From 19218c025a0cc6128eacc2101d2607f1fa5a9610 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 14 Mar 2024 01:28:57 +0000 Subject: [PATCH] build based on 2cf1223 --- previews/PR166/.documenter-siteinfo.json | 1 + previews/PR166/api/index.html | 14 + previews/PR166/assets/documenter.js | 1050 +++++++++++++++++ previews/PR166/assets/intro_plot.png | Bin 0 -> 69996 bytes .../PR166/assets/themes/documenter-dark.css | 7 + .../PR166/assets/themes/documenter-light.css | 9 + previews/PR166/assets/themeswap.js | 84 ++ previews/PR166/assets/warner.js | 52 + previews/PR166/index.html | 17 + previews/PR166/objects.inv | Bin 0 -> 1449 bytes previews/PR166/range_bounding/d6a20fe1.svg | 54 + previews/PR166/range_bounding/index.html | 30 + previews/PR166/search_index.js | 3 + previews/PR166/siteinfo.js | 1 + 14 files changed, 1322 insertions(+) create mode 100644 previews/PR166/.documenter-siteinfo.json create mode 100644 previews/PR166/api/index.html create mode 100644 previews/PR166/assets/documenter.js create mode 100644 previews/PR166/assets/intro_plot.png create mode 100644 previews/PR166/assets/themes/documenter-dark.css create mode 100644 previews/PR166/assets/themes/documenter-light.css create mode 100644 previews/PR166/assets/themeswap.js create mode 100644 previews/PR166/assets/warner.js create mode 100644 previews/PR166/index.html create mode 100644 previews/PR166/objects.inv create mode 100644 previews/PR166/range_bounding/d6a20fe1.svg create mode 100644 previews/PR166/range_bounding/index.html create mode 100644 previews/PR166/search_index.js create mode 100644 previews/PR166/siteinfo.js diff --git a/previews/PR166/.documenter-siteinfo.json b/previews/PR166/.documenter-siteinfo.json new file mode 100644 index 00000000..19e272a6 --- /dev/null +++ b/previews/PR166/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.2","generation_timestamp":"2024-03-14T01:28:53","documenter_version":"1.3.0"}} \ No newline at end of file diff --git a/previews/PR166/api/index.html b/previews/PR166/api/index.html new file mode 100644 index 00000000..2c95ee6a --- /dev/null +++ b/previews/PR166/api/index.html @@ -0,0 +1,14 @@ + +API · TaylorModels.jl
GitHub

TaylorModels.jl

TaylorModels.RTaylorModel1 — Type
RTaylorModel1{T,S}

Relative Taylor model in 1 variable, providing a rigurous polynomial approximation given by a Taylor polynomial pol (around x0) and a relative remainder rem for a function f(x) in one variable, valid in the interval dom. This corresponds to definition 2.3.2 of Mioara Joldes' thesis.

Fields:

  • pol: polynomial approximation, represented as TaylorSeries.Taylor1
  • rem: the interval bound
  • x0 : expansion point
  • dom: domain, interval over which the Taylor model is defined / valid

The approximation $f(x) = p(x) + \delta (x - x_0)^{n+1}$ is satisfied for all $x\in \mathcal{D}$; n is the order (degree) of the polynomial $p(x)=\sum_{i=0}^n p_i (x - x_0)^i$.

source
TaylorModels.TMSol — Type
TMSol{N,T,V1,V2,M}

Structure containing the solution of a validated integration.

Fields

`time :: AbstractVector{T}`  Vector containing the expansion time of the `TaylorModel1` solutions
+
+`fp   :: AbstractVector{IntervalBox{N,T}}`  IntervalBox vector representing the flowpipe
+
+`xTMv :: AbstractMatrix{TaylorModel1{TaylorN{T},T}}`  Matrix whose entry `xTMv[i,t]` represents
+the `TaylorModel1` of the i-th dependent variable, obtained at time time[t].
source
TaylorModels.TaylorModel1 — Type
TaylorModel1{T,S}

Absolute Taylor model in 1 variable, providing a rigurous polynomial approximation given by a Taylor polynomial pol (around x0) and an absolute remainder rem for a function f(x) in one variable, valid in the interval dom. This corresponds to definition 2.1.3 of Mioara Joldes' thesis.

Fields:

  • pol: polynomial approximation, represented as TaylorSeries.Taylor1
  • rem: the interval bound
  • x0 : expansion point
  • dom: domain, interval over which the Taylor model is defined / valid

The approximation $f(x) = p(x) + \Delta$ is satisfied for all $x\in \mathcal{D}$ ($0\in \Delta$); n is the order (degree) of the polynomial $p(x)=\sum_{i=0}^n p_i (x - x_0)^i$.

source
TaylorModels._monot_bound_remainder — Method
_monot_bound_remainder(::Type{RTaylorModel1}, ::Val{true}, f::Function, polf::Taylor1, polfI::Taylor1, x0, I::Interval)

Computes the remainder exploiting monotonicity; see Prop 2.3.7 in Mioara Joldes' PhD thesis (pp 67).

source
TaylorModels._monot_bound_remainder — Method
_monot_bound_remainder(::Type{TaylorModel1}, ::Val{false}, f::Function, polf::Taylor1, polfI::Taylor1, x0, I::Interval)

Computes the remainder using Lagrange bound

source
TaylorModels._monot_bound_remainder — Method
_monot_bound_remainder(::Type{TaylorModel1}, ::Val{true}, f::Function, polf::Taylor1, polfI::Taylor1, x0, I::Interval)

