Optimal conditioning of Vandermonde-like matrices anda measurement problem
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ndltd-OhioLink-oai-etd.ohiolink.edu-kent15559499154179762021-08-03T07:10:49Z Optimal conditioning of Vandermonde-like matrices anda measurement problem Kuian, Mykhailo Mathematics In this work several aspects of conditioning optimization are presented. Firstly, weanalyzed the conditioning of rectangular Vandermonde and Vandermonde-like matrices.Vandermonde matrices are known to be highly ill-conditioned when the nodes arereal. One approach to reduce ill-conditioning is based on using a basis of orthogonalpolynomials. The matrices so obtained are commonly referred to as Vandermonde-like.Gautschi analyzed optimally conditioned and optimally scaled square Vandermonde andVandermonde-like matrices with real nodes. In Chapter 2 Gautschi's analysis is extendedto rectangular Vandermonde-like matrices with real nodes, as well as to Vandermonde-likematrices with nodes on the unit circle in the complex plane. There we generalize classicalPosse theorem for Gauss and Gauss-Szego quadrature rules. Using these results existenceand uniquenceness of optimally conditioned Vandermonde-like matrices is analyzed. Inthe nal part of Chapter 2 properties of rectangular general Vandermonde-type matriceswith Chebyshev nodes or with equidistant nodes on the unit circle in the complex planeare discussed. We show that condition number of these matrices is independent of thenumbers of nodes.In Chapter 3 an explicit QR and QR-like factorization for rectangular Vandermondematrices with Chebyshev nodes are presented. The use of Chebyshev nodes is known toreduce the conditioning of Vandermonde matrices. Based on QR and QR-like factorizationstwo new fast methods for solving least squares problem for Vandermonde matrices withChebyshev nodes are derived.Chapter 4 discuss conditioning analysis of the measurement problem. There we investigatecombined multi-measuring systems that determine several unknown quantities frommeasurements of a single variable at different preprogrammed conditions determined bycontrol parameters. Such measurements are described by nonlinear systems of equationswhere perturbations are present simultaneously in both the control parameters and measureddata. The errors in the measured quantities are caused by measurement errors anderrors in the setting of the control parameters. To provide better accuracy for the entirerange of the unknown quantities, a model of conditioning of combined multi-measuringsystems is derived analytically. The set of control parameters is detemined by optimizingthe conditioning. To demonstrate the capability of the proposed method, we apply it tothe polarized light microscopy technique called LC-PolScope. We compare the computedoptimal set of control parameters with other sets including those used in the PolScopeand demonstrate that our computed set works very well for the entire range of determinedquantities. We believe that the described method can be applied to a wide range ofmeasurement systems. 2019-05-03 English text Kent State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=kent1555949915417976 http://rave.ohiolink.edu/etdc/view?acc_num=kent1555949915417976 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
| collection |
NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics |
| spellingShingle |
Mathematics Kuian, Mykhailo Optimal conditioning of Vandermonde-like matrices anda measurement problem |
| author |
Kuian, Mykhailo |
| author_facet |
Kuian, Mykhailo |
| author_sort |
Kuian, Mykhailo |
| title |
Optimal conditioning of Vandermonde-like matrices anda measurement problem |
| title_short |
Optimal conditioning of Vandermonde-like matrices anda measurement problem |
| title_full |
Optimal conditioning of Vandermonde-like matrices anda measurement problem |
| title_fullStr |
Optimal conditioning of Vandermonde-like matrices anda measurement problem |
| title_full_unstemmed |
Optimal conditioning of Vandermonde-like matrices anda measurement problem |
| title_sort |
optimal conditioning of vandermonde-like matrices anda measurement problem |
| publisher |
Kent State University / OhioLINK |
| publishDate |
2019 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=kent1555949915417976 |
| work_keys_str_mv |
AT kuianmykhailo optimalconditioningofvandermondelikematricesandameasurementproblem |
| _version_ |
1719455459220389888 |