thesis on rational interpolation

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We provide an introduction to Lagrange rational interpolation: an effective way of constructing rational functions that interpolate given data sets.

We start with the connection between the well-known Lagrange polynomial interpolant and the Lagrange rational interpolant. More precisely, both interpolants rely on an