Computes the remainder exploiting monotonicity; see Prop 2.2.1 in Mioara Joldes' PhD thesis (pp 52).

source
TaylorModels._picard — Method
picard(dx, x0, box)

Computes the picard (integral) operator for the initial condition x0. dx must be the rhs of the differential equation.

source
TaylorModels._rpa — Function
   _rpa(::Type{TaylorModel1}, f::Function, x0::Interval, I::Interval, _order::Integer)
+   _rpa(::Type{RTaylorModel1}, f::Function, x0::Interval, I::Interval, _order::Integer)

Rigurous polynomial approximation (RPA) with absolute/relative remainder for the function f on the interval I, using a Taylor expansion around the interval x0 of order _order. The bound is computed by bound_remainder(@ref) exploiting monotonicity if possible, otherwise, it uses Lagrange bound.

source
TaylorModels._validate_step! — Method
_validate_step(xTM1K, f!, dx, x0, params, t, box, dof; ε=1e-10, maxsteps=20, extrasteps=50)

Validate the Taylor Model solution for the current integration time step. This function implements the epsilon inflation algorithm proposed by Bünger with some custom adaptations.

Ref: Florian B"unger, A Taylor model toolbox for solving ODEs implemented in MATLAB/INTLAB, J. Comput. Appl. Math. 368, 112511, https://doi.org/10.1016/j.cam.2019.112511

source
TaylorModels.absorb_remainder — Method
absorb_remainder(a::TaylorModelN)

Returns a TaylorModelN, equivalent to a, such that the remainder is mostly absorbed in the constant and linear coefficients. The linear shift assumes that a is normalized to the IntervalBox(-1..1, Val(N)).

Ref: Xin Chen, Erika Abraham, and Sriram Sankaranarayanan, "Taylor Model Flowpipe Construction for Non-linear Hybrid Systems", in Real Time Systems Symposium (RTSS), pp. 183-192 (2012), IEEE Press.

source
TaylorModels.bound_integration — Function
bound_integration(xTM::TaylorModel1{T,S}, δ)
+bound_integration(xTM::Vector{TaylorModel1{T,S}}, δ)

Bound the remainder of the integration of a xTM::TaylorModel1, where δ is the domain used to bound the integration. The remainder corresponds to $δ * remainder(a) + a.pol[order] * δ^(order+1) / (order+1)$. This is tighter that the one used by Berz+Makino, which corresponds to $Δ = δ * remainder(a) + a.pol[order] * δ^(order+1)$.

bound_integration(xTM::RTaylorModel1{T,S}, δ)
+bound_integration(xTM::Vector{RTaylorModel1{T,S}, δ)

Remainder bound for the integration of a xTM::RTaylorModel1, where δ is the domain used to bound the integration. The remainder corresponds to $Δ = δ * remainder(a)/(order+2) + getcoeff(polynomial(a), order)/(order+1)$.

source
TaylorModels.bound_remainder — Method

bound_remainder(::Type{RTaylorModel1}, f::Function, polf::Taylor1, polfI::Taylor1, x0::Interval, I::Interval)

Bound the relative remainder of the polynomial approximation of f given by the Taylor polynomial polf around x0 on the interval I. It requires an the interval extension polfI of a polynomial that approximates f for the whole interval I, in order to compute the Lagrange remainder.

If polfI[end] has a definite sign, then it is monotonic in the interval I, which is exploited; otherwise, the last coefficients bounds the relative remainder. This corresponds to Prop 2.3.7 in Mioara Joldes' PhD thesis (pp 67).

source
TaylorModels.bound_remainder — Method

bound_remainder(::Type{TaylorModel1}f::Function, polf::Taylor1, polfI::Taylor1, x0::Interval, I::Interval)

Bound the absolute remainder of the polynomial approximation of f given by the Taylor polynomial polf around x0 on the interval I. It requires the interval extension polfI of the polynomial that approximates f for the whole interval I, in order to compute the Lagrange remainder.

If polfI[end] has a definite sign, then it is monotonic in the intervals [I.lo, x0] and [x0.hi, I.hi], which is exploited; otherwise, it is used to compute the Lagrange remainder. This corresponds to Prop 2.2.1 in Mioara Joldes PhD thesis (pp 52).

source
TaylorModels.bound_taylor1 — Function
bound_taylor1(fT::TaylorModel1, I=domain(fT)::Interval)

Compute a tight polynomial bound for the Taylor model fT in the interval I, considering whether its derivative ftd has a definite sign.

source
TaylorModels.bound_taylor1 — Method
bound_taylor1(fT::Taylor1, I::Interval)

Compute a tight polynomial bound for the Taylor polynomial fT in the interval I.

Note: Algorithm 2.1.1 corresponds to evaluate(fT, I) or simply fT(I). This function uses the roots of the derivative offt` to obtain a tighter bound.

source
TaylorModels.bound_taylor1 — Method
bound_taylor1(fT::Taylor1, fTd::Taylor1, I::Interval)

Compute a tight polynomial bound for the Taylor polynomial fT in the interval I, considering whether its derivative ftd has a definite sign.

source
TaylorModels.fp_rpa — Function
fp_rpa(tm::TaylorModel1{Interval{T},T})
+fp_rpa(tm::RTaylorModel1{Interval{T},T})

Convert a tm TaylorModel1 to a TaylorModel1 whose polynomial coefficients are Float64. The accumulated error is added to the remainder. The mid point of the expansion interval is preferentially rounded down if it is not an exactly representable value.

