Risch algorithm

Risch algorithm

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The Risch algorithm, named after Robert H. Risch, is an algorithm for the calculus operation of indefinite integration (i.e., finding antiderivatives). The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch, who developed the algorithm in 1968, called it a decision procedure, because it is a method for deciding if a function has an elementary function as an indefinite integral; and also, if it does, determining it. The Risch algorithm is described (in more than 100 pages) in "Algorithms for Computer Algebra" by Keith O. Geddes, Stephen R. Czapor and George Labahn. The RischNorman algorithm, a faster but less powerful technique, was developed in 1976.

Contents

[hide] 1 Description

2 Problem examples

3 Implementation

4 Decidability

5 See also

6 References

7 Notes

[edit] DescriptionThe Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, trigonometry, and the four operations (+ ). 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 but was only implemented in the 1960s.

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 i and elementary functions ui and v the solution is of the form

g = v +iln(ui)

i < n

Risch developed a method that allows one to only consider 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 eg, where f and g are differentiable functions, we have

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

then if lnng were in the result of an integration, then only a few powers of the logarithm should be expected.

An important consequence of Risch's result is that the Gaussian integral has no elementary antiderivative.

[edit] Problem examplesFinding an elementary antiderivative is very sensitive to details. For instance, the following function has an elementary antiderivative:

But if 71 is changed to 72, it is not possible to represent the antiderivative using elementary functions. The reason is that the Galois group of

is D(4), e.g. generated by permutations (1 2 3 4) and (1 3), and contains 8 elements (same as in x42), while the Galois group of

is S(4), e.g. generated by permutations (1 2), (1 3), (1 4), and contains 24 elements.

[edit] ImplementationTransforming the Risch decision procedure into an algorithm that can be executed by a computer is a complex task that requires the use of heuristics and many refinements. No software (as of March 2008[update]) is known to implement the full Risch algorithm, although several computer algebra systems have partial implementations. The only software that claims it has implemented in full the negative part is Axiom (e.g. if Axiom says "no" this means that antiderivative cannot be represented using elementary functions, but in many cases Axiom says "error").

For example, of all known programs, only Axiom[citation needed] can find an elementary antiderivative for the following:

Axiom can solve the above case, the result being:

Many programs (including Maple and Mathematica) can find the antiderivative for the above function using non-elementary functions (which is not the topic for Risch algorithm).

The following is a more complex example, which most software cannot find an elementary antiderivative for,[1]

while the antiderivative for the above has a short form

[edit] DecidabilityThe Risch algorithm is not an algorithm but a semi-algorithm because it needs as a part to check if some expression is equivalent to zero (constant problem). And for a common meaning of an elementary function it is not known whether such an algorithm exists or not (current computer algebra systems use heuristics); moreover, if one adds the absolute value function to the list of elementary functions, it is known that no such algorithm exists; see Richardson's theorem.

Symbolic integration

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In calculus, a branch of mathematics, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find the differentiable function F(x) such that

This is also denoted

The term symbolic is used to distinguish this problem from that of numerical integration, where the value of F at a particular input or set of inputs, rather than a general formula for F, is sought.

Both problems were held to be of practical and theoretical importance long before the time of digital computers, but they are now generally considered the domain of computer science, as computers are most often used currently to tackle individual instances.

Finding the derivative of an expression is a straightforward process for which it is easy to construct an algorithm. The reverse question of finding the integral is much more difficult. Many expressions which are relatively simple do not have integrals that can be expressed in closed form. See antiderivative for more details.

A procedure called the Risch algorithm exists which is capable of determining if an integral exists and returning it if it does, for many classes of expressions. Such algorithms are still being expanded.

Contents

[hide] 1 Example

2 See also

3 References

4 External links

[edit] ExampleFor example:

is a symbolic result for an indefinite integral (here C is a constant of integration),

is a symbolic result for a definite integral, and

is a numerical result for the same definite integral.

[edit] See also Antiderivative

Elementary function

Risch algorithm

[edit] References Symbolic Integration 1 (transcendental functions) by Manuel Bronstein, 1997 by Springer-Verlag, ISBN 3-540-60521-5

Joel Moses, Symbolic integration: the stormy decade, Proceedings of the second ACM symposium on Symbolic and algebraic manipulation, p.427-440, March 23-25, 1971, Los Angeles, California, United States

Risch Algorithm

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The Risch algorithm is a decision procedure for indefinite integration that determines whether a given integral is elementary, and if so, returns a closed-form result for the integral. It builds a tower of logarithmic, exponential, and algebraic extensions. The case of algebraic extensions is quite complicated and is therefore not completely implemented in any computer algebra system. Liouville's principle, which dates back to the 19th century, is an important part of the Risch algorithm. There are extensions to the Risch algorithm, notably by Cherry, to be able to handle some special functions.

Principio del formulario

SEE ALSO: Elementary Function, Horowitz Reduction, Indefinite Integral, Liouville's Principle

Final del formulario

This entry contributed by Bhuvanesh Bhatt

REFERENCES:

Bronstein, M. Symbolic Integration I: Transcendental Functions. New York: Springer-Verlag, 1997.

Cherry, G.W. Algorithms for Integrating Elementary Functions in Terms of Logarithmic Integrals and Error Functions. Ph.D. thesis. University of Delaware, 1983.

Cherry, G.W. "Integration in Finite Terms with Special Functions: The Logarithmic Integral." SIAM J. Computing 15, 1-12, 1986.

Cherry, G.W. "An Analysis of the Rational Exponential Integral." SIAM J. Computing 18, 893-905, 1989.

Davenport, J.H. On the Integration of Algebraic Functions. Berlin: Springer-Verlag, 1981.

Geddes, K.O.; Czapor, S.R.; and Labahn, G. "The Risch Integration Algorithm." Ch.12 in Algorithms for Computer Algebra. Amsterdam, Netherlands: Kluwer, pp.511-573, 1992.

Risch, R. "On the Integration of Elementary Functions Which are Built Up using Algebraic Operations." Report SP-2801/002/00. Santa Monica, CA: Sys. Dev. Corp., 1968.

Risch, R. "The Problem of Integration in Finite Terms." Trans. Amer. Math. Soc. 139, 167-189, 1969.

Risch, R. "The Solution of the Problem of Integration in Finite Terms." Bull. Amer. Math. Soc., 1-76, 605-608, 1970.

Risch, R. "Algebraic Properties of Elementary Functions of Analysis." Amer. J. Math. 101, 743-759, 1979.