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Topic: Risch algorithm

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Antiderivative

Symbolic integration

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	List of all algorithms, classified by purpose
		Algorithm to convert nondeterministic automaton to deterministic automaton.
		It is a general-purpose algorithm that is simpler than the number field sieve and the fastest for integers under 100 decimal digits.
		Algorithm to allocate memory such that fragmentation is less.
	www.scriptol.org /list-of-algorithms.html (2730 words)

	Risch algorithm - Wikipedia, the free encyclopedia
		The Risch algorithm is an algorithm for the calculus operation of indefinite integration (i.e.
		The Risch algorithm is used to integrate elementary functions.
		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.
	en.wikipedia.org /wiki/Risch_algorithm (379 words)

	The Integrator: History of Integration
		In the 1940s Ostrowski extended this algorithm to rational expressions involving the logarithm.
		In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions.
		His algorithm does not automatically apply to all classes of elementary functions because at the heart of it there is a hard differential equation that needs to be solved.
	integrals.wolfram.com /about/history (465 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		For at least the following integrals, There is a fallacy in claiming the Risch "algorithm" is an algorithm at all: it depends, for solution of subproblems, on heuristics to tell if expressions are equivalent to zero.
		And in cases where approximate solutions are easily obtained for the corresponding definite integral, the Risch algorithm may grind on for a while and then say there is no closed form.
		The theory of algebraic integration, and the history of this problem is interesting, but the connection with computer algebra systems aimed at solving applied mathematics problems in complex analysis is not nearly as central as one might think.
	www.math.niu.edu /~rusin/known-math/99/risch (382 words)

	Symbolisk integrering
		After a review of some polynomal factorization algorithms like Yun's algorithm, we outline the major algorithm which is the Risch's algorithm.
		The Risch's algorithm always looks for an integral which is elementary, that is can be obtained from the rational functions in X by repeatedly adjoining a finite number of nested logarithms, exponentials, and algebraic numbers of functions.
		If the algorithm doesn't find such an integral then the answer is that the integral is not elementary.
	epubl.luth.se /1402-1781/1998/02/index.html (131 words)

	Algebra and Calculus
		For general polynomial inequalities, a version of the Collins algorithm is used with McCallum's improved projection operator.
		For linear inequalities, methods based on either the simplex algorithm or the Loos-Weispfenning linear quantifier elimination algorithm are used.
		For indefinite integrals, an extended version of the Risch algorithm is used whenever both the integrand and integral can be expressed in terms of elementary functions, exponential integral functions, polylogarithms and other related functions.
	documents.wolfram.com /v4/MainBook/A.9.5.html (790 words)

	Miscellaneous functions
		Notes: Risch uses a partial implementation of the Risch algorithm, for logarithmic and exponential extensions.
		The Risch algorithm builds a tower of logarithmic, exponential, and algebraic extensions.
		At this time, this (Risch) function is experimental and should be seen as a framework for future functionality rather than a currently useful tool.
	www.technicalc.org /packages/mathtools/misc.htm (926 words)

	Symbolic Methods
		In 1970, Robert Risch solved the problem, providing a provably correct and finite method for integrating any elementary function whose indefinite integral is elementary.
		Many applications for simple operations (add, multiply), and surprising applications for more complicated operations (division, gcd), e.g., Sturm's algorithm for find the number of real roots of a polynomial in a given interval, solving systems of polynomial equations, Groebner bases.
		Sturm's algorithm is an elegant method to determine the number of real roots of a rational polynomial over a given interval.
	www.cs.princeton.edu /introcs/92symbolic/index.php (2902 words)

	Differential Galois theory and algorithms for linear ODE's. (Site not responding. Last check: 2007-10-09)
		The Risch' algorithm involves the study of solutions of the Risch differential equation y'=ay+b over some differential field K, which are either in K or are algebraic over K.
		Algorithms for order two and order three equations have been developed by M. van Hoeij, F.
		Apart from [F], the only known method is that of Liouvillian solutions combined with the algebraic case of Risch' algorithm.
	www-lmc.imag.fr /CATHODE2/Cirm2000/extended/Vanderput/Vanderput.html (2569 words)