source
TaylorModels.initialize! — Method
initialize!(X0::Vector{TaylorModel1{TaylorN{T}, T}}, orderQ, orderT, x, xI)
+initialize!(X0::Vector{TaylorModel1{TaylorN{T}, T}}, orderQ, orderT, x)

Initialize the auxiliary integration variables assuming that the given vector of taylor models X0 is normalized to the domain [-1, 1]^n in space.

source
TaylorModels.initialize! — Method
initialize!(X0::IntervalBox, orderQ, orderT, x, xI)
+initialize!(X0::IntervalBox, orderQ, orderT, x)

Initialize the internal integration variables and normalize the given interval box to the domain [-1, 1]^n.

source
TaylorModels.iscontractive — Method
iscontractive(Δ, Δx)

Checks if Δ .⊂ Δx is satisfied. If `Δ ⊆ Δx is satisfied, it returns true if all cases where == holds corresponds to the zero Interval.

source
TaylorModels.linear_dominated_bounder — Method
linear_dominated_bounder(fT::TaylorModel1, ϵ=1e-3::Float64, max_iter=5::Int)

Compute a tighter polynomial bound for the Taylor model fT by the linear dominated bounder algorithm. The linear dominated algorithm is applied until the bound of fT gets tighter than ϵ or the number of steps reachs max_iter. The returned bound corresponds to the improved polynomial bound with the remainder of the TaylorModel1 included.

source
TaylorModels.linear_dominated_bounder — Method
linear_dominated_bounder(fT::TaylorModelN, ϵ=1e-3::Float64, max_iter=5::Int)

Compute a tighter polynomial bound for the Taylor model fT by the linear dominated bounder algorithm. The linear dominated algorithm is applied until the bound of fT gets tighter than ϵ or the number of steps reachs max_iter. The returned bound corresponds to the improved polynomial bound with the remainder of the TaylorModelN included.

source
TaylorModels.mince_in_time — Method
mince_in_time(sol::TMSol; var=0, timediv=1) --> ::Vector{Interval}

For var=0, this function divides the time domain of each entry of sol in timediv parts (timediv==1 is the initial domain), and returns the time intervals where the solution holds. This is useful for plotting or finding specific events. For var ≥ 1, this function evaluates the flowpipes at the split domain intervals, which is useful to decrease the overapproximations associated to the whole time domain.

source
TaylorModels.picard_iteration — Method
picard_iteration(f!, dx, xTM1K, params, t, x0, box)

Computes the picard (integral) operator for the set of equations f! and the initial conditionx0. This method returns the remainder of the resulting Taylor Model without the remainder of the initial condition.

source
TaylorModels.picard_iteration — Method
picard_iteration(f!, dx, xTM1K, params, t, x0, box, ::Val{true})

Computes the picard (integral) operator for the set of equations f! and the initial conditionx0. The Val parameter enables the selection of the desired method. This one returns the remainder of the resulting Taylor Model with the remainder of the initial condition included.

source
TaylorModels.quadratic_fast_bounder — Method
quadratic_fast_bounder(fT::TaylorModel1)

Compute a tighter polynomial bound by the quadratic fast bounder. The returned bound corresponds to the "improved" polynomial bound with the remainder of the TaylorModel1 included. This "improved" bound can be one of the following two: 1) An improved bound: if the domain of fT has a local minimizer, then an improved bound is returned. 2) Original TaylorModel bound: if the local minimizer criteria is not satisfied, then the original bound of fT is returned.

This algorithm is a slight modification to the Makino & Berz algorithm. For this algorithm the linear part is bounded by solving a simple set of linear equations (compared to the iterative procedure by Makino & Berz).

source
TaylorModels.quadratic_fast_bounder — Method
quadratic_fast_bounder(fT::TaylorModelN)

Compute a tighter polynomial bound by the quadratic fast bounder. The returned bound corresponds to the "improved" polynomial bound with the remainder of the TaylorModelN included. This "improved" bound can be one of the following two: 1) An improved bound: if the domain of fT has a local minimizer, then an improved bound is returned. 2) Original TaylorModel bound: if the local minimizer criteria is not satisfied, then the original bound of fT is returned.

This algorithm is a slight modification to the Makino & Berz algorithm. For this algorithm the linear part is bounded by solving a simple set of linear equations (compared to the iterative procedure by Makino & Berz).

source
TaylorModels.remainder_taylorstep! — Method
remainder_taylorstep!(f!, t, x, dx, xI, dxI, δI, δt, params)

Returns a remainder for the integration step for the dependent variables (x) checking that the solution satisfies the criteria for existence and uniqueness.

source
TaylorModels.rpa — Function

rpa(g::Function, tmf::TaylorModel1) rpa(g::Function, tmf::RTaylorModel1) rpa(g::Function, tmf::TaylorModelN)

Rigurous polynomial approximation (RPA) for the function g using the Taylor Model with absolute/relative remainder tmf. The bound is computed exploiting monotonicity if possible, otherwise, it uses Lagrange bound.

source
TaylorModels.shrink_wrapping! — Method
shrink_wrapping!(xTMN::TaylorModelN)

Returns a modified inplace xTMN, which has absorbed the remainder by the modified shrink-wrapping method of Florian Bünger. The domain of xTMN is the normalized interval box [-1,1]^N.

