On Thu, Aug 31, 2023 at 8:22 PM Shahriar Iravanian <[email protected]> wrote: > > Thank you very much for your comments. > > While hyint ansatz generation is similar in spirit to heurisch, it is less > rigorous and more > heuristic. One reason is that the numerical portion of hyint is much more > forgiving and > can filter out many undesirable ansatzes (by using QR decomposition to remove > linearly > dependent ones and sparse regression to solve the integral). Therefore, we > can generate > many different forms of ansatz, which allows hyint to solve integrals beyond > what > heurisch can do. > > For example, I use a set of 170 test cases (listed in test.py) to test hyint. > The current version > can solve 154 of them; heurisch solves 126 (of course, this list is curated > for hyint). > > Some integrals that hyint can solve and heurisch cannot: > > x/sqrt(x - 1) => 2*x*sqrt(x - 1)/3 + 4*sqrt(x - 1)/3 > 1/sqrt(x**2 + 4) => asinh(x/2) > sqrt(x)*log(x) => 2*x**(3/2)*log(x)/3 - 4*x**(3/2)/9 > log(log(x))/x => log(x)*log(log(x)) - log(x) > log(cos(x))*tan(x) => -log(cos(x))**2/2 > sin(x + 1)/(x + 1) => Si(x + 1) > sqrt(1 - sin(x)) => 2*cos(x)/sqrt(1 - sin(x))
By the way, a big problem with heurisch is that it's very sensitive to the way the derivative of the expression is represented, since that's how it generates the ansatz. This is fundamental to the approach, but an issue is that it just directly uses whatever sympy.diff() returns. This is a big problem for algebraic functions because they can be represented in multiple different ways, and that representation can affect the result of integration (a big piece of the "real" Risch algorithm for algebraic functions is normalizing things so that this isn't a problem). For example, the derivative of s = sqrt(x) can be represented as 1/(2*s) or s/(2*x). SymPy happens to use the former form, because of the way the core combines exponents, but there are integrals with square roots where heurisch can't find the answer but would if it used the latter form (not so much because the latter form is better, although in some ways it is since it's a polynomial in s, but usually just by dumb luck of the way it generates the ansatz). > > Some integrals that heurisch can solve and hyint cannot: > > exp(x)/(exp(2*x) - 1) => log(exp(x) - 1)/2 - log(exp(x) + 1)/2 > tan(x)**3 => -log(tan(x)**2 + 1)/2 + tan(x)**2/2 > > > Actually it wouldn't be much work to generalize this to something that > > gives an exact answer at least some of the time. > > I completely agree. I believe the main application of hyint is as an ansatz > discovery > engine rather than a standalone integrator. This is a line of research I'm > actively > pursuing and would appreciate help, comments, and collaboration! I like the idea of using some faster linear algebra as a preprocessor to reduce the size of the problem to be solved. You could then take the reduced problem and solve it with exact symbolic arithmetic. The worst that can happen is you might drop an ansatz that shouldn't have been dropped, but you would still always give a symbolically correct answer when you do. If heurisch could support much larger, possibly linearly dependent ansatz, then that would open up a lot of possibilities, like for instance, trying to use ansatz coming from both sqrt(x) and x/sqrt(x) simultaneously. But right now it's way too slow even with just the linear system it generates. I'm also wondering if you've thought about symbolic constants and if there's any tricks you could do to support them. I think there might be, especially when still thinking about things in terms of just using faster linear algebra as a pre-processor to a symbolic solve. Aaron Meurer > > -- Shahriar > > On Thursday, August 31, 2023 at 6:48:24 PM UTC-4 [email protected] wrote: >> >> On Thu, Aug 31, 2023 at 5:07 AM Oscar Benjamin >> <[email protected]> wrote: >> > >> > On Tue, 29 Aug 2023 at 07:09, Shahriar Iravanian <[email protected]> >> > wrote: >> > > >> > > Hi All, >> > > >> > >> > Hi Shahriar, >> > >> > > I have uploaded a new package, hyint, to github and PyPi. It is a hybrid >> > > (symbolic-numeric) integration package on top of SymPy and numpy/scipy. >> > > I would appreciate it if it is added to the list of projects using SymPy. >> > > >> > > You can find it at https://github.com/siravan/hyint or install it as >> > > 'pip install hyint'. >> > > >> > > hyint uses an ansatz generation algorithm similar to the Risch-Bronstein >> > > poor man's integrator combined with a sparse regression algorithm >> > > adopted from the Sparse identification of nonlinear dynamics (SINDy) to >> > > solve indefinite integration problems to univariate expressions. It can >> > > solve a large subset of standard elementary integrals despite a very >> > > small size (a few hundred lines of code). >> > > >> > > I'm one of the principal contributors to SymbolicNumericIntegartion.jl >> > > (https://github.com/SciML/SymbolicNumericIntegration.jl), which is a >> > > Julia package for symbolic-numeric integration and is the basis of hyint. >> > >> > This sounds excellent. Yes, you can add it to the list of projects >> > using SymPy using a pull request, I think to the website repo. >> > >> > How exactly is it different from Bronstein's poor man's integrator? >> > >> > SymPy has an integration algorithm called heurisch which is based on >> > the poor man's integrator but uses exact rather than approximate >> > solutions to the linear equations. Is exact vs approximate the main >> > distinction here between what hyint does and what heurisch does? >> >> Judging from the examples, it looks like it >> >> In: hyint.integrate(1 / (x**3 - 2*x + 1), x) >> Out: 2.17082039324994*log(x - 1) + 1.34164078649988*log(x + 1/2 + >> sqrt(5)/2) - 1.17082039324993*log(x**3 - 2*x + 1) >> >> And looking at the code, I'd say a difference is that the Bronstein >> algorithm is more rigorous and general in how it generates its ansatz. >> But actually one of the reasons heurisch() is so slow is that it can >> generate ansatz that are huge (especially when it's given an integrand >> that doesn't have an answer to begin with). Employing some heuristics >> like the ones used here in heurisch could help it out. >> >> Actually it wouldn't be much work to generalize this to something that >> gives an exact answer at least some of the time. For example, you >> could use nsimplify() to guess at exact values for numeric >> coefficients. As long as diff(answer) - integrand symbolically >> simplifies to 0 you know you have a correct answer. >> >> Aaron Meurer >> >> > >> > -- >> > Oscar >> > >> > -- >> > You received this message because you are subscribed to the Google Groups >> > "sympy" group. >> > To unsubscribe from this group and stop receiving emails from it, send an >> > email to [email protected]. >> > To view this discussion on the web visit >> > https://groups.google.com/d/msgid/sympy/CAHVvXxRyJ3FUZ1pN6GACMoHZBYGbUJAavKq5Y1RW5__1nFAT8w%40mail.gmail.com. > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/c0d397a1-640f-477e-9897-e1157bb6ca67n%40googlegroups.com. -- You received this message because you are subscribed to the Google Groups "sympy" group. 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