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Show HN: Rust library for numerical integration of real-valued functions
Integrate is a fast, small, lightweight Rust library for performing numerical integration of real-valued functions. It is designed to integrate functions, providing a simple and efficient way to approximate definite integrals using various numerical methods.
Integrate supports a variety of numerical integration techniques:
- Newton-Cotes methods:
- Rectangle Rule.
- Trapezoidal Rule.
- Simpson's Rule.
- Newton's 3/8 Rule.
- Gauss quadrature methods:
- Gauss-Legendre.
- Gauss-Laguerre.
- Gauss-Hermite.
- Gauss-Chebyshev First Kind.
- Gauss-Chebyshev Second Kind.
- Adaptive Methods:
- Adaptive Simpson's method
- Romberg’s method.
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- JanisErdmanis 2y agoIt looks a bit sloppy to hardcode so many constants in a single file: `src/gauss_quadrature/legendre.rs`. Isn't it possible to generate them with the help of rust macros in the same way Julia uses metaprogramming?
- ok123456 2y agoGaussian quadrature points are typically solved numerically. There's a good chance these ultimately came from a table. Additionally, compile time floating-point evaluation is limited. When I looked around recently, I didn't see a rust equivalent of gcem; any kind of transcendental function evaluation (which finding Gaussian quadrature points absolutely would require) would not allow compile-time evaluation.
- AlotOfReading 2y agoSupport for float const fns was merged just a couple months ago and hasn't been officially announced yet.
- ok123456 2y agoIIRC, that only supports elementary arithmetic operations. Useful but not general.
- AlotOfReading 2y agoIt's relatively straightforward to build transcendental functions out of the basic operations and the stdlib support will eventually get there, but rust's float story is still a work in progress. They're trying to do things more properly and document semantics better than C and C++ have.
- BD103 2y agoSupport for constant float operations was released in Rust 1.82! https://blog.rust-lang.org/2024/10/17/Rust-1.82.0.html https://blog.rust-lang.org/2024/10/17/Rust-1.82.0.html
- zokier 2y agoI was under the impression that macros can execute arbitrary code, surely some FP would not be big problem. And if not macros then build.rs script certainly could do the trick.
- dhosek 2y agobuild.rs can definitely execute arbitrary code, which means that a lot of places (including, IIRC crates.io) will forbid crates that rely on build.rs. I ended up refactoring my build.rs into a separate sub-application in finl_unicode that built data tables which are then checked into git and used pre-built. I include the sub-app in the source code so that anyone with access to the repo could continue development if I were to die tomorrow.
- n_plus_1_acc 2y agoThere are many crates with build.rs scripts on crates.io, because they host just the source code with the Cargo.{toml,lock}.
- dhosek 2y agoI ran into some issues with crates.io and my build.rs when I first released the crate, although it’s long enough ago, that I don’t remember exactly what the issue was. It might also have been that the build.rs script downloaded raw data files from unicode.org
- Arnavion 2y agocrates.io doesn't care what your build.rs does because it doesn't try to compile your code, neither now or ever in the past. There would be no point in it trying to compile your code; there are lots of crates that are bindings to C libraries that crates.io's builders can't be expected to have, crates that target architectures which crates.io can't be expected to have builders for, etc.
- two_handfuls 2y agoProbably but that would slow down compilation a lot.
- simlevesque 2y agoExactly, it's not like the constants are gonna change.
- blharr 2y agoYou wouldn't have to recompile them every time. What if you didn't necessarily use macros but auto-generated it in a file that you keep separate from the other code at the bottom?
- dataflow 2y agoWhat I would do in these cases is to define the general computation function, but special-case it to return the hard-coded value for specific common inputs if it's being evaluated at compile time. Then add a test to verify both behaviors.
- mtantaoui 2y agothis was mainly to use an Iteration free method in this paper: https://www.cfm.brown.edu/faculty/gk/APMA2560/Handouts/GL_quad_Bogaert_2014.pdf https://www.cfm.brown.edu/faculty/gk/APMA2560/Handouts/GL_qu... this method is much faster and simpler.
- wjholden 2y agoI was always amazed that R can do: > integrate(dnorm, -Inf, +Inf) 1 with absolute error < 9.4e-05 Can we do the same in this library?
- Buttons840 2y agoHow many evaluations of the underlying function does it make? (Hoping someone will fire up their R interpreter and find out.) Or, probably, dnorm is a probability distribution which includes a likeliness function, and a cumulative likeliness function, etc. I bet it doesn't work on arbitrary functions.