Ref: Florian B"unger, Shrink wrapping for Taylor models revisited, Numer Algor 78:1001–1017 (2018), https://doi.org/10.1007/s11075-017-0410-1

source
TaylorSeries.integrate — Function
integrate(a::TM{T,S}, c0)
+integrate(a::TM{TaylorN{T},S}, c0, cc0)

Integrates the one-variable Taylor Model (TaylorModel1 or RTaylorModel1) with respect to the independent variable. c0 is the integration constant; if omitted it is taken as zero. When the coefficients of a are TaylorN variables, the domain is specified by cc0::IntervalBox.

integrate(fT, which)

Integrates a fT::TaylorModelN with respect to which variable. The returned TaylorModelN corresponds to the Taylor Model of the definite integral ∫f(x) - ∫f(expansion_point).

source
diff --git a/previews/PR166/assets/documenter.js b/previews/PR166/assets/documenter.js new file mode 100644 index 00000000..c6562b55 --- /dev/null +++ b/previews/PR166/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 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+$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR166/assets/intro_plot.png b/previews/PR166/assets/intro_plot.png new file mode 100644 index 0000000000000000000000000000000000000000..e48dfdd9ddd55dfc4658653596449b8ba939a83f GIT binary patch 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html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark 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.button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark 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.docstring>section>a.button.is-inverted.docs-sourcelink[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted,fieldset[disabled] html.theme--documenter-dark .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;border-color:transparent;box-shadow:none;color:#375a7f}html.theme--documenter-dark .button.is-primary.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-loading.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-primary.is-outlined,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink{background-color:transparent;border-color:#375a7f;color:#375a7f}html.theme--documenter-dark .button.is-primary.is-outlined:hover,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-outlined.is-hovered,html.theme--documenter-dark 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.button.is-danger:hover,html.theme--documenter-dark .button.is-danger.is-hovered{background-color:#92190c;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus,html.theme--documenter-dark .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus:not(:active),html.theme--documenter-dark .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(158,27,13,0.25)}html.theme--documenter-dark .button.is-danger:active,html.theme--documenter-dark .button.is-danger.is-active{background-color:#86170b;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:#9e1b0d;box-shadow:none}html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#9e1b0d;border-color:#9e1b0d;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--documenter-dark .icon.is-medium{height:2rem;width:2rem}html.theme--documenter-dark .icon.is-large{height:3rem;width:3rem}html.theme--documenter-dark .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--documenter-dark .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--documenter-dark .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--documenter-dark .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--documenter-dark div.icon-text{display:flex}html.theme--documenter-dark .image,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--documenter-dark .image img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--documenter-dark .image.is-4by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--documenter-dark .image.is-4by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--documenter-dark .image.is-3by2 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--documenter-dark .image.is-3by2 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--documenter-dark .image.is-5by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--documenter-dark .image.is-5by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--documenter-dark .image.is-16by9 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--documenter-dark .image.is-16by9 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--documenter-dark .image.is-2by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--documenter-dark .image.is-2by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--documenter-dark .image.is-3by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--documenter-dark .image.is-3by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--documenter-dark .image.is-4by5 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--documenter-dark .image.is-4by5 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--documenter-dark .image.is-3by4 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TaylorModels.jl

This package combines the IntervalArithmetic.jl and TaylorSeries.jl packages to provide Taylor models, i.e. Taylor polynomials with guaranteed error bounds to approximate functions.

An introduction is available in this video from JuliaCon 2018.

Introduction to Taylor Models

Taylor models provide a way to rigorously manipulate and evaluate functions using floating-point arithmetic. They have been widely used for validated computing: in global optimization and range bounding, for validated solutions of ODEs, rigorous quadrature, etc.

A Taylor model (TM) of order $n$ for a function $f$ which is supposed to be $n + 1$ times continuously differentiable over an interval $[a,b]$, is a rigorous polynomial approximation of $f$. Specifically, it is a couple $(P, \Delta)$ formed by a polynomial $P$ of degree $n$, and an interval part $\Delta$, such that $f(x) − P(x) \in \Delta$, $\forall x ∈ [a,b]$. Roughly speaking, as their name suggests, the polynomial can be seen as a Taylor expansion of the function at a given point. The interval $\Delta$ (also called interval remainder) provides the validation of the approximation, meaning that it provides an enclosure of all the approximation errors encountered (truncation, roundings).

Here we generate TMs of order 6 and 7 over $I = [-0.5,1.0]$. We can view a TM as a a tube around the actual function.

using TaylorModels
+
+f(x) = x*(x-1.1)*(x+2)*(x+2.2)*(x+2.5)*(x+3)*sin(1.7*x+0.5)
+a =  -0.5 .. 1.0 # Domain
+x0 = mid(a)     # Expansion point
+tm6 = TaylorModel1(6, interval(x0), a) # Independent variable for Taylor models, order 6
+tm7 = TaylorModel1(7, interval(x0), a)  # Independent variable for Taylor models, order 7
+# Taylor models corresponding to f(x) of order 6 and 7
+ftm6 = f(tm6)
+ftm7 = f(tm7)
+
+# Now the plot
+using Plots; gr()
+plot(range(inf(a), stop=sup(a), length=1000), x->f(x), label="f(x)", lw=2, xaxis="x", yaxis="f(x)")
+plot!(ftm6, label="6th order")
+plot!(ftm7, label="7th order")

intro_plot

Authors

  • Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)
  • David P. Sanders, Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM)

Bibliography

Acknowledgements

Financial support is acknowledged from DGAPA-UNAM PAPIIT grants IN-117117, IG-100616 and IG-100819. DPS acknowledges support through a Cátedra Marcos Moshinsky (2018).

