Talk:Risch algorithm

Constant undecidability

I'm removing the following statement from the page because I'm not sure it's true:

The Risch decision procedure is not formally an algorithm because it requires an oracle that decides whether a constant expression is zero, a problem shown by Daniel Richardson to be undecidable.

According to MathWorld, Richardson's theorem states:

Let $R$ be the class of expressions generated by

1. The rational numbers and the two real numbers $\pi$ and $\ln 2$,
2. The variable x,
3. The operations of addition, multiplication, and composition, and
4. The sine, exponential, and absolute value functions.

Then if $E$ in $R$, the predicate $E=0$ is recursively undecidable.

This is based on a simple algebraic system that includes, among other things, an absolute value function. This implies an ordered field, while Risch integration is typically (always?) done over an unordered field. And this seems to make a big difference! My logic goes as follows:

1. Any constant expression involving nothing but rational numbers and the standard field operators (+ - * /) should be decidable. In sums and differences, multiply denominators by common multiples to make them equal, then combine fractions, throw away the denominator, and test for zero on the numerator. Multiplication is even simpler — just multiple the numerators and denominators seperately, and for division just interchange numerator and denominator.
2. Any simple algebraic extension over a field can be reduced using Euclidean long division to testing if a remainder in the underlying field is zero.
3. Any simple transcendental extension over a field requires all coefficients to be identically zero — each is testable in the underlying field.

Therefore, any algebraic system that consists purely of the rationals plus a finite number of algebraic and/or transcendental extensions should be decidably testable for equality to zero. This is the world of the Risch algorithm.

And the presence of the absolute value function makes all the difference for the Richardson theorem, right?

Baccala@freesoft.org 05:49, 14 January 2006 (UTC)

Now that Baccala@freesoft.org mentions it, it does make sense. I wrote that line in because I remember reading it in one of the introductory articles on computer algebra, but I have never read Richardson's result. These things can be tricky, and given that I am not a professional in computer algebra, I agree with the removal. XaosBits 17:46, 14 January 2006 (UTC)

I found the reference. It's from Moses (of Macsyma fame) paper, Symbolic integration: the stormy decade, in Communications of the ACM, volume 14. The quote about deciding if an expression is zero in on page 557. XaosBits

OK, thanks, I've looked over that reference now. It would seem to me that 1) I was wrong, and 2) the original statement was not entirely correct. My argument above only works if you can figure out if an extension is algebraic or transcendental to begin with, which seems to be the problem. Moses says "There exists no known general algorithm for determining whether a constant involving exponentials and logarithms is 0" (p. 557). But I still don't think Richardson's theorem applies here — there might be such an algorithm; I don't think it's been proven undecidable. I'll go read Richardson's paper in detail before I say more.

Baccala@freesoft.org 21:49, 26 January 2006 (UTC)

I think the following statement applies, and is confusingly written:

Also, the Risch algorithm is not an "algorithm" literally, because it needs as a part to check if some expression is equivalent to zero. And for a common meaning of what an "elementary function" is it's not known whether such an algorithm exists or not

That sentence needs to be put into the intro if its correct, it needs to be sourced, and if its true we need to stop calling it an algorithm. Fresheneesz (talk) 07:40, 18 November 2008 (UTC)

Basically, these problems arise in the constant field. So you can have some function c*f(x), and c == 0 will be undecidable. Therefore, the Risch Algorithm requires that the constant field be computable. For example, for Q, the rational numbers it is decidable if a == 0, but this also the case, for example, in Q(a), the field or rational functions on a with rational coefficients, where a is some variable not dependent on x, the integration variable, or even Q(a1, a2, …, an), where the ai do not depend on each other or on x. The actual result of course does not apply to the function field, because abs() is non-elementary. Even if you get a function that is really 0, and try to Risch integrate it, you will get as your result a function that is really an element of the constant field. For example, the integral of sin(x)^2 + cos(x)^2 - 1 is (integrating term wise), x/2 - sin(x)*cos(x)/2 + sin(x)*cos(x)/2 + x/2 - x, which is of course 0. But the algorithm never needed to know or care that the integrand was really identically 0 to get to that. Also, you should know that basic things like the division algorithm, which are essential to algorithms like the Risch Algorithm or even Euclid's gcd algorithm, do not work correctly if they cannot determine zero-equivalence in their coefficients. --Asmeurer (talk ♬ contribs) 03:37, 17 July 2010 (UTC)

"An important consequence of Risch's result is that the Gaussian integral has no elementary antiderivative." Actually, this was proven by Liouville years before Risch. —Preceding unsigned comment added by 82.244.80.154 (talk) 20:22, 5 October 2010 (UTC)

Possibly an error?

