## Almost tight recursion tree bounds for the Descartes method (2006)

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Venue: | In Proc. Int. Symp. on Symbolic and Algebraic Computation |

Citations: | 29 - 3 self |

### BibTeX

@INPROCEEDINGS{Eigenwillig06almosttight,

author = {Arno Eigenwillig and Vikram Sharma and Chee K. Yap},

title = {Almost tight recursion tree bounds for the Descartes method},

booktitle = {In Proc. Int. Symp. on Symbolic and Algebraic Computation},

year = {2006},

pages = {71--78},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give a unified (“basis free”) framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiXi with integer coefficients |ai | < 2 L, this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.

### Citations

429 |
zur Gathen and
- von
- 1991
(Show Context)
Citation Context ...cients of length O(nL). Both the H and TH transformations increase the length of the coefficients by O(n) bits on each level. It is known that we can perform the T-transformation in O(n 2 ) additions =-=[11, 10, 26]-=-; the H-transformation needs O(n) shift operations. Hence a node at recursion depth h has bit cost O(n 2 (nL+nh)) for the power basis. In the Bernstein basis, we need O(n 2 ) additions and O(n) shifts... |

277 |
Algorithms in real algebraic geometry
- Basu, Pollack, et al.
- 2003
(Show Context)
Citation Context ...of TR(A). The test whether AR(0) = 0 amounts to inspection of the constant term. We call the resulting algorithm the power basis variant of the Descartes method. An alternative choice of basis is the =-=[0, 1]-=--Bernstein basis (B n 0 (X), B n 1 (X), . . . , B n n(X)), with B n i (X) := B n i [0, 1](X) where B n i [c, d](X) := n i ! (X − c) i (d − X) n−i (d − c) n , 0 ≤ i ≤ n. Its usefulness for the Descarte... |

140 |
Fundamental Problems of Algorithmic Algebra
- Yap
- 2000
(Show Context)
Citation Context ...of the nodes have in-degree 1, then Y (v i,v j)∈E |vi−vj| ≥ p |discr(A)|·M(A) −(n−1) ·(n/ √ 3) −m ·n −n/2 . Proof. This proof is not self-contained, but refers to the standard argument from Davenport =-=[5, 27]-=-. Let (v1, . . . , vk) be the topologically sorted list of the vertices of G, where (vi, vj) ∈ E implies j < i. Given such an ordering we modify )j,i as follows: the n × n Vandermonde matrix WA = (α j... |

70 |
Bézier and BSpline Techniques
- Prautzsch, Boehm, et al.
- 2002
(Show Context)
Citation Context ...that DescartesTest(A,(0,1)) = Var(b0, . . . , bn), without any additional transformation. To obtain AL and AR from A(X) = Pn i=0 biBn i (X), we use a fraction-free variant of de Casteljau’s algorithm =-=[22]-=-: For 0 ≤ i ≤ n set b0,i := bi. For 1 ≤ j ≤ n and 0 ≤ i ≤ n −j set bj,i := bj−1,i + bj−1,i+1. From this, one obtains coefficients of 2 n A(X) = Pn i=0 b′ iB n i [0, 1 2 ](X) = Pn i=0 b′′ i B n i [ 1 b... |

64 | Polynomial real root isolation using Descartes’ rule of signs
- Collins, Akritas
- 1976
(Show Context)
Citation Context ...s a straightforward derivation of the best known bit complexity bound (Section 4).1.1 Previous work Root isolation using Descartes’ rule of signs was cast into its modern form by Collins and Akritas =-=[3]-=-, using a representation of polynomials in the usual power basis. Rouillier and Zimmermann [25] summarize various improvements of this method until 2004. The algorithm’s equivalent formulation using t... |

41 |
Sylvester-Habicht Sequences and Fast Cauchy Index Computation
- Lickteig, Roy
(Show Context)
Citation Context ...3.5); and the work at each node of this tree can be done with e O(n 3 L) bit operations (using asymptotically fast basic operations), where e O indicates that we are omitting logarithmic factors (see =-=[23, 14, 6]-=- or Theorem 4.2, respectively). The connection between root isolation in the power basis using the Descartes method, and in the Bernstein basis using de Casteljau’s algorithm and the variation-diminis... |

39 | On the complexity of isolating real roots and computing with certainty the topological degree
- Mourrain, Vrahatis, et al.
(Show Context)
Citation Context ...lgorithm’s equivalent formulation using the Bernstein basis was first described by Lane and Riesenfeld [13] and more recently by Mourrain, Rouillier and Roy [19] and Mourrain, Vrahatis and Yakoubsohn =-=[20]-=-; see also [1, §10.2]. The crucial tool for our bound on the size of the recursion tree is Davenport’s generalization [5] of Mahler’s bound [15] on root separation. Davenport used his bound for an ana... |