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GitHub

Taylor model variables and range bounding

Taylor models can be applied to the problem of range bounding, that is to find an interval $I \subseteq \mathbb{R}$ such that $f(x) \in I$ on a given domain $x \in D$ (including possible floating point errors, see examples below). There are several ways to construct a Taylor model. A convenient way is to define a "Taylor model variable", which is then passed as argument to Julia functions. The following examples should help to clarify this method. To simplify the presentation we have only considered the univariate case, but this package can also handle multivariate Taylor models with the TaylorModelN type.

Here we construct a Taylor model variable specifying that:

  • The truncation order is 3.
  • The expansion is around the origin (interval(0)).
  • The domain is the real interval centered around the origin $[-0.5, 0.5]$.
using TaylorModels
+
+t = TaylorModel1(3, interval(0), -0.5..0.5)
 [1, 1] t + [0, 0]

Here the polynomial part is (the interval) $1$, and the remainder is zero. We can pass this Taylor model variable to any Julia function, for example:

texp = exp(t)
 [1, 1] + [1, 1] t + [0.5, 0.5] t² + [0.166666, 0.166667] t³ + [0, 0.00288794]

This expression is a polynomial of order 3 (in agreement with the truncation order specified in the construction of t), whose coefficients are intervals that are guaranteed to contain the exact coefficient of the Taylor expansion of the function $t \mapsto e^t$ in $D : [-0.5, 0.5] \subset \mathbb{R}$. Similarly, we can expand trigonometric functions such as $t \mapsto \sin(t)$:

tsin = sin(t)
 [1, 1] t + [-0.166667, -0.166666] t³ + [-0.00124851, 0.00124851]

Common arithmetic operators, such as addition (+) and multiplication (*) work with Taylor model variables out-of-the-box:

s = texp + tsin
 [1, 1] + [2, 2] t + [0.5, 0.5] t² + [-2.77556e-17, 2.77556e-17] t³ + [-0.00124851, 0.00413645]
p = texp * tsin
 [1, 1] t + [1, 1] t² + [0.333333, 0.333334] t³ + [-0.0065406, 0.00610658]

One-dimensional bounding

To bound the range of a Taylor model in one variable, use the function bound_taylor1:

using TaylorModels: bound_taylor1
+
+[bound_taylor1(x) for x in [s, p]]
2-element Vector{Interval{Float64}}:
+  [0.124999, 2.12501]
+ [-0.291667, 0.791667]

This shows in particular that $0.12499 \leq e^t + \sin(t) \leq 2.12501$ and that $-0.291667 \leq e^t \sin(t) \leq 0.791667$ for all $t \in D$. Such bounds are in general not tight.

Mincing

If desired, the common approach to improve the bounds is to evaluate the Taylor model on a smaller interval, e.g.

D = domain(s) # domain -0.5 .. 0.5
+
+E = evaluate(s, D) # original, no mincing
[-0.00124851, 2.12914]
Dm = mince(D, 8) # split the domain into 8 smaller chunks
+
+Em = evaluate.(s, Dm) # evaluate the Taylor model on each sub-domain
+
+Rm = reduce(hull, Em) # take the convex hull, i.e. the smallest interval that contains them all
[0.0690639, 2.12914]

Here the lower bound has been improved by mincing (or splitting) the domain, and it may improve by repeating such operation recursively on smaller domains. In particular, the fact that the lower bound is greater than zero constitutes an algorithmic proof that $s : t \mapsto e^t + \sin(t)$ is positive on $D$. Let's visualize the function $s(t)$ and the bounds obtained so far.

using Plots
+
+Dt = range(-0.5, 0.5, length=100)
+
+fig = plot(xlab="t", ylab="s(t)", legend=:topleft)
+plot!(fig, Dt, t -> exp(t) + sin(t), lab="", c=:black)
+
+# range bounds
+plot!(fig, Dt, t -> sup(E), lab="N = 1", c=:blue, style=:dash)
+plot!(fig, Dt, t -> inf(E), c=:blue, lab="", style=:dash)
+
+plot!(fig, Dt, t -> sup(Rm), lab="N = 8", c=:red, style=:dash)
+plot!(fig, Dt, t -> inf(Rm), c=:red, lab="", style=:dash)
+
+R16 = reduce(hull, evaluate.(s, mince(D, 16)))
+plot!(fig, Dt, t -> sup(R16), lab="N = 16", c=:orange, style=:dash)
+plot!(fig, Dt, t -> inf(R16), c=:orange, lab="", style=:dash)
Example block output

Internal representation

Consider again the Taylor model variable from the Taylor model variables and range bounding example.

t
 [1, 1] t + [0, 0]

Such constructor is an alias for

    TaylorModel1(x0 + Taylor1(eltype(x0), ord), zero(dom), interval(x0), dom)

Taylor models in one variable are internally represented using four fields: a Taylor series (pol) in one variable that holds the polynomial approximation of order ord; the interval remainder (rem); the expansion point (x0), and the interval domain of interest (dom). Getter functions are defined for each of these fields:

get_order(t)
3
remainder(t)
[0, 0]
polynomial(t)
 [1, 1] t + 𝒪(t⁴)
domain(t)
[-0.5, 0.5]
expansion_point(t)
[0, 0]

Finally, note that the Taylor model type has two parameters, T and S. The first parameter, T, refers to the numeric type of the coefficients of the polynomial, in this case an interval with double precision floating point values (Interval{Float64}). The second parameter, S, refers to the numeric type of the interval that holds the remainder, expansion point and domain of interest (in this case Float64).

typeof(t)
TaylorModel1{Interval{Float64}, Float64}

If we had defined the expansion point using 0.0 instead of interval(0), the coefficients of (the polynomial part of) this Taylor model variable would be floats instead of intervals.

z = TaylorModel1(3, 0.0, -0.5..0.5)
 1.0 t + [0, 0]
typeof(z)
TaylorModel1{Float64, Float64}
polynomial(z)
 1.0 t + 𝒪(t⁴)

Using a polynomial with interval coefficients guarantees that all arithmetic operations involving t are conservative, or rigorous, with respect to floating point arithmetic.

