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move readme to docs
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‎.github/workflows/Documentation.yml‎

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push:
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branches:
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- main
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- 'release-'
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tags: '*'
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pull_request:
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‎README.md‎

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**SymbolicNumericIntegration.jl** is a hybrid symbolic/numerical integration package that works on the [Julia Symbolics](https://github.com/JuliaSymbolics/Symbolics.jl) expressions.
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**SymbolicNumericIntegration.jl** uses a randomized algorithm based on a hybrid of the *method of undetermined coefficients* and *sparse regression* and is able to solve a large subset of basic standard integrals (polynomials, exponential/logarithmic, trigonometric and hyperbolic, inverse trigonometric and hyperbolic, rational and square root).
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The basis of how it works and the theory of integration using the Symbolic-Numeric methods refer to [Basis of Symbolic-Numeric Integration](docs/theory.ipynb).
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Function `integrate` returns the integral of a univariate expression with *constant* real or complex coefficients. `integrate` returns a tuple with three values. The first one is the solved integral, the second one is the sum of the unsolved terms, and the third value is the residual error. If `integrate` is successful, the unsolved portion is reported as 0.
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```julia
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using Symbolics
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using SymbolicNumericIntegration
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julia> integrate(3x^3 + 2x - 5)
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(x^2 + (3//4)*(x^4) - (5x), 0, 0)
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julia> integrate((5 + 2x)^-1)
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((1//2)*log((5//2) + x), 0, 0.0)
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julia> integrate(1 / (6 + x^2 - (5x)))
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(log(x - 3) - log(x - 2), 0, 3.339372764128952e-16)
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y = integrate(1 / (x^2 - 16))
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((1//8)*log(x - 4) - ((1//8)*log(4 + x)), 0, 1.546926788028958e-16)
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julia> integrate(x^2/(16 + x^2))
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(x + 4atan((-1//4)*x), 0, 1.3318788420751984e-16)
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julia> integrate(x^2/sqrt(4 + x^2))
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((1//2)*x*((4 + x^2)^0.5) - ((2//1)*log(x + sqrt(4 + x^2))), 0, 8.702422633074313e-17)
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julia> integrate(x^2*log(x))
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((1//3)*log(x)*(x^3) - ((1//9)*(x^3)), 0, 0)
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julia> integrate(x^2*exp(x))
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(2exp(x) + exp(x)*(x^2) - (2x*exp(x)), 0, 0)
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julia> integrate(tan(2x))
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((-1//2)*log(cos(2x)), 0, 0)
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julia> integrate(sec(x)*tan(x))
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(cos(x)^-1, 0, 0)
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julia> integrate(cosh(2x)*exp(x))
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((2//3)*exp(x)*sinh(2x) - ((1//3)*exp(x)*cosh(2x)), 0, 7.073930088880992e-8)
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julia> integrate(cosh(x)*sin(x))
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((1//2)*sin(x)*sinh(x) - ((1//2)*cos(x)*cosh(x)), 0, 4.8956233716268386e-17)
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julia> integrate(cosh(2x)*sin(3x))
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(0.153845sinh(2x)*sin(3x) - (0.23077cosh(2x)*cos(3x)), 0, 4.9807620877373405e-6)
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julia> integrate(log(log(x))*(x^-1))
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(log(x)*log(log(x)) - log(x), 0, 0)
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julia> integrate(exp(x^2))
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(0, exp(x^2), Inf) # as expected!
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```
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`integrate` has the form `integrate(y; kw...)` or `integrate(y, x; kw...)`, where `y` is the integrand and the optional `x` is the variable of integration. The keyword parameters are:
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* `abstol` (default `1e-6`): the error tolerance to accept a solution.
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* `symbolic` (default `true`): if true, pure symbolic integration is attempted first.
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* `bypass` (default `false`): if true, the whole expression is considered at once and not per term.
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* `num_steps` (default `2`): one plus the number of expanded basis to check (if `num_steps` is 1, only the main basis is checked).
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* `num_trials` (default `5`): the number of attempts to solve the integration numerically for each basis set.
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* `show_basis` (default `false`): print the basis set, useful for debugging. Only works if `verbose` is also set.
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* `homotopy` (default: `true` as of version 0.7.0): uses the continuous Homotopy operators to generate the integration candidates.
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* `verbose` (default `false`): if true, prints extra (and voluminous!) debugging information.
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* `radius` (default `1.0`): the starting radius to generate random test points.
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* `opt` (default `STLSQ(exp.(-10:1:0))`): the optimizer passed to `sparse_regression!`.
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* `max_basis` (default `110`): the maximum number of expression in the basis.
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* `complex_plane` (default `true`): random test points are generated on the complex plane (only over the real axis if `complex_plane` is `false`).
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## Testing
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`test/runtests.jl` contains a test suite of 160 easy to moderate test integrals (can be run by calling `test_integrals`). Currently, **SymbolicNumericIntegration.jl** solves more than 90% of its test suite.
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Additionally, 12 test suites from the *Rule-based Integrator* ([Rubi](https://rulebasedintegration.org/)) are included in the `/test` directory. For example, we can test the first one as below ([Axiom](http://www.axiom-developer.org/) refers to the format of the test files)
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```julia
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using SymbolicNumericIntegration
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include("test/axiom.jl") # note, you may need to use the correct path
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L = convert_axiom(:Apostle) # can also use L = convert_axiom(1)
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test_axiom(L, false; bypass=false, verbose=false, homotopy=true)
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```
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The test suites description based on the header of the files in the Rubi directory are
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| name | id | comment |
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|-------------|----|------------------------------------------|
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|:Apostle | 1 | Tom M. Apostol - Calculus, Volume I, Second Edition (1967) |
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|:Bondarenko | 2 | Vladimir Bondarenko Integration Problems |
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|:Bronstein | 3 | Manuel Bronstein - Symbolic Integration Tutorial (1998) |
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|:Charlwood | 4 | Kevin Charlwood - Integration on Computer Algebra Systems (2008) |
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|:Hearn | 5 | Anthony Hearn - Reduce Integration Test Package |
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|:Hebisch | 6 | Waldek Hebisch - email May 2013 |
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|:Jeffrey | 7 | David Jeffrey - Rectifying Transformations for Trig Integration (1997) |
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|:Moses | 8 | Joel Moses - Symbolic Integration Ph.D. Thesis (1967) |
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|:Stewart | 9 | James Stewart - Calculus (1987) |
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|:Timofeev | 10 | A. F. Timofeev - Integration of Functions (1948) |
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|:Welz | 11 | Martin Welz - posts on Sci.Math.Symbolic |
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|:Webster | 12 | Michael Wester |
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## Citation
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If you use **SymbolicNumericIntegration.jl**, please cite [Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression](https://arxiv.org/abs/2201.12468):
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```
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@article{Iravanian2022,
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author = {Shahriar Iravanian and Carl Julius Martensen and Alessandro Cheli and Shashi Gowda and Anand Jain and Julia Computing and Yingbo Ma and Chris Rackauckas},
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doi = {10.48550/arxiv.2201.12468},
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month = {1},
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title = {Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression},
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url = {https://arxiv.org/abs/2201.12468v2},
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year = {2022},
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}
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```