34 |
Bounds on a polynomial
- Lane, Riesenfeld
- 1981
(Show Context)
Citation Context ...ower basis using the Descartes method, and in the Bernstein basis using de Casteljau’s algorithm and the variation-diminishing property of Bézier curves was already pointed out by Lane and Riesenfeld =-=[13]-=-, but this connection is often unclear in the literature. In Section 2, we provide a general framework for viewing both as a form of the Descartes method. In Section 3, we present the main result, whi... |

32 |
An inequality for the discriminant of a polynomial
- Mahler
- 1964
(Show Context)
Citation Context ...lier and Roy [19] and Mourrain, Vrahatis and Yakoubsohn [20]; see also [1, §10.2]. The crucial tool for our bound on the size of the recursion tree is Davenport’s generalization [5] of Mahler’s bound =-=[15]-=- on root separation. Davenport used his bound for an analysis of Sturm’s method (see [6]). He mentioned a relation to the Descartes method but did not work it out. This has been done later by Johnson ... |

30 |
Algorithms for Polynomial Real Root Isolation
- Johnson
- 1991
(Show Context)
Citation Context ...r. Despite the apparent inferiority of Descartes’ rule as compared to Sturm sequences, there is considerable recent interest in the Descartes approach because of its excellent performance in practice =-=[9, 24, 19, 25]-=-. This paper shows that the asymptotic worst case bound on recursion tree size for the Descartes method (Theorem 3.4) is no worse than the best known bound for Sturm’s method (Theorem 6 of [6]). For t... |

28 | Real Algebraic Numbers: Complexity Analysis and Experimentation
- Emiris, Mourrain, et al.
- 2008
(Show Context)
Citation Context ... 2 L + n 2 h) [26]. Substituting h = O(n(L + log n)), we get the bound O(M(n 3 (L + log n)). Multiplied by tree size O(n(L + log n)), we obtain the theorem. Remark. 3 Emiris, Mourrain, and Tsigaridas =-=[7]-=- describe the following approach to obtain a similar speedup for the Bernstein basis variant: Suppose the vector (bi)i of Bernstein coefficients of A(X) = Pn i=0 biBn i (X) is given and the Bernstein ... |

28 | Polynomials: An algorithmic approach - Mignotte, Stefanescu - 1999 |

26 | Bernstein’s basis and real root isolation
- Mourrain, Rouillier, et al.
- 2004
(Show Context)
Citation Context ...r. Despite the apparent inferiority of Descartes’ rule as compared to Sturm sequences, there is considerable recent interest in the Descartes approach because of its excellent performance in practice =-=[9, 24, 19, 25]-=-. This paper shows that the asymptotic worst case bound on recursion tree size for the Descartes method (Theorem 3.4) is no worse than the best known bound for Sturm’s method (Theorem 6 of [6]). For t... |

26 |
Asymptotically fast computation of subresultants
- Reischert
- 1997
(Show Context)
Citation Context ...3.5); and the work at each node of this tree can be done with e O(n 3 L) bit operations (using asymptotically fast basic operations), where e O indicates that we are omitting logarithmic factors (see =-=[23, 14, 6]-=- or Theorem 4.2, respectively). The connection between root isolation in the power basis using the Descartes method, and in the Bernstein basis using de Casteljau’s algorithm and the variation-diminis... |

23 |
Amortized bound for root isolation via Sturm sequences. Symbolic-Numeric Computation, edited by Dongming Wang and Lihong
- Du, Sharma, et al.
- 2007
(Show Context)
Citation Context ...24, 19, 25]. This paper shows that the asymptotic worst case bound on recursion tree size for the Descartes method (Theorem 3.4) is no worse than the best known bound for Sturm’s method (Theorem 6 of =-=[6]-=-). For the particular case of polynomials with integer coefficients of magnitude less than L, the recursion tree is O(n(L + log n)) both for Sturm’s method [5, 6] and the Descartes method (Corollary 3... |

22 |
Isolierung reeller Nullstellen von Polynomen. Wissenschaftliches Rechnen, edited by Jürgen Herzberger
- Krandick
- 1995
(Show Context)
Citation Context ...sis of Sturm’s method (see [6]). He mentioned a relation to the Descartes method but did not work it out. This has been done later by Johnson [9] and, filling a gap in Johnson’s argument, by Krandick =-=[11]-=-. However, they bound the number of internal nodes at each level of the recursion tree separately. This leads to bounds that imply 1 a tree size of O(n log n(log n + L)) and a bit complexity of O(n 5 ... |

20 |
New bounds for the Descartes method
- Krandick, Mehlhorn
(Show Context)
Citation Context ...log n + L) 2 ) for a polynomial of degree n with L-bit integer coefficients. Their argument uses a termination criterion for the Descartes method due to Collins and Johnson [4]. Krandick and Mehlhorn =-=[12]-=- employ a theorem by Ostrowski [21] that yields a sharper termination criterion. However, they just use it to improve on the constants of the bounds in [11] 2 . We will show that Ostrowski’s result al... |