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For var ≥ 1, this function evaluates the flowpipes at the split domain intervals, which is useful to decrease the overapproximations associated to the whole time domain.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.picard_iteration-NTuple{7, Any}","page":"API","title":"TaylorModels.picard_iteration","text":"picard_iteration(f!, dx, xTM1K, params, t, x0, box)\n\nComputes the picard (integral) operator for the set of equations f! and the initial conditionx0. This method returns the remainder of the resulting Taylor Model without the remainder of the initial condition.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.picard_iteration-Tuple{Any, Any, Any, Any, Any, Any, Any, Val{true}}","page":"API","title":"TaylorModels.picard_iteration","text":"picard_iteration(f!, dx, xTM1K, params, t, x0, box, ::Val{true})\n\nComputes the picard (integral) operator for the set of equations f! and the initial conditionx0. The Val parameter enables the selection of the desired method. This one returns the remainder of the resulting Taylor Model with the remainder of the initial condition included.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.picard_remainder!-Union{Tuple{T}, Tuple{N}, Tuple{Function, Taylor1{T}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{TaylorN{Interval{T}}}, 1}, Array{Taylor1{TaylorN{Interval{T}}}, 1}, IntervalBox{N, T}, Interval{T}, IntervalBox{N, T}, IntervalBox{N, T}, Any}} where {N, T}","page":"API","title":"TaylorModels.picard_remainder!","text":"picard_remainder!(f!, t, x, dx, xxI, dxxI, δI, δt, Δx, Δ0, params)\n\nReturn the remainder of Picard operator\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.quadratic_fast_bounder-Tuple{TaylorModel1}","page":"API","title":"TaylorModels.quadratic_fast_bounder","text":"quadratic_fast_bounder(fT::TaylorModel1)\n\nCompute a tighter polynomial bound by the quadratic fast bounder. The returned bound corresponds to the \"improved\" polynomial bound with the remainder of the TaylorModel1 included. This \"improved\" bound can be one of the following two: 1) An improved bound: if the domain of fT has a local minimizer, then an improved bound is returned. 2) Original TaylorModel bound: if the local minimizer criteria is not satisfied, then the original bound of fT is returned.\n\nThis algorithm is a slight modification to the Makino & Berz algorithm. For this algorithm the linear part is bounded by solving a simple set of linear equations (compared to the iterative procedure by Makino & Berz).\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.quadratic_fast_bounder-Tuple{TaylorModelN}","page":"API","title":"TaylorModels.quadratic_fast_bounder","text":"quadratic_fast_bounder(fT::TaylorModelN)\n\nCompute a tighter polynomial bound by the quadratic fast bounder. The returned bound corresponds to the \"improved\" polynomial bound with the remainder of the TaylorModelN included. This \"improved\" bound can be one of the following two: 1) An improved bound: if the domain of fT has a local minimizer, then an improved bound is returned. 2) Original TaylorModel bound: if the local minimizer criteria is not satisfied, then the original bound of fT is returned.\n\nThis algorithm is a slight modification to the Makino & Berz algorithm. For this algorithm the linear part is bounded by solving a simple set of linear equations (compared to the iterative procedure by Makino & Berz).\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.remainder_taylorstep!-Union{Tuple{T}, Tuple{N}, Tuple{Function, Taylor1{T}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{Interval{T}}, 1}, Array{Taylor1{Interval{T}}, 1}, IntervalBox{N, T}, Interval{T}, Any}} where {N, T}","page":"API","title":"TaylorModels.remainder_taylorstep!","text":"remainder_taylorstep!(f!, t, x, dx, xI, dxI, δI, δt, params)\n\nReturns a remainder for the integration step for the dependent variables (x) checking that the solution satisfies the criteria for existence and uniqueness.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.rpa","page":"API","title":"TaylorModels.rpa","text":"rpa(g::Function, tmf::TaylorModel1) rpa(g::Function, tmf::RTaylorModel1) rpa(g::Function, tmf::TaylorModelN)\n\nRigurous polynomial approximation (RPA) for the function g using the Taylor Model with absolute/relative remainder tmf. The bound is computed exploiting monotonicity if possible, otherwise, it uses Lagrange bound.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorModels.shrink_wrapping!-Union{Tuple{Array{TaylorModelN{N, T, T}, 1}}, Tuple{T}, Tuple{N}} where {N, T}","page":"API","title":"TaylorModels.shrink_wrapping!","text":"shrink_wrapping!(xTMN::TaylorModelN)\n\nReturns a modified inplace xTMN, which has absorbed the remainder by the modified shrink-wrapping method of Florian Bünger. The domain of xTMN is the normalized interval box [-1,1]^N.\n\nRef: Florian B\"unger, Shrink wrapping for Taylor models revisited, Numer Algor 78:1001–1017 (2018), https://doi.org/10.1007/s11075-017-0410-1\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.truncate_taylormodel-Tuple{RTaylorModel1, Integer}","page":"API","title":"TaylorModels.truncate_taylormodel","text":"truncate_taylormodel(a::RTaylorModel1, m::Integer)\n\nTruncates a::RTaylorModel1 to order m.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorModels.validated_step!-Union{Tuple{T}, Tuple{N}, Tuple{Any, Taylor1{T}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{TaylorN{T}}, 1}, Taylor1{T}, Array{Taylor1{Interval{T}}, 1}, Array{Taylor1{Interval{T}}, 1}, Array{Taylor1{Interval{T}}, 1}, T, T, Int64, Array{TaylorModelN{N, T, T}, 1}, Array{Interval{T}, 1}, IntervalBox{N, T}, IntervalBox{N, T}, Int64, T, Any, Bool, T, Bool}, Tuple{Any, Taylor1{T}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{TaylorN{T}}, 1}, Array{Taylor1{TaylorN{T}}, 1}, Taylor1{T}, Array{Taylor1{Interval{T}}, 1}, Array{Taylor1{Interval{T}}, 1}, Array{Taylor1{Interval{T}}, 1}, T, T, Int64, Array{TaylorModelN{N, T, T}, 1}, Array{Interval{T}, 1}, IntervalBox{N, T}, IntervalBox{N, T}, Int64, T, Any, Bool, T, Bool, Function}} where {N, T}","page":"API","title":"TaylorModels.validated_step!","text":"validated-step!