Shouldn't the first equation be g = ... (instead of f = ...)? Ariel

Looks that way---I've changed it to g. Michael Hardy 23:39, 8 May 2006 (UTC)

If we have the equation g ′ = f, then the anti-derivative of f is g (up to a constant): ʃf = ʃg ′ = g. In the case that

$f=v^{\prime }+\sum _{i<n}\alpha _{i}{\frac {u_{i}^{\prime }}{u_{i}}}\,.$

the anti-derivative is

$g=v+\sum \alpha _{i}\ln u_{i}$.

Why not write it with the logarithms? Because I went to check Rosenlicht, and he stated the result in terms of f.

XaosBits 01:33, 10 May 2006 (UTC)

In the current state of the article this formula is a little overcomplicated: why are there logarithms in the first place? If the formula is correct, then there's no need for logarithms, since the logarithm of an elementary function is an elementary function.

85.187.35.160 (talk) 15:27, 13 January 2011 (UTC)

As it is stated, the sum is not needed at all: if you skip the sum, it says that if there is an elementary solution g then for some elementary function v the solution is of the form g=v, so there's obviously something wrong with the statement. /Pontus (talk) 09:25, 31 May 2011 (UTC)

typo of some sort?

This text is in the article: "...For the function f*e^g, where f and g [what?], we have..."

I don't know what's missing... "are elementary functions"? Thought I should point it out. 71.191.51.134 19:04, 29 July 2007 (UTC)

This is incorrect

The article says:

For example, all known programs (except Axiom) cannot find the antiderivative for

$f(x)=x/{\sqrt {x^{4}+10x^{2}-96x-71}}$

This is simply not true. Both Mathematica and Maple can calculate the antiderivative (even not-so-recent versions). Anyone can try this at http://integrals.wolfram.com/ . (The Root[] objects represent roots of polynomials, i.e. numbers, and can be easily expanded (written in explicit form) using the ToRadicals[] command). I won't edit the article because I am not familiar with the topic (perhaps these programs don't use the Risch-algorithm for this specific problem?), but this needs to be cleaned up or rephrased ... —Preceding unsigned comment added by 129.177.44.25 (talk) 08:51, 8 April 2008 (UTC)

Sorry to tell this, but this seems to be correct. Try http://integrals.wolfram.com with input "x/sqrt(x^4+10*x^2-96*x-71)". You will receive large answer. But the problem is not in the Root[] objects (of course, you are right, they are just numbers, so the fuction is elementary even if it use such numbers - no matter whether they can be transferred to radicals or not). The problem that the answer contains "F", and "Π". F is which is elliptic ingegral of the first kind, and Π is elliptic integral of the third kind. Both F and Π cannot be written using elementary functions. And the question is not whether x/sqrt(x^4+10*x^2-96*x-71) has an antiderivative - the question is whether this antiderivative can be written using elementary functions.

Starting from this point to the end, I am not sure, whether this is 100% correct.

Try to change 71 to 72 in the polynomial. Integrals.wolfram.com will give you the answer which will look near the same. This is because integrals.wolfram.com cannot understand, that there is very big difference between

x/sqrt(x^4+10*x^2-96*x-71) and

x/sqrt(x^4+10*x^2-96*x-72)

Antiderivative of the first can be written using elementary functions, and antiderivative of the second cannot. This is because Galois groups of these polynomials are different:

x^4+10*x^2-96*x-71 Galois group is D(4), e.g. generated by permutations (1 2 3 4) and (1 3), and contains 8 elements (same as in "x^4-2")

x^4+10*x^2-96*x-72 Galois group is S(4), e.g. generated by permutations (1 2), (1 3), (1 4) and contains 24 elements

So x^4+10*x^2-96*x-71 is very special case of quadric polynomials, and this is the reason why Risch algorithm in the 71 case gives the answer "yes", and in the 72 case gives the answer "no".