‎docs/Project.toml‎

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[deps]
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SymbolicNumericIntegration = "78aadeae-fbc0-11eb-17b6-c7ec0477ba9e"
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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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[compat]
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Documenter = "0.27"

‎docs/make.jl‎

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using Documenter
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using SymbolicNumericIntegration
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using Documenter, SymbolicNumericIntegration
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include("pages.jl")
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makedocs(
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sitename="SymbolicNumericIntegration",
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format=Documenter.HTML(),
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modules=[SymbolicNumericIntegration]
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sitename="SymbolicNumericIntegration.jl",
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authors="Shahriar Iravanian",
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modules=[SymbolicNumericIntegration],
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clean=true,doctest=false,
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format = Documenter.HTML(analytics = "UA-90474609-3",
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assets = ["assets/favicon.ico"],
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canonical="https://symbolicnumericintegration.sciml.ai/stable/"),
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pages=pages
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)
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# Documenter can also automatically deploy documentation to gh-pages.
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# See "Hosting Documentation" and deploydocs() in the Documenter manual
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# for more information.
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deploydocs(
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repo="https://github.com/SciML/SymbolicNumericIntegration.jl";
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push_preview=true
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repo = "github.com/SciML/SymbolicNumericIntegration.jl.git";
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push_preview = true
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)

