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2013-10-17T02:21:48Z

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The '''Risch algorithm''', named after [[Robert Henry Risch]], is an [[algorithm]] for the [[calculus]] operation of indefinite integration (i.e., finding [[antiderivative]]s). The algorithm transforms the problem of integration into a problem in [[differential algebra|algebra]]. It is based on the form of the function being integrated and on methods for integrating [[rational function]]s, [[Nth root|radical]]s, [[logarithm]]s, and [[exponential function]]s. Risch, who developed the algorithm in 1968, called it a [[decision procedure]], because it is a method for deciding ''whether'' a function has an [[elementary function (differential algebra)|elementary function]] as an indefinite integral; and also, if it does, determining it. The Risch algorithm is summarized (in more than 100 pages) in ''Algorithms for Computer Algebra'' by [[Keith Geddes|Keith O. Geddes]], Stephen R. Czapor and George Labahn. The Risch–Norman algorithm (after A. C. Norman), a faster but less powerful technique, was developed in 1976.

==Description==
The Risch algorithm is used to integrate [[elementary function]]s. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic operations (+ − × ÷). [[Pierre-Simon Laplace|Laplace]] solved this problem for the case of [[rational functions]], as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions. The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program it was finally implemented in the 1960s.

[[Joseph Liouville|Liouville]] formulated the problem that is 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 ''ui'' and ''v'' the solution is of the form

:<math> g = v + \sum_{i<n} \alpha_i \,\ln (u_i) </math>

Risch developed a method that allows one to consider only a finite set of elementary functions of Liouville's form.

The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function ''f'' e''g'', where ''f'' and ''g'' are [[differentiable function]]s, we have

: <math> (f \cdot e^g)' = (f^\prime + f\cdot g^\prime)\cdot e^g, \, </math>

so if e''g'' were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as

: <math> (f \cdot(\ln g)^n)' = f^\prime (\ln {g})^n + n f \frac{g^\prime}{g} (\ln{g})^{n-1} </math>

then if (ln ''g'')''n'' were in the result of an integration, then only a few powers of the logarithm should be expected.

==Problem examples==
Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function<ref>This example was posted by Manuel Bronstein to the usenet forum ''comp.soft-sys.math.maple'' on 24 Nov. 2000.[https://groups.google.com/d/msg/comp.soft-sys.math.maple/5CcPIR9Ft-Y/xYfGiyJauuoJ]</ref> has an elementary antiderivative:

: <math> f(x) = \frac{x}{\sqrt{x^4 + 10 x^2 - 96 x - 71}},</math>

namely:

: <math> \begin{align} F(x) = - \frac{1}{8}\ln &\,\Big( (x^6+15 x^4-80 x^3+27 x^2-528 x+781) \sqrt{ x^4+10 x^2-96 x-71} \Big. \\ & {} - \Big .(x^8 + 20 x^6 - 128 x^5 + 54 x^4 - 1408 x^3 + 3124 x^2 + 10001) \Big) + C. \end{align}</math>

(Some [[computer algebra system]]s may here return an antiderivative in terms of ''non-elementary'' functions (i.e. [[elliptic integral]]s), which however are outside the scope of the Risch algorithm.) But if the coefficient 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions.

The following is a more complex example<ref>This example comes from Manuel Bronstein's "Symbolic Integration Tutorial". See the references.</ref> that involves both algebraic and transcendental functions:

: <math>f(x) = \frac{x^2+2x+1+ (3x+1)\sqrt{x+\ln x}}{x\,\sqrt{x+\ln x}(x+\sqrt{x+\ln x})}.</math>

In fact, the antiderivative of this function has a fairly short form:

: <math>F(x) = 2 (\sqrt{x+\ln x} + \ln(x+\sqrt{x+\ln x})) + C.</math>

==Implementation==
Transforming Risch's theoretical algorithm into an algorithm that can be effectively executed by a computer was a complex task which took a long time.