\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.integrate","page":"API","title":"TaylorSeries.integrate","text":"integrate(a::TM{T,S}, c0)\nintegrate(a::TM{TaylorN{T},S}, c0, cc0)\n\nIntegrates the one-variable Taylor Model (TaylorModel1 or RTaylorModel1) with respect to the independent variable. c0 is the integration constant; if omitted it is taken as zero. When the coefficients of a are TaylorN variables, the domain is specified by cc0::IntervalBox.\n\n\n\nintegrate(fT, which)\n\nIntegrates a fT::TaylorModelN with respect to which variable. The returned TaylorModelN corresponds to the Taylor Model of the definite integral ∫f(x) - ∫f(expansion_point).\n\n\n\n\n\n","category":"function"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"DocTestSetup = :(using TaylorModels)\nCurrentModule = TaylorModels","category":"page"},{"location":"range_bounding/#Taylor-model-variables-and-range-bounding","page":"Range bounding","title":"Taylor model variables and range bounding","text":"","category":"section"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Taylor models can be applied to the problem of range bounding, that is to find an interval I subseteq mathbbR such that f(x) in I on a given domain x in D (including possible floating point errors, see examples below). There are several ways to construct a Taylor model. A convenient way is to define a \"Taylor model variable\", which is then passed as argument to Julia functions. The following examples should help to clarify this method. To simplify the presentation we have only considered the univariate case, but this package can also handle multivariate Taylor models with the TaylorModelN type.","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Here we construct a Taylor model variable specifying that:","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"The truncation order is 3.\nThe expansion is around the origin (interval(0)).\nThe domain is the real interval centered around the origin -05 05.","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"using TaylorModels\n\nt = TaylorModel1(3, interval(0), -0.5..0.5)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Here the polynomial part is (the interval) 1, and the remainder is zero. We can pass this Taylor model variable to any Julia function, for example:","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"texp = exp(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"This expression is a polynomial of order 3 (in agreement with the truncation order specified in the construction of t), whose coefficients are intervals that are guaranteed to contain the exact coefficient of the Taylor expansion of the function t mapsto e^t in D -05 05 subset mathbbR. Similarly, we can expand trigonometric functions such as t mapsto sin(t):","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"tsin = sin(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Common arithmetic operators, such as addition (+) and multiplication (*) work with Taylor model variables out-of-the-box:","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"s = texp + tsin","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"p = texp * tsin","category":"page"},{"location":"range_bounding/#One-dimensional-bounding","page":"Range bounding","title":"One-dimensional bounding","text":"","category":"section"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"To bound the range of a Taylor model in one variable, use the function bound_taylor1:","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"using TaylorModels: bound_taylor1\n\n[bound_taylor1(x) for x in [s, p]]","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"This shows in particular that 012499 leq e^t + sin(t) leq 212501 and that -0291667 leq e^t sin(t) leq 0791667 for all t in D. Such bounds are in general not tight.","category":"page"},{"location":"range_bounding/#Mincing","page":"Range bounding","title":"Mincing","text":"","category":"section"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"If desired, the common approach to improve the bounds is to evaluate the Taylor model on a smaller interval, e.g.","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"D = domain(s) # domain -0.5 .. 0.5\n\nE = evaluate(s, D) # original, no mincing","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Dm = mince(D, 8) # split the domain into 8 smaller chunks\n\nEm = evaluate.(s, Dm) # evaluate the Taylor model on each sub-domain\n\nRm = reduce(hull, Em) # take the convex hull, i.e. the smallest interval that contains them all","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Here the lower bound has been improved by mincing (or splitting) the domain, and it may improve by repeating such operation recursively on smaller domains. In particular, the fact that the lower bound is greater than zero constitutes an algorithmic proof that s t mapsto e^t + sin(t) is positive on D. Let's visualize the function s(t) and the bounds obtained so far.","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"using Plots\n\nDt = range(-0.5, 0.5, length=100)\n\nfig = plot(xlab=\"t\", ylab=\"s(t)\", legend=:topleft)\nplot!(fig, Dt, t -> exp(t) + sin(t), lab=\"\", c=:black)\n\n# range bounds\nplot!(fig, Dt, t -> sup(E), lab=\"N = 1\", c=:blue, style=:dash)\nplot!(fig, Dt, t -> inf(E), c=:blue, lab=\"\", style=:dash)\n\nplot!(fig, Dt, t -> sup(Rm), lab=\"N = 8\", c=:red, style=:dash)\nplot!(fig, Dt, t -> inf(Rm), c=:red, lab=\"\", style=:dash)\n\nR16 = reduce(hull, evaluate.(s, mince(D, 16)))\nplot!(fig, Dt, t -> sup(R16), lab=\"N = 16\", c=:orange, style=:dash)\nplot!