Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization

2015 ◽  
Vol 37 (5) ◽  
pp. S581-S601 ◽  
Author(s):  
Sarah W. Gaaf ◽  
Michiel E. Hochstenbach
Keyword(s):  
Condition Number ◽  
Lanczos Bidiagonalization ◽  
The Matrix ◽  
Matrix Condition Number ◽  
Matrix Condition

Tuning the axial singularity method for accurate calculation of potential flow around axisymmetric bodies

1994 ◽  
Vol 98 (976) ◽  
pp. 215-226
Author(s):  
M. F. Zedan
Keyword(s):  
Condition Number ◽  
Intensity Variation ◽  
Axial Line ◽  
Control Points ◽  
Singularity Method ◽  
The Matrix ◽  
Matrix Condition Number ◽  
Order Of Magnitude ◽  
Matrix Condition ◽  
Solution Parameters

Abstract The performance of axial line singularity methods has been investigated systematically for various solution parameters using carefully chosen test cases. The results indicate that increasing the number of elements and using stretched node distribution improves the solution accuracy until the matrix becomes near-singular. The matrix condition number increases with these parameters as well as with the order of intensity variation and profile thickness. For moderate fineness ratios, the linear methods outperform zero-order methods. The linear doublet method performs best with control points at the x-locations of nodes while the source methods perform best with control points mid-way between nodes. The doublet method has a condition number an order of magnitude lower than the source method and generally provides more accurate results and handles a wider range of bodies. With appropriate solution parameters, the method provides excellent accuracy for bodies without slope discontinuity. The smoothing technique proposed recently by Hemsch has been shown to reduce the condition number of the matrix; however it should be used with caution. It is recommended to use it only when the solution is highly oscillatory with a near-singular matrix. A criterion for the optimum value of the smoothing parameter is proposed.

The inverse problem of bimorph mirror tuning on a beamline

2011 ◽  
Vol 18 (6) ◽  
pp. 930-937 ◽  
Author(s):  
Rong Huang
Keyword(s):  
Inverse Problem ◽  
Tikhonov Regularization ◽  
Measurement Errors ◽  
Regularization Parameter ◽  
X Rays ◽  
The Matrix ◽  
Matrix Condition Number ◽  
Matrix Condition ◽  
Safety Limit

One of the challenges of tuning bimorph mirrors with many electrodes is that the calculated focusing voltages can be different by more than the safety limit (such as 500 V for the mirrors used at 17-ID at the Advanced Photon Source) between adjacent electrodes. A study of this problem at 17-ID revealed that the inverse problem of the tuningin situ, using X-rays, became ill-conditioned when the number of electrodes was large and the calculated focusing voltages were contaminated with measurement errors. Increasing the number of beamlets during the tuning could reduce the matrix condition number in the problem, but obtaining voltages with variation below the safety limit was still not always guaranteed and multiple iterations of tuning were often required. Applying Tikhonov regularization and using the L-curve criterion for the determination of the regularization parameter made it straightforward to obtain focusing voltages with well behaved variations. Some characteristics of the tuning results obtained using Tikhonov regularization are given in this paper.

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