	MUG: Disappearing cursor and showing steps (9.2.00)
		Though there is no reason not to use the Risch algorithm by hand.
		A little off topic, but what would be a good reference for the Risch algorithm.
		Another of my favourite books has a good exposition of Risch's algorithm: 'Algorithms for Computer Algebra' by Keith Geddes, S Czapor, and G Labahn.
	www.math.rwth-aachen.de /mapleAnswers/html/924.html (577 words)

	[Maxima] Bug in the integrate. (Site not responding. Last check: 2007-10-09)
		The reason is just it tries to use the sin algorithm to evaluate the integral instead of the risch algorithm, which is more general but produces results which are harder to simplify.
		So I have made a hack which tests for the presence of some exponential function with a base different from $%E (neper constant) in the integrand to call the risch algorithm instead of the sin one.
		The risch algorithm produces good results, but in a funny format (it is not aware of simplification).
	www.ma.utexas.edu /pipermail/maxima/2002/001880.html (237 words)

	Re: [obm-l] Risch algorithm
		o ALGORITMO DE RISCH e um desenvolvimento do teorema de um trabalho de Laplace que permite fazer da integracao analitica um algoritmo assim como fazemos hoje com a diferenciacao.
		I >haven't done it, but I believe that the Risch algorithm will show that the >antiderivative of u/u^u is not an elementary function.
		Of course, one can >re-express the series sum as an integral; a quick calculation gives > >\int_0^1 x^{x+1} dx, > >and I am confident that one prove (using the Risch algorithm) that >x^{x+1} has no antiderivative in elementary terms.
	www.mail-archive.com /obm-l@mat.puc-rio.br/msg07483.html (666 words)

	An Implementation of Karr's Summation Algorithm in Mathematica -- from Mathematica Information Center
		Implementations of the celebrated Gosper algorithm (1978) for indefinite summation are available on almost any computer algebra platform.
		Karr's algorithm is, in a sense, the summation couterpart of Risch's algorithm for indefinite integration.
		This is the first implementation of this algorithm in a major computer algebra system.
	library.wolfram.com /infocenter/Articles/3392 (130 words)

	Alchemy :: Maths : Integral Calculus (Site not responding. Last check: 2007-10-09)
		The Risch-Norman algorithm is able to compute any integral of such a shape; that is, if the antiderivative involves polynomials, sines, cosines, etc..., the Risch-Norman algorithm will be able to compute it.
		Extended versions of this algorithm are implemented in Mathematica and the Maple computer algebra system.
		In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function and so on).
	www.alchemy-education.com /Maths/Integral.html (1023 words)

	Gosper Algorithm
		The Gosper algorithm [1] is a decision procedure, that decides by algebraic calculations whether a given hypergeometric term
		The Gosper algorithm is the discrete analogue of the Risch algorithm for integration in terms of elementary functions.
		It takes care of the necessary simplifications, and therefore (in principle) provides you with the solution of the decision problem as long as the memory or time requirements are not too high for the computer used.
	www.uni-koeln.de /REDUCE/zeilberg/node2.html (218 words)

	Risch Integration - Spring 2006
		In 1968, Robert Risch, using abstract algebra, solved a centuries-old problem in mathematics - given an elementary function f(x), can its integral be written as another elementary function F(x)?
		Risch's technique either finds the integral or proves that it doesn't exist.
		By the end of the class, we'll be into various research papers.
	www.freesoft.org /Classes/Risch2006 (430 words)

	How the computer works?
		Only since the fundamental work of Robert Risch we dispose of an algorithm which decides whether the primitive of an elementary function is again an elementary function, and if it is, it will be computed.
		His algorithm is (partly) implemented in the bigger computer algebra systems (like Maple).
		However, because of the Risch algorithm there is less need of exercises with the bag of tricks.
	hej.sze.hu /ANM/ANM-010201-A/anm010201a/node5.html (380 words)

	COMPUTER ALGEBRA: LECTURE #13 (Site not responding. Last check: 2007-10-09)
		The answer is stated as Liouville's principle which, however, is not algorithmic in nature.
		The last part of the course will introduce you to an algorithmic version of that theorem, the Risch algorithm.
		The Risch algorithm for logarithmic and exponential extensions.
	www.math.aau.dk /~raussen/CA/99/lp13/lp13.html (194 words)

