Robert Risch. Algebraic properties of the elementary functions of analysis. American Journal of Mathematics, 101:743759, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ring operations and power series solutions to algebraic di erential equations. The correctness and non ambiguity of expressions may be ensured by structural induction. This may involve zerotesting for the series represented by subexpressions. Several classical approaches for zero testing exist [9, 6, 10, 11, 7, 12] and we provide a quick survey of them in section 2. Our new zero test, which is described in section 5, is based on a similar approach as [10, 11, 12] We believe the algorithm to be interesting for ve main reasons: We treat di erential equations of arbitrary order. Our method accommodates ....

....series under consideration. This is clearly the case if we restrict the di erentially algebraic power series to be algebraic. A more interesting example is obtained when we also allow left composition with log(1 z) and exp z. In this case, the Ax theorem [1] and the Risch structure theorem [9] may be used to design a fast zero test. The structural approach may yield very ecient algorithms when it works. However, it requires the characterization of all possible relations in a given context, where we merely asked for a test whether a particular one holds. Consequently, the approach ....

Risch, R. Algebraic properties of elementary functions in analysis. Amer. Journ. of Math. 4, 101 (1975), 743-759.

.... distinguish the following main subproblems: ffl How to test whether f 2F locally represents the zero power series ffl How to evaluate f 2F safely and eOEciently at any point where f may be dened The rst subproblem is called the zero test problem and it has been studied before in several works [Ris75, DL84, DL89, Kho91, Sha89, Sha93, PG97, SvdH01, vdH02b]. For large classes of special functions, it turns out that the zero test problem for power series can be reduced to the zero test problem for constants (see also [vdH01b] for a discussion of this latter problem) In this paper, we will focus on the second subproblem, while leaving aside the ....

R.H. Risch. Algebraic properties of elementary functions in analysis. Amer. Journ. of Math., 4(101):743759, 1975.

....Some successes of this approach are the following: ffl Computations in algebraic extensions of a eld using Groebner basis techniques. In this case an element of the algebraic extension is represented uniquely by its reduction modulo the Groebner basis. ffl The Risch structure theorem [Ris75] allows computations in dioeerential eld extensions by exponentials, logarithms, or integrals. This technique may be adapted to a few other cases [SSC85] ffl Richardson designed a zero test for elementary constants (i.e. constants which may be dened implicitly using rational numbers, the eld ....

R.H. Risch. Algebraic properties of elementary functions in analysis. Amer. Journ. of Math., 4(101):743759, 1975.

....now. First, can we reduce the problem of deciding whether an exp log function is zero at infinity to the corresponding problem for exp log constants Although several algorithms exist for deciding whether a given exp log function f is locally zero in the neighbourhood of a point of analyticity [26, 6, 28, 20], no one considered the problem of deciding whether the germ of f at infinity is zero. Indeed, these problems are not equivalent: consider the function p Gamma x: In computer algebra, f is usually represented by y Gamma x in the ring Q[x; y] y ) whence f 6= 0. Performing a local ....

R.H. Risch. Algebraic properties of elementary functions in analysis. Amer. Journ. of Math., 4(101):743--759, 1975.

....the n th root. These functions are built in, to a greater or lesser extent, into many computer algebra systems (not to mention other programming languages [12, 17] and are heavily used. As abstract algebraic solutions to integrals and or di erential equations, they satisfy well known properties [16]. However, reasoning with them as functions C C is more dicult than is usually acknowledged, and all algebra systems have one, sometimes both, of the following defects: they make mistakes, be it the traditional schoolchild one 1 = p 1 = p ( 1) 2 = 1 (1) or more subtle ones; they ....

Risch,R.H., Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) pp. 743-759.

....elementary constant is zero; and which is guaranteed to terminate (eventually) unless it is working on a problem which includes a counterexample to Schanuel s conjecture. The zero equivalence problem for exp log functions is then usually reduced to the exp log constants case via Risch s algorithm [11] or differential algebra methods [15, 19, 8] Special consideration of algebraic extensions based on error estimation and numerical evaluation is described in [21] Another issue we have not mentioned is that of complexity. The algorithm as we have described it has terrible worst case complexity. ....

Risch, R. H. Algebraic properties of the elementary functions of analysis. American Journal of Mathematics 4, 101 (1975), 743--759.

....rationally dependent terms and equate them to zero. For testing whether two terms are rationally dependent, we divide one by the other and test whether the quotient is a rational function in x or not. This is an easy and fast approach. Of course, we could also use the Risch normalization procedure [15, 2] to generate the rationally independent terms, but first this normalization is rather expensive and only works for elementary functions and second, our simplified approach will not lead to wrong results, it may at most happen that we miss a simpler solution, which, in practice, rarely happens, ....

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

.... a logarithm could in fact adjoin only a new constant, and an exponential could in fact be algebraic, for example Q(x) log(x) log(2x) Q(log(2) x) log(x) and Q(x) e log(x) 2 ) Q(x) p x) There are however algorithms that detect all such occurences and modify the tower accordingly [16], so we can assume that all the logarithms and exponentials appearing in E are monomials, and that Const(E) C. Let now k 0 be the largest index such that t k0 is transcendental over K = C(x) t 1 ; t k0 Gamma1 ) and t = t k0 . Then E is a nitely generated algebraic extension of K(t) ....

Robert Risch. Algebraic properties of the elementary functions of analysis. American Journal of Mathematics, 101:743759, 1979.

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Robert Risch. Algebraic properties of the elementary functions of analysis. American Journal of Mathematics, 101:743759, 1979.

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