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Postponing Representation Errors Through a Mix of Symbolic and Numerical Computation in Data Fitting

[+] Author Affiliations
P. Venkataraman

Rochester Institute of Technology, Rochester, NY

Paper No. DETC2012-70341, pp. 81-87; 7 pages
doi:10.1115/DETC2012-70341
From:
  • ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 2: 32nd Computers and Information in Engineering Conference, Parts A and B
  • Chicago, Illinois, USA, August 12–15, 2012
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-4501-1
  • Copyright © 2012 by ASME

abstract

Simple data fitting schemes involve tradeoff between the number of terms in the representation and the accuracy of the fit due to finite number of digits used to represent real numbers. This is also true for functional representation of data, where a known family of functions must be chosen to represent the data. Often these functions must also establish higher order continuity in the data. In practice, polynomial representation of data is very popular, either directly or indirectly through parameterized description such as NURBS. In this case the function approximation requires determining the constants that multiply the polynomial terms. Bezier functions, a special B-spline, can represent the data and the derivatives with fidelity. These functions are established by minimizing the squared error between the data and the representation. The Bezier representation can be determined numerically through simple matrix operations. The accuracy increases with the order of the polynomials until round-off errors becomes a factor. A completely numerical approach will compute faster but will limit the order of the polynomial due to round-off error. Symbolic processing on the other hand extends precision with more terms in representation, but is difficult to determine explicitly. Current symbolic software systems allow translation of the symbolic representation into efficient numerical representation for the application of optimization. This combination improves accuracy of the representation by postponing round off errors. This is demonstrated by application to two examples, one two-dimensional and the second, three-dimensional.

Copyright © 2012 by ASME

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