Modular Algorithms in Symbolic Summation and Symbolic Integration

Gerhard, Jürgen

doi:10.1007/b104035

Cited by 32 publications

(24 citation statements)

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“…A similar algorithm for integer polynomials is given by the first author in [7], Section 7.2. The idea is related to that of Man & Wright [13], who describe an algorithm for dispersion based on polynomial factorization in [x].…”

Section: Computing the Dispersion Set Over [X]mentioning

confidence: 96%

Shiftless decomposition and polynomial-time rational summation

Gerhard

Giesbrecht

Storjohann

et al. 2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

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Section: Computing the Dispersion Set Over [X]mentioning

confidence: 96%

Shiftless decomposition and polynomial-time rational summation

Gerhard

Giesbrecht

Storjohann

et al. 2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

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“…By Lemma 2, the kernel K is differential-reduced. Note that the DRCF of a rational function also appears in [7,Chap. 8].…”

Section: A Differential Canonical Formmentioning

confidence: 98%

Differential rational normal forms and a reduction algorithm for hyperexponential func

Geddes

2004

Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation

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We describe differential rational normal forms of a rational function and their properties. Based on these normal forms, we present an algorithm which, given a hyperexponential function T (x), constructs two hyperexponential functions T1(x) and T2(x) such that T (x) = T 1 (x) + T2(x) and T2(x) is minimal in some sense. The algorithm can be used to accelerate the differential Gosper's algorithm and to compute right factors of the telescopers.

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“…The more costly step is the Taylor shift by 1, that is, x → x + 1. Using asymptotically fast Taylor shift [14], we therefor need O(n(n + τ(f I ))) =Õ(n(n(L + h) + Σ( f ))) =Õ(n 2 (L − log σ ( f ))) bit operations at a node of level h = O(log n − log σ ( f )) because −n log σ ( f ) ≥ Σ( f ). For the cost of computing the polynomials at all nodes, we thus get the bound (|T apx | + |T ex |) ·Õ(n 2 (L − log σ ( f ))) =Õ(n 2 (L − log σ ( f ))(n + Σ( f ))) due to Theorem 21.…”

Section: Bit Complexitymentioning

confidence: 99%

A General Approach to Isolating Roots of a Bitstream Polynomial

Sagraloff

2010

Math.Comput.Sci.

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Abstract. We describe a new approach to isolate the roots (either real or complex) of a square-free polynomial F with real coefficients. It is assumed that each coefficient of F can be approximated to any specified error bound and refer to such coefficients as bitstream coefficients. The presented method is exact, complete and deterministic. Compared to previous approaches [10,12,23] we improve in two aspects. Firstly, our approach can be combined with any existing subdivision method for isolating the roots of a polynomial with rational coefficients. Secondly, the approximation demand on the coefficients and the bit complexity of our approach is considerably smaller. In particular, we can replace the worst-case quantity σ (F) by the average-case quantity ∏ n i=1 n √ σ i , where σ i denotes the minimal distance of the i−th root ξ i of F to any other root of F, σ (F) := min i σ i , and n = deg F. For polynomials with integer coefficients, our method matches the best bounds known for existing practical algorithms that perform exact operations on the input coefficients.

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Modular Algorithms in Symbolic Summation and Symbolic Integration

Cited by 32 publications

References 0 publications

Shiftless decomposition and polynomial-time rational summation

Shiftless decomposition and polynomial-time rational summation

Differential rational normal forms and a reduction algorithm for hyperexponential func

A General Approach to Isolating Roots of a Bitstream Polynomial

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