- thrasibule 2y agoR integrate is just a wrapper around quadpack. It works with arbitrary functions, but arguably dnorm is pretty well behaved.
- legobmw99 2y agoIt seems like it is lacking the functionality R's integrate has for handling infinite boundaries, but I suppose you could implement that yourself on the outside. For what it's worth, use integrate::adaptive_quadrature::simpson::adaptive_simpson_method; use statrs::distribution::{Continuous, Normal}; fn dnorm(x: f64) -> f64 { Normal::new(0.0, 1.0).unwrap().pdf(x) } fn main() { let result = adaptive_simpson_method(dnorm, -100.0, 100.0, 1e-2, 1e-8); println!("Result: {:?}", result); } prints Result: Ok(1.000000000053865) It does seem to be a usability hazard that the function being integrated is defined as a fn, rather than a Fn, as you can't pass closures that capture variables, requiring the weird dnorm definition
- antononcube 2y agoYou will be completely blown away, then, from what Wolfram Language (aka Mathematica) can do. (When it comes to numerical integration.) https://reference.wolfram.com/language/tutorial/NIntegrateOverview.html https://reference.wolfram.com/language/tutorial/NIntegrateOv...
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- Philpax 2y agoUm... clicking through to each integration method, they clearly have examples and documentation on how they work: https://docs.rs/integrate/latest/integrate/adaptive_quadrature/simpson/fn.adaptive_simpson_method.html https://docs.rs/integrate/latest/integrate/adaptive_quadratu... https://docs.rs/integrate/latest/integrate/gauss_quadrature/legendre/fn.legendre_rule.html https://docs.rs/integrate/latest/integrate/gauss_quadrature/... https://docs.rs/integrate/latest/integrate/romberg/fn.romberg_method.html https://docs.rs/integrate/latest/integrate/romberg/fn.romber...
- antononcube 2y agoI think the project should be called "NIntegrate". BTW, that is not a serious suggestion; it is just that Wolfram Language (aka Mathematica) has both `Integrate` and `NIntegrate` for symbolic and numeric integration, respectively.
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- antononcube 2y agoThanks for showing this! It is very motivating to develop (and finish) my Raku numerical integration project.
- mtantaoui 2y agoThanks! That’s awesome to hear—I’d love to see how your Raku numerical integration project turns out! You can email me if you want to, I'll be happy to help.
- cozzyd 2y agoI don't see any explicit SIMD in here. Is the rust compiler able to emit SIMD instructions automatically in cases like this? (I guess I could compile and disassemble to check... )
- jvanderbot 2y agoIn my experience Rust is very good about using simd for loading and not great at using it automatically for math. This is from some experimentation at work and checking disassembly so ymmv There are common library extensions for that.
- cozzyd 2y agoymmv Or zmmv if avx-512 is supported?
- jvanderbot 2y agoThis is a good joke, if anyone is wondering.
- zppln 2y agoFast, small, lightweight compared to what?
- akkad33 2y agoThose are adjectives not comparatives.
- aDyslecticCrow 2y agoIt's a minimal implementation (small, lightweight) of well-known and established algorithms (fast).
- wiz21c 2y agoIn the rectangle method, there is "let x = a + i * h + (h / F1::from(2)...)" I didn't check, but I wonder if it is not better to do x = a + (i+0.5)*h... My reasoning is that if "i" is very big, then i * h can be much bigger than h/2 and thus h/2 may be eaten by numerical precision when added to i*h... And then you have to convince the optimizer to do it the way you want. But well, it's late...
- zackangelo 2y ago> Does the function oscillate over the region of integration? If it does, then make sure that the step size is chosen to be smaller than the wave length of the function. Nyquist limit, but for numerical integration?
- im3w1l 2y agoWell in both cases it comes down to aliasing I think (high frequency wave presents as low frequency wave with too big step size)
- dawnofdusk 2y agoThe Nyquist sampling theorem is of course proved by considering a Fourier transform, which is given by an integral, so the relation to integration in general should not be surprising.
- alleycat5000 2y agoOne way to think about is that these techniques work by integrating exactly the polynomial that interpolates the function where you're sampling it, so you need to resolve the features of the function to get good accuracy.
- legobmw99 2y agoIs there a technical reason to now allow closures as the integrand?
- n_plus_1_acc 2y agoMayve because they aren't guaranteed to be actual functions (in the mathematical sense) and could return random values
- legobmw99 2y agoThe Fn trait could be used, which prevents mutation, but allows a lot of useful closures. I should note, a motivated user could provide a junk function no matter what the type accepted is