	COMPUTER ALGEBRA: LECTURE #14 (Site not responding. Last check: 2007-10-09)
		Part of the algorithm consists in modifacations of the methods of Hermite and Rothstein-Trager.
		It is no longer trivial to integrate polynomial expressions (involving logarithms or exponentials as the latest variable); in fact, such polynomials most likely do not posess elementary integrals.
		The roots of the resultant polynomial in the Rothstein-Trager algorithm will most likely not be constants.
	www.math.aau.dk /~raussen/CA/99/lp14/lp14.html (179 words)

	June27.html
		(By "symbolic" we mean an algorithmic implementation of the process, probably with a computer).
		Most implementations make use of what is called the Risch algorithm.
		Even with the appearance of the algorithm, it took several years before software interfaces could implement the algorithm effectively.
	www.uwec.edu /smithaj/Summer710/June27.html (473 words)

	ACM Sigplan Not. 12, 8 (Aug. 1977), 55-59.
		If the Minsky copying algorithm is used [8], the collector has its own stack to keep track of grey nodes; the Cheney algorithm [4] uses a "scan pointer" to linearly scan the new semispace, while updating the pointers of newly moved nodes by moving the nodes they point to.
		The correspondence between our coloring scheme and these algorithms is this: white nodes are those which reside in the old semispace; grey nodes are those which have been copied to the new semispace, but whose outgoing pointers have not been updated to point into the new semispace (i.e.
		Since their algorithm does not mark a user process by coloring it fl (thereby prohibiting it from directly touching white nodes), and allows these white processes to run, the proof that the algorithm collects only and all garbage is long and very subtle (see [15]).
	home.pipeline.com /~hbaker1/Futures.html (3971 words)

	Maxima Manual: 20. Integration
		If none of the preceding heuristics find the indefinite integral, the Risch algorithm is executed.
		is applied on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.
		The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
	maxima.sourceforge.net /docs/manual/en/maxima_20.html (3627 words)

	JSC Special issue on Differential Equations and Differential Algebra (Site not responding. Last check: 2007-10-09)
		An early application of their theory is the well-known Risch algorithm for integration in finite terms.
		While these algorithms (and their generalizations) work in theory, implementation posts many challenging algorithmic subproblems, many of which are purely algebraic in nature.
		After the phenomenal success of Buchberger's algorithm and its many generalizations, some researchers began to turn to similar techniques for algebraic differential equations.
	www.eecis.udel.edu /~caviness/jsc/specialIssues/diff-eqs-diff-alg.html (488 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		These are some questions that this course will address, the focus being on algorithms and data structures for the integers, polynomials, and general mathematical formulae.
		Maple will be used interactively as a algebraic calculator and for programming some of the algorithms.
		o - Brown's modular algorithm for polynomial greatest common divisors o - The P-adic Newton iteration and Hensel lifting o - Polynomial factorization over finite fields and the integers o - The Risch integration algorithm.
	www.cs.sfu.ca /gradpgm/Outlines/2000-1/CMPT-881-Monagan-00-1.txt (328 words)

	Mathematica CalcCenter Features: Calculus
		The following table lists the principal algorithms used by Mathematica CalcCenter for symbolic and numeric integration.
		An extended version of the Risch algorithm is used whenever both the integrand and the integral can be expressed in terms of elementary functions.
		The algorithms in Mathematica CalcCenter cover all of the indefinite integrals resulting in elementary functions that are found in standard reference books such as Gradshteyn-Ryzhik.
	www.wolfram.com /products/calccenter/features/calculus.html?print_this_page=1 (142 words)

	Research (Site not responding. Last check: 2007-10-09)
		Simplification is a fundamental problem in symbolic mathematical computation as it is a mechanism that enables one to determine equality among expressions and to compute with them effectively.
		It is also fundamental in obtaining algorithmic methods for other problems such as integration in finite terms.
		Our work on integration in finite terms has ranged from new methods for determining the minimal algebraic field extensions needed for expressing the logarithmic part of an elementary integral to extensions of parts of the Risch algorithm for integration in terms of certain special functions.
	www.cis.udel.edu /~caviness/node3.html (397 words)

	CiteULike: Tag risch (Site not responding. Last check: 2007-10-09)
		The importance of race and ethnic background in biomedical research and clinical practice.
		Commentary: considerations for use of racial/ethnic classification in etiologic research.
		Categorization of humans in biomedical research: genes, race and disease.
	www.citeulike.org /tag/risch (254 words)

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