On the Complexity of the Plantinga–Vegter Algorithm

Cucker, Felipe; Ergür, Alperen A.; Tonelli-Cueto, Josué

doi:10.1007/s00454-022-00403-x

Cited by 1 publication

(12 citation statements)

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“…The condition-based estimates in Theorem 4.19 are very similar to those of [24,Theorem 6.3]. It is important to observe that only one norm computation is performed by PV-Interval ∞ (in its very first step) and that the cost of this computation is already included in the cost bound in Theorem 4.…”

Section: The Number Of Arithmetic Operations Performed By Pv-interval...mentioning

confidence: 64%

“…factor vanishes from these estimates when we pass from 𝜅 to K. This shows that the version of Plantinga-Vegter using K yields better cost bounds than the one using 𝜅: that is, the one in [24]. Theorem 4.34.…”

Section: Complexity Of the Plantinga-vegter Algorithmmentioning

confidence: 93%

“…For , we define where is the homogenisation of f . Taking the maps (2.3), (2.4), (2.5) in [24] and substituting on them the Weyl norm by the real -norm, we get together with and …”

Section: Numerical Algorithms In Real Algebraic Geometrymentioning

confidence: 99%

“…One can use these maps to produce interval approximations as we do in [24]. For 𝑋 ⊆ R 𝑚 , we denote by 𝑋 the set of boxes contained in X.…”

Section: The Plantinga-vegter Subdivision Algorithmmentioning

confidence: 99%

“… 5 There is a slight difference in the way the anti-concentration constant is defined in [24] and here. …”

mentioning

confidence: 99%

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Functional norms, condition numbers and numerical algorithms in algebraic geometry

Cucker

Ergür

Tonelli-Cueto

2022

Forum of Mathematics, Sigma

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In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand to optimise accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty }$ . This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty }$ -norm, the use of $L_p$ -norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale’s 17th problem).

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Section: The Number Of Arithmetic Operations Performed By Pv-interval...mentioning

confidence: 64%

Section: Complexity Of the Plantinga-vegter Algorithmmentioning

confidence: 93%

“…For , we define where is the homogenisation of f . Taking the maps (2.3), (2.4), (2.5) in [24] and substituting on them the Weyl norm by the real -norm, we get together with and …”

Section: Numerical Algorithms In Real Algebraic Geometrymentioning

confidence: 99%

“…One can use these maps to produce interval approximations as we do in [24]. For 𝑋 ⊆ R 𝑚 , we denote by 𝑋 the set of boxes contained in X.…”

Section: The Plantinga-vegter Subdivision Algorithmmentioning

confidence: 99%

“… 5 There is a slight difference in the way the anti-concentration constant is defined in [24] and here. …”

mentioning

confidence: 99%

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