The case of the purely transcendental functions (which do not involve roots of polynomials) is relatively easy and was implemented early in most [[computer algebra system]]s. The first implementation was done by [[Joel Moses]] in [[Macsyma]] soon after the publication of Risch's paper.<ref>{{citation|author= Joel Moses|title= Macsyma: A personal history|journal= Journal of Symbolic Computation|volume= 47|year= 2012|pages= 123–130|doi= 10.1016/j.jsc.2010.08.018}}</ref>

The case of purely algebraic functions was solved and implemented in [[Reduce (computer algebra system)|Reduce]] by [[James H. Davenport]].<ref>Not to be confused with his father [[Harold Davenport]]</ref><ref>{{cite book|author= James H. Davenport|title= On the integration of algebraic functions|year= 1981|publisher= Springer|isbn= 0-387-10290-6, 3-540-10290-6|series= Lecture notes in computer science|volume= 102}}</ref>

The general case was solved and implemented in Scratchpad, a precursor of [[Axiom (computer algebra system)|Axiom]], by Manuel Bronstein.<ref>{{citation|author= Manuel Bronstein|title= Integration of elementary functions|journal= Journal of Symbolic Computation|volume= 9|issue= 2|year= 1990|pages= 117–173}}</ref>

==Decidability==
The Risch algorithm applied to general elementary functions is not an algorithm but a [[RE (complexity)|semi-algorithm]] because it needs to check, as a part of its operation, if certain expressions are equivalent to zero ([[constant problem]]), in particular in the constant field. For expressions that involve only functions commonly taken to be [[elementary function|elementary]] it is not known whether an algorithm performing such a check exists or not (current [[computer algebra system]]s use heuristics); moreover, if one adds the [[absolute value|absolute value function]] to the list of elementary functions, it is known that no such algorithm exists; see [[Richardson's theorem]].

Note that this issue also arises in the [[polynomial division algorithm]]; this algorithm will fail if it cannot correctly determine whether coefficients vanish identically.<ref>{{cite web| title= Mathematica 7 Documentation: PolynomialQuotient| url= http://reference.wolfram.com/mathematica/ref/PolynomialQuotient.html| work= Section: Possible Issues| accessdate= 17 July 2010}}</ref> Virtually every non-trivial algorithm relating to polynomials uses the polynomial division algorithm, the Risch algorithm included. If the constant field is [[computable]], i.e., for elements not dependent on ''x'', the problem of zero-equivalence is decidable, then the Risch algorithm is a complete algorithm. Examples of computable constant fields are <math>\mathbb{Q}</math> and <math>\mathbb{Q}(y)</math>, i.e., rational numbers and rational functions in y with rational number coefficients, respectively, where ''y'' is an indeterminate that does not depend on ''x''.

This is also an issue in the [[Gaussian elimination]] matrix algorithm (or any algorithm that can compute the nullspace of a matrix), which is also necessary for many parts of the Risch algorithm. Gaussian elimination will produce incorrect results if it cannot correctly determine if a pivot is identically zero{{Citation needed|date=January 2012}}).

==See also==
*[[Lists of integrals]]
*[[Liouville's theorem (differential algebra)]]
*[[Symbolic integration]]
*[[Axiom (computer algebra system)]]
*[[Incomplete gamma function]]

==References==
* {{cite journal | author=R. H. Risch | title= The problem of integration in finite terms | journal= Transactions of the American Mathematical Society | year= 1969 | volume=139 | pages= 167–189 | doi=10.2307/1995313 | publisher=American Mathematical Society | jstor=1995313}}
* {{cite journal | author=R. H. Risch | title= The solution of the problem of integration in finite terms | journal= Bulletin of the American Mathematical Society | year= 1970 | volume=76 | issue=3 | pages= 605–608 | doi=10.1090/S0002-9904-1970-12454-5 }}
* {{cite journal | author=Maxwell Rosenlicht | title=Integration in finite terms | journal=American Mathematical Monthly | year= 1972 | volume=79 | issue=9 | pages=963–972 | doi=10.2307/2318066 | publisher=Mathematical Association of America | jstor=2318066}}
* {{cite book | author=Geddes, Czapor, Labahn| title=Algorithms for Computer Algebra | publisher=Kluwer Academic Publishers | year=1992 | isbn=0-7923-9259-0}}
* {{cite book | author=Manuel Bronstein | title=Symbolic Integration I | publisher=Springer | year=2005 | isbn=3-540-21493-3}}
* {{cite paper | author=Manuel Bronstein | title=Symbolic Integration Tutorial | year=1998 | url=http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf }}
*{{MathWorld|urlname=RischAlgorithm|title=Risch Algorithm|author=Bhatt, Bhuvanesh}}

==Notes==
{{reflist}}

{{DEFAULTSORT:Risch Algorithm}}
[[Category:Integral calculus]]
[[Category:Differential algebra]]
[[Category:Computer algebra]]

[[hy:Ռիսքի ալգորիթմ]]
[[hu:Risch-algoritmus]]

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