I have tried Maple v. 11 with input:

simplify(convert(int(x/sqrt(x^4+10*x^2-96*x-71),x),radical));

The answer also contains "EllipticF" and "EllipticPi". So Maple also does not understand that antiderivative for x/sqrt(x^4+10*x^2-96*x-71) can be written using elementary functions.

Do you agree with my argumentation?

Sorry for my bad English.

Gaz v pol (talk) 18:14, 18 April 2008 (UTC)

What the original writer of the article was trying to say is probably correct. What the article is really saying (i.e. that no system can find an antiderivative---no mention of elementary functions) is incorrect (Mathematica does find an antiderivative, but not in terms of elementary functions). This article needs a big cleanup, but I dare not touch it because this is a very complicated mathematical topic that I know nothing about (I came here to learn a little about it :) ). If you are familiar with the topic, it would be great if you cleaned it up a bit (and made it more precise)! —Preceding unsigned comment added by 83.166.219.47 (talk) 15:41, 6 June 2008 (UTC)

Just FYI, this is how complex this is: https://math.stackexchange.com/questions/681893/how-to-integrate-int-fracx-sqrtx410x2-96x-71dx Stackexchange cannot be used on wikipedia though. 109.252.90.67 (talk) 03:54, 28 November 2021 (UTC)

This craziness can be finally solved by Mathematica 13: https://www.wolframcloud.com/obj/d9af14f6-3b98-43c4-b996-11dedc9d9f10 2A00:1370:812D:5272:8FD:D915:7268:189C (talk) 16:55, 11 December 2021 (UTC)

Unsolved Integral

The last example says that no software is known to be able to solve it, however it appears sympy(which uses the Risch algorithm), can: http://dpaste.com/81951/ —Preceding unsigned comment added by 128.113.195.217 (talk) 16:52, 2 October 2008 (UTC)

See http://live.sympy.org/ to try it. --91.39.86.160 (talk) 10:22, 3 October 2008 (UTC)

I profiled the code and it seems that it is indeed the rich algorithm that computes this one (i.e., it is not some special case in the code). This is interesting, because SymPy cannot even do the other one on this page (maybe because it is not so easy in the elementary case and it doesn't know about the special functions that Maple and Mathematica produce). I will update the article. Asmeurer (talk ♬ contribs) 16:09, 9 October 2009 (UTC)

Issues with section Description?

The section says:

"Liouville formulated the problem solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution g to the equation g ′ = f then for constants α_i and elementary functions u_i and v the solution is of the form

$f=\sum _{i<n}\alpha _{i}{\frac {u_{i}^{\prime }}{u_{i}}}+v^{\prime }\,.$"

1. $f$ should be already known, we seek $g$ (see also above).
2. What does $n$ (the upper bound for the sum index) stand for?? I assume it just indicates that the index set is finite, but in that case I don't understand the following: "Risch developed a method that allows one to only consider a finite set of elementary functions of Liouville's form."

Can anybody enlighten me on this? --Berntie (talk) 20:19, 4 October 2008 (UTC)

Claims

The article claims that only Axiom can solve a certain integral, and that a second is unsolvable by any present CAS. I made a page User:CRGreathouse/Risch on my userspace to record my test of those claims: regardless of the need for WP:V, I'd be happier if I knew the claims in the article were true.

It seems that the first claim holds: I was not able to find any CAS other than Axiom solving the first integral, not even the new version of Mathematica. The second claim is false, as SymPy solves it fairly quickly.

I would prefer to take examples out of published papers, because those may have claims like "no CAS known to the authors can solve this". For the time being, I think we should leave the first integral but remove the claim on the second (and maybe remove it outright).

CRGreathouse (t | c) 05:43, 14 May 2009 (UTC)

Hello CRGreathouse! I was the person who added both these examples. At the time of adding it was true (we have tested almost all available CAS's). However, as you wrote, now the second example can be solved by SymPy. That's very interesting. I do not trust that any software has implemented the Risch algorighm in full now. But I will test current version of SymPy and try to give you better example. Thank you.Gaz v pol (talk) 22:28, 17 May 2009 (UTC)

I'm almost certain that SymPy doesn't have a full implementation of the Risch algorithm -- and as you can see in my page or try yourself, it can't handle your first example.