‎docs/pages.jl‎

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# Put in a separate page so it can be used by SciMLDocs.jl
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pages=[
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"Home" => "index.md",
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"symbolicnumericintegration.md"
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]

‎docs/src/assets/favicon.ico‎

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‎docs/src/assets/logo.png‎

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‎docs/src/index.md‎

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# SymbolicNumericIntegration.jl
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**SymbolicNumericIntegration.jl** is a hybrid symbolic/numerical integration package that works on the [Julia Symbolics](https://github.com/JuliaSymbolics/Symbolics.jl) expressions.
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Documentation for SymbolicNumericIntegration.jl
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**SymbolicNumericIntegration.jl** uses a randomized algorithm based on a hybrid of the *method of undetermined coefficients* and *sparse regression* and is able to solve a large subset of basic standard integrals (polynomials, exponential/logarithmic, trigonometric and hyperbolic, inverse trigonometric and hyperbolic, rational and square root).
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The basis of how it works and the theory of integration using the Symbolic-Numeric methods refer to [Basis of Symbolic-Numeric Integration](docs/theory.ipynb).
6+
7+
Function `integrate` returns the integral of a univariate expression with *constant* real or complex coefficients. `integrate` returns a tuple with three values. The first one is the solved integral, the second one is the sum of the unsolved terms, and the third value is the residual error. If `integrate` is successful, the unsolved portion is reported as 0.
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```julia
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using Symbolics
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using SymbolicNumericIntegration
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@variables x
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julia> integrate(3x^3 + 2x - 5)
16+
(x^2 + (3//4)*(x^4) - (5x), 0, 0)
17+
18+
julia> integrate((5 + 2x)^-1)
19+
((1//2)*log((5//2) + x), 0, 0.0)
20+
21+
julia> integrate(1 / (6 + x^2 - (5x)))
22+
(log(x - 3) - log(x - 2), 0, 3.339372764128952e-16)
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24+
y = integrate(1 / (x^2 - 16))
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((1//8)*log(x - 4) - ((1//8)*log(4 + x)), 0, 1.546926788028958e-16)
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julia> integrate(x^2/(16 + x^2))
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(x + 4atan((-1//4)*x), 0, 1.3318788420751984e-16)
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julia> integrate(x^2/sqrt(4 + x^2))
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((1//2)*x*((4 + x^2)^0.5) - ((2//1)*log(x + sqrt(4 + x^2))), 0, 8.702422633074313e-17)
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julia> integrate(x^2*log(x))
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((1//3)*log(x)*(x^3) - ((1//9)*(x^3)), 0, 0)
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julia> integrate(x^2*exp(x))
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(2exp(x) + exp(x)*(x^2) - (2x*exp(x)), 0, 0)
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julia> integrate(tan(2x))
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((-1//2)*log(cos(2x)), 0, 0)
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julia> integrate(sec(x)*tan(x))
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(cos(x)^-1, 0, 0)
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julia> integrate(cosh(2x)*exp(x))
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((2//3)*exp(x)*sinh(2x) - ((1//3)*exp(x)*cosh(2x)), 0, 7.073930088880992e-8)
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julia> integrate(cosh(x)*sin(x))
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((1//2)*sin(x)*sinh(x) - ((1//2)*cos(x)*cosh(x)), 0, 4.8956233716268386e-17)
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julia> integrate(cosh(2x)*sin(3x))
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(0.153845sinh(2x)*sin(3x) - (0.23077cosh(2x)*cos(3x)), 0, 4.9807620877373405e-6)
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julia> integrate(log(log(x))*(x^-1))
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(log(x)*log(log(x)) - log(x), 0, 0)
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julia> integrate(exp(x^2))
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(0, exp(x^2), Inf) # as expected!
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```
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`integrate` has the form `integrate(y; kw...)` or `integrate(y, x; kw...)`, where `y` is the integrand and the optional `x` is the variable of integration. The keyword parameters are:
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* `abstol` (default `1e-6`): the error tolerance to accept a solution.
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* `symbolic` (default `true`): if true, pure symbolic integration is attempted first.
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* `bypass` (default `false`): if true, the whole expression is considered at once and not per term.
66+
* `num_steps` (default `2`): one plus the number of expanded basis to check (if `num_steps` is 1, only the main basis is checked).
