Computing eigenvalues of quasi‐rational Bernstein–Vandermonde matrices to high relative accuracy

2021 ◽  
Author(s):  
Zhao Yang ◽  
Xiao‐Xiao Ma
Keyword(s):  
Relative Accuracy ◽  
High Relative Accuracy ◽  
Vandermonde Matrices

scholarly journals Trajectory Definition with High Relative Accuracy (HRA) by Parametric Representation of Curves in Nano-Positioning Systems

Micromachines ◽  
2019 ◽  
Vol 10 (9) ◽  
pp. 597
Author(s):  
Lucía Díaz Pérez ◽  
Beatriz Rubio Serrano ◽  
José A. Albajez García ◽  
José A. Yagu¨e Fabra ◽  
Esmeralda Mainar Maza ◽  
...  
Keyword(s):  
Curve Fitting ◽  
Control Strategies ◽  
Parametric Representation ◽  
Relative Accuracy ◽  
Positioning Control ◽  
Positioning Systems ◽  
High Relative Accuracy ◽  
Cad Cam ◽  
Data Points ◽  
Novel Algorithms

Nanotechnology applications demand high accuracy positioning systems. Therefore, in order to achieve sub-micrometer accuracy, positioning uncertainty contributions must be minimized by implementing precision positioning control strategies. The positioning control system accuracy must be analyzed and optimized, especially when the system is required to follow a predefined trajectory. In this line of research, this work studies the contribution of the trajectory definition errors to the final positioning uncertainty of a large-range 2D nanopositioning stage. The curve trajectory is defined by curve fitting using two methods: traditional CAD/CAM systems and novel algorithms for accurate curve fitting. This novel method has an interest in computer-aided geometric design and approximation theory, and allows high relative accuracy (HRA) in the computation of the representations of parametric curves while minimizing the numerical errors. It is verified that the HRA method offers better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting: When fitting a curve by interpolation with the HRA method, fewer data points are required to achieve the precision requirements. Similarly, when fitting a curve by a least-squares approximation, for the same set of given data points, the HRA method is capable of obtaining an accurate approximation curve with fewer control points.

Cleaving optical fibre lengths to high relative accuracy

1998 ◽  
Vol 157 (1-6) ◽  
pp. 282-290 ◽  
Author(s):  
Mark P.J.L Chang ◽  
David F Buscher
Keyword(s):  
Optical Fibre ◽  
Relative Accuracy ◽  
High Relative Accuracy
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