17 | Architectureaware classical Taylor shift by 1
- Johnson, Krandick, et al.
- 2005
(Show Context)
Citation Context ...cients of length O(nL). Both the H and TH transformations increase the length of the coefficients by O(n) bits on each level. It is known that we can perform the T-transformation in O(n 2 ) additions =-=[11, 10, 26]-=-; the H-transformation needs O(n) shift operations. Hence a node at recursion depth h has bit cost O(n 2 (nL+nh)) for the power basis. In the Bernstein basis, we need O(n 2 ) additions and O(n) shifts... |

11 |
Computer algebra for cylindrical algebraic decomposition
- Davenport
- 1985
(Show Context)
Citation Context ...n bound for Sturm’s method (Theorem 6 of [6]). For the particular case of polynomials with integer coefficients of magnitude less than L, the recursion tree is O(n(L + log n)) both for Sturm’s method =-=[5, 6]-=- and the Descartes method (Corollary 3.5); and the work at each node of this tree can be done with e O(n 3 L) bit operations (using asymptotically fast basic operations), where e O indicates that we a... |

10 |
Note on Vincent’s theorem
- Ostrowski
- 1983
(Show Context)
Citation Context ...degree n with L-bit integer coefficients. Their argument uses a termination criterion for the Descartes method due to Collins and Johnson [4]. Krandick and Mehlhorn [12] employ a theorem by Ostrowski =-=[21]-=- that yields a sharper termination criterion. However, they just use it to improve on the constants of the bounds in [11] 2 . We will show that Ostrowski’s result allows an immediate bound on the numb... |

9 |
Some inequalities about univariate polynomials
- Mignotte
- 1981
(Show Context)
Citation Context ...ls is optimal under the assumption L = Ω(log n). To do so, we construct a family of inputs of unbounded degree n and coefficient length L for which the height of the recursion tree is Ω(nL). Mignotte =-=[16]-=- gave a family of polynomials P(X) = X n − 2(aX − 1) 2 parameterized by integers n ≥ 3 and a ≥ 3. By Eisenstein’s criterion, P(X) is irreducible (use the prime number 2). Let h = a −n/2−1 . Since P(a ... |

7 |
Quantifier Elimination and the Sign Variation Method for Real Root Isolation
- Collins, Johnson
(Show Context)
Citation Context ...a bit complexity of O(n 5 (log n + L) 2 ) for a polynomial of degree n with L-bit integer coefficients. Their argument uses a termination criterion for the Descartes method due to Collins and Johnson =-=[4]-=-. Krandick and Mehlhorn [12] employ a theorem by Ostrowski [21] that yields a sharper termination criterion. However, they just use it to improve on the constants of the bounds in [11] 2 . We will sho... |

7 |
On the Distance Between the Roots of a Polynomial
- Mignotte
- 1995
(Show Context)
Citation Context ... |det WA|, thus giving us the desired result. Remark. The bound in Theorem 3.1 is invariant under replacing A(X) by a non-zero scalar multiple λA(X). Remark. A bound similar to Theorem 3.1 appears in =-=[17]-=-. Instead of M(A) n−1 , it uses a product of root magnitudes with varying exponents of n − 1 or less. 3.2 The Recursion Tree Our application of the Davenport-Mahler theorem rests on the following lemm... |

5 |
Abschätzungen und Iterationsverfahren für PolynomNullstellen
- Batra
- 1999
(Show Context)
Citation Context ...e v − p is non-negative and even, the 1 Personal communication, Krandick and Mehlhorn. 2 This potential use of Ostrowski’s result is mentioned but not carried out in the 1999 Ph.D. thesis of P. Batra =-=[2]-=-. Descartes test yields the exact number of roots whenever its result is 0 or 1. The Descartes method for isolating the real roots of an input polynomial Ain(X) in an open interval J consists of a rec... |

4 | Observatiunculae ad theoriam aequationum pertinentes. J. für die reine und angewandte Mathematik 13 - Jacobi |

4 |
Efficient Isolation of a Polynomial Real Roots. Rapport de recherche
- Rouillier, Zimmermann
- 2001
(Show Context)
Citation Context ...r. Despite the apparent inferiority of Descartes’ rule as compared to Sturm sequences, there is considerable recent interest in the Descartes approach because of its excellent performance in practice =-=[9, 24, 19, 25]-=-. This paper shows that the asymptotic worst case bound on recursion tree size for the Descartes method (Theorem 3.4) is no worse than the best known bound for Sturm’s method (Theorem 6 of [6]). For t... |

4 |
Efficient Isolation of
- Rouillier, Zimmermann
- 2001
(Show Context)
Citation Context |