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* `num_trials` (default `5`): the number of attempts to solve the integration numerically for each basis set.
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* `show_basis` (default `false`): print the basis set, useful for debugging. Only works if `verbose` is also set.
69+
* `homotopy` (default: `true` as of version 0.7.0): uses the continuous Homotopy operators to generate the integration candidates.
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* `verbose` (default `false`): if true, prints extra (and voluminous!) debugging information.
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* `radius` (default `1.0`): the starting radius to generate random test points.
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* `opt` (default `STLSQ(exp.(-10:1:0))`): the optimizer passed to `sparse_regression!`.
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* `max_basis` (default `110`): the maximum number of expression in the basis.
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* `complex_plane` (default `true`): random test points are generated on the complex plane (only over the real axis if `complex_plane` is `false`).
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## Testing
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`test/runtests.jl` contains a test suite of 160 easy to moderate test integrals (can be run by calling `test_integrals`). Currently, **SymbolicNumericIntegration.jl** solves more than 90% of its test suite.
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Additionally, 12 test suites from the *Rule-based Integrator* ([Rubi](https://rulebasedintegration.org/)) are included in the `/test` directory. For example, we can test the first one as below ([Axiom](http://www.axiom-developer.org/) refers to the format of the test files)
81+
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```julia
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using SymbolicNumericIntegration
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include("test/axiom.jl") # note, you may need to use the correct path
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L = convert_axiom(:Apostle) # can also use L = convert_axiom(1)
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test_axiom(L, false; bypass=false, verbose=false, homotopy=true)
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```
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The test suites description based on the header of the files in the Rubi directory are
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| name | id | comment |
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|-------------|----|------------------------------------------|
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|:Apostle | 1 | Tom M. Apostol - Calculus, Volume I, Second Edition (1967) |
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|:Bondarenko | 2 | Vladimir Bondarenko Integration Problems |
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|:Bronstein | 3 | Manuel Bronstein - Symbolic Integration Tutorial (1998) |
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|:Charlwood | 4 | Kevin Charlwood - Integration on Computer Algebra Systems (2008) |
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|:Hearn | 5 | Anthony Hearn - Reduce Integration Test Package |
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|:Hebisch | 6 | Waldek Hebisch - email May 2013 |
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|:Jeffrey | 7 | David Jeffrey - Rectifying Transformations for Trig Integration (1997) |
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|:Moses | 8 | Joel Moses - Symbolic Integration Ph.D. Thesis (1967) |
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|:Stewart | 9 | James Stewart - Calculus (1987) |
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|:Timofeev | 10 | A. F. Timofeev - Integration of Functions (1948) |
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|:Welz | 11 | Martin Welz - posts on Sci.Math.Symbolic |
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|:Webster | 12 | Michael Wester |
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## Citation
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If you use **SymbolicNumericIntegration.jl**, please cite [Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression](https://arxiv.org/abs/2201.12468):
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```@autodocs
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Modules = [SymbolicNumericIntegration]
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```
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@article{Iravanian2022,
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author = {Shahriar Iravanian and Carl Julius Martensen and Alessandro Cheli and Shashi Gowda and Anand Jain and Julia Computing and Yingbo Ma and Chris Rackauckas},
114+
doi = {10.48550/arxiv.2201.12468},
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month = {1},
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title = {Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression},
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url = {https://arxiv.org/abs/2201.12468v2},
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year = {2022},
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}
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```

‎docs/src/symbolicnumericintegration.md‎

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# API
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```@docs
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integrate
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```

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