(fig, Dt, t -> inf(R16), c=:orange, lab=\"\", style=:dash)","category":"page"},{"location":"range_bounding/#Internal-representation","page":"Range bounding","title":"Internal representation","text":"","category":"section"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Consider again the Taylor model variable from the Taylor model variables and range bounding example.","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"t","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Such constructor is an alias for","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":" TaylorModel1(x0 + Taylor1(eltype(x0), ord), zero(dom), interval(x0), dom)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Taylor models in one variable are internally represented using four fields: a Taylor series (pol) in one variable that holds the polynomial approximation of order ord; the interval remainder (rem); the expansion point (x0), and the interval domain of interest (dom). Getter functions are defined for each of these fields:","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"get_order(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"remainder(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"polynomial(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"domain(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"expansion_point(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Finally, note that the Taylor model type has two parameters, T and S. The first parameter, T, refers to the numeric type of the coefficients of the polynomial, in this case an interval with double precision floating point values (Interval{Float64}). The second parameter, S, refers to the numeric type of the interval that holds the remainder, expansion point and domain of interest (in this case Float64).","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"typeof(t)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"If we had defined the expansion point using 0.0 instead of interval(0), the coefficients of (the polynomial part of) this Taylor model variable would be floats instead of intervals.","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"z = TaylorModel1(3, 0.0, -0.5..0.5)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"typeof(z)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"polynomial(z)","category":"page"},{"location":"range_bounding/","page":"Range bounding","title":"Range bounding","text":"Using a polynomial with interval coefficients guarantees that all arithmetic operations involving t are conservative, or rigorous, with respect to floating point arithmetic.","category":"page"},{"location":"#TaylorModels.jl","page":"Home","title":"TaylorModels.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package combines the IntervalArithmetic.jl and TaylorSeries.jl packages to provide Taylor models, i.e. Taylor polynomials with guaranteed error bounds to approximate functions.","category":"page"},{"location":"","page":"Home","title":"Home","text":"An introduction is available in this video from JuliaCon 2018.","category":"page"},{"location":"#Introduction-to-Taylor-Models","page":"Home","title":"Introduction to Taylor Models","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Taylor models provide a way to rigorously manipulate and evaluate functions using floating-point arithmetic. They have been widely used for validated computing: in global optimization and range bounding, for validated solutions of ODEs, rigorous quadrature, etc.","category":"page"},{"location":"","page":"Home","title":"Home","text":"A Taylor model (TM) of order n for a function f which is supposed to be n + 1 times continuously differentiable over an interval ab, is a rigorous polynomial approximation of f. Specifically, it is a couple (P Delta) formed by a polynomial P of degree n, and an interval part Delta, such that f(x) P(x) in Delta, forall x ab. Roughly speaking, as their name suggests, the polynomial can be seen as a Taylor expansion of the function at a given point. The interval Delta (also called interval remainder) provides the validation of the approximation, meaning that it provides an enclosure of all the approximation errors encountered (truncation, roundings).","category":"page"},{"location":"","page":"Home","title":"Home","text":"Here we generate TMs of order 6 and 7 over I = -0510. We can view a TM as a a tube around the actual function.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using TaylorModels\n\nf(x) = x*(x-1.1)*(x+2)*(x+2.2)*(x+2.5)*(x+3)*sin(1.7*x+0.5)\na = -0.5 .. 1.0 # Domain\nx0 = mid(a) # Expansion point\ntm6 = TaylorModel1(6, interval(x0), a) # Independent variable for Taylor models, order 6\ntm7 = TaylorModel1(7, interval(x0), a) # Independent variable for Taylor models, order 7\n# Taylor models corresponding to f(x) of order 6 and 7\nftm6 = f(tm6)\nftm7 = f(tm7)\n\n# Now the plot\nusing Plots; gr()\nplot(range(inf(a), stop=sup(a), length=1000), x->f(x), label=\"f(x)\", lw=2, xaxis=\"x\", yaxis=\"f(x)\")\nplot!(ftm6, label=\"6th order\")\nplot!(ftm7, label=\"7th order\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: intro_plot)","category":"page"},{"location":"#Authors","page":"Home","title":"Authors","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)\nDavid P. Sanders, Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM)","category":"page"},{"location":"#Bibliography","page":"Home","title":"Bibliography","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Rigorous Polynomial Approximations and Applications, Mioara Maria Joldes, Ecole normale supérieure de lyon - ENS LYON (2011)","category":"page"},{"location":"#Acknowledgements","page":"Home","title":"Acknowledgements","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Financial support is acknowledged from DGAPA-UNAM PAPIIT grants IN-117117, IG-100616 and IG-100819. DPS acknowledges support through a Cátedra Marcos Moshinsky (2018).","category":"page"}] +} diff --git a/previews/PR166/siteinfo.js b/previews/PR166/siteinfo.js new file mode 100644 index 00000000..84b96211 --- /dev/null +++ b/previews/PR166/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR166";