alternatives/SciML/execute-results/html.json
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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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