Would you give the results (here or on my page) for Maple and MathCad? Those two seem like the only other ones with a reasonable chance at solving hard integrals.

CRGreathouse (t | c) 00:56, 18 May 2009 (UTC)

It seems that I will have access to Maple 13 only at about 20'th of June (Maple 12 cannot solve this). I will send you info about whether Maple 13 can do this when I will have access. Sorry for the delay. Gaz v pol (talk) 11:02, 1 June 2009 (UTC)

Not a problem. Would you also paste the Maple command you're using on my page? That way others can verify it more easily. I look forward to seeing what Maple 13 can do when you get access. CRGreathouse (t | c) 15:08, 1 June 2009 (UTC)

I checked with Maple 13 under Windows 7 today. Results: first integral - with EllipticF (so Maple does not make use of the Galois group of the polynomial under sqrt), second - no success, third - full success. I have added results to your page, but not to the table (sorry, I feel not comfort with editing html). I can also check any other integral on Maple 13 by your request. Also I have tried SymPy with simular integrals but with change x to x-1. It gives me error. Either I have done something wrong with syntax (that's possible), or they just added this particular integral to their table :-). I suggest the following to be written on the page: 1'st integral can be done only by Axiom, second only by SymPy, and their sum can be done by no software as of June 2008. What do you think about this idea?Gaz v pol (talk) 15:38, 10 June 2009 (UTC)

Sounds good to me. CRGreathouse (t | c) 16:17, 11 June 2009 (UTC)

That integral is NOT in SymPy's heuristic table (otherwise, it wouldn't take so long to compute it :). It is being computed by SymPy's implementation of Bronstein's poor-man's integrator, which is based on the Risch-Norman algorithm. Actually, I wonder if we should be looking at the poor-man's integrator here (the original code is in Maple syntax).

But if you are looking for more examples of hard integrals, you might try some of the other ones from Bronstein's integration tutorial (see the references of this article, it's freely downloadable on his website).

And by the way, I got the integral with x + 1 to work just fine.Asmeurer (talk ♬ contribs) 00:57, 14 November 2010 (UTC)

Decidability unknown for elementary functions

But isn’t $|x|={\sqrt {x^{2}}}$ elementary presentable? --Chricho ∀ (talk) 23:07, 26 June 2012 (UTC)

The problem is that there are 2 square roots to each number, and the choice between them cannot be controlled. So ${\sqrt {x^{2}}}$ could also mean $x$, or $-x$--77.126.235.228 (talk) 14:34, 13 May 2013 (UTC)

I agree with Chriko. Nowadays the ${\sqrt {}}$ symbol is most widely used to mean "the positive square root of". An example of this is the quadratic formula. If the symbol were used to signify both the positive and negative square root we wouldn't need the plus or minus sign before it. Also ${\sqrt {x}}$ would not be a function if that symbol were used for both the positive and negative square root. — Preceding unsigned comment added by 98.232.93.206 (talk) 05:22, 20 October 2013 (UTC)

A New Algorithm

It's possible we need a new algorithm. Since this topic about symbolic integration algorithms is connected to differential Galois theory, maybe we need one that uses Galois groups and it's generated permutations. Galois group would be relatively easy to implement, perhaps using set from C++ would be a good idea. We also need to generate a constructor which tests that E is an extension to F. For more information look at this Gave232 (talk) 22:49, 15 July 2013 (UTC)Gave232

Inaccurate statement

"the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions."

Unless they are including complex logarithms, you also need to consider inverse tangents, which are the result of irreduceable quadratic factors in the denominator.

For example, the integral of (x^3+x-1)/(x^4+x^2) is ln|x|+1/x+arctanx+C

98.232.93.206 (talk) 05:14, 20 October 2013 (UTC)

Complex logarithms are included. They are also allowed in the definition of an elementary function. — Pt _(T) 13:20, 20 April 2015 (UTC)

Missing Algorithm

An adequate description of the Risch Algorithm seems to be missing ... from the article about the Risch Algorithm. Bah, minor detail. 88.130.29.105 (talk) 07:27, 18 April 2015 (UTC)