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Condition: the geometry of numerical algorithms - Lecture 1
Post-edited
Authors : Bürgisser, Peter (Author of the conference)
CIRM (Publisher )
15A12 - Conditioning of matrices
65F10 - Iterative methods for linear systems
65F35 - Matrix norms, conditioning, scaling (numerical linear algebra)
65H10 - Systems of equations
65K05 - Mathematical programming methods
68Q15 - Complexity classes (hierarchies, relations among complexity classes, etc.)
90C51 - Interior-point methods
CIRM (Publisher )
Abstract :
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.
MSC Codes : 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.
15A12 - Conditioning of matrices
65F10 - Iterative methods for linear systems
65F35 - Matrix norms, conditioning, scaling (numerical linear algebra)
65H10 - Systems of equations
65K05 - Mathematical programming methods
68Q15 - Complexity classes (hierarchies, relations among complexity classes, etc.)
90C51 - Interior-point methods
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 26/01/2017 Conference Date : 16/01/2017 Subseries : Research talks arXiv category : Numerical Analysis ; Computer Science Mathematical Area(s) : Computer Science ; Numerical Analysis & Scientific Computing Format : MP4 (.mp4) - HD Video Time : 01:32:36 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2017-01-16_Burgisser_part1.mp4 
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Information on the Event
Event Title : French computer algebra days / JNCF - Journées nationales de calcul formel / Event Organizers : Brisebarre, Nicolas ; El Bacha, Carole ; Giorgi, Pascal ; Mezzarobba, Marc ; Moroz, Guillaume Dates : 16/01/17 - 20/01/17 Event Year : 2017 Event URL : http://conferences.cirm-math.fr/1584.html
Citation Data
DOI : 10.24350/CIRM.V.19110103Cite this video as: Bürgisser, Peter (2017). Condition: the geometry of numerical algorithms - Lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19110103 URI : http://dx.doi.org/10.24350/CIRM.V.19110103 |
See Also
- [Multi angle] Condition: the geometry of numerical algorithms - Lecture 2 / Author of the conference Bürgisser, Peter.
Bibliography
- Bürgisser, P., Cucker, F. (2013). Condition. The geometry of numerical algorithms. Berlin: Springer - http://dx.doi.org/10.1007/978-3-642-38896-5
