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The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.
The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral ...
65F35 ; 65K05 ; 68Q15 ; 68W01 ; 15A12 ; 65F10 ; 90C51 ; 65H10
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Déposez votre fichier ici pour le déplacer vers cet enregistrement.
The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.
The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral ...
65F35 ; 65K05 ; 68Q15 ; 68W01 ; 15A12 ; 65F10 ; 90C51 ; 65H10
... Lire [+]
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- xxxi; 554 p.
ISBN 978-3-642-38895-8
Grundlehren der mathematischen wissenschaften , 0349
Localisation : Collection 1er étage
analyse numérique # algorithme # nombre de conditionnement # algèbre linéaire # optimisation linéaire # résolution d'équation polynômiale # analyse d'erreur # système linéaire triangulaire # algorithme itératif pour résolution d'équation linéaire # méthode des ellipsoides # méthode des points intérieurs # analyse probabilistique de nombre conditionnel # méthode de Newton # 17ème problème de Smale
15A12 ; 52A22 ; 60D05 ; 65-02 ; 65F22 ; 65F35 ; 65G50 ; 65H10 ; 65H20 ; 90-02 ; 90C05 ; 90C31 ; 90C51 ; 90C60 ; 68Q25 ; 68W40
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- 310 p.
ISBN 978-0-387-29570-1
CMS Books in mathematics
Localisation : Ouvrage RdC (BORW)
fonction convexe # optimisation # théorie non-linéaire # calcul de variations # point intérieur # programmation convexe # dualité de Frenckel # théorème de Karush-Kuhn-Tucker # point fixe # théorème de Radenacker
90-01 ; 49-01 ; 90C51 ; 90C25 ; 49J53 ; 52A41 ; 46N10 ; 47H10
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- vii; 324 p.
ISBN 978-3-540-63183-5
Mathématiques & applications , 0027
Localisation : Collection 1er étage
algorithme optimal # programmation linéaire # programmation quadratique # programmation non linéaire # programmation avec contraintes # processus stochastique # optimisation numérique # méthode de points intérieurs # optimisation non contrainte # programmation non différentiable # méthode de Newton
65K05 ; 65-01 ; 90C05 ; 90C20 ; 90C30 ; 90-01 ; 90C51 ; 90C53
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- xviii; 402 p.
ISBN 978-0-387-78976-7
Springer undergraduate texts in mathematics and technology
Localisation : Ouvrage RdC (FORS)
optimisation # programmation mathématique
90-01 ; 68W30 ; 90C30 ; 90C46 ; 90C51 ; 90Cxx
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- x; 219 p.
ISBN 978-0-8218-3352-0
Fields institute monographs , 0027
Localisation : Collection 1er étage
programmation semidéfinie # optmisation combinatoire # optimisation combinatoire # inégalité linéaire de matrices # géométrie convexe
90C22 ; 90C27 ; 15A39 ; 52A41 ; 65Y20 ; 90C05 ; 90C25 ; 90C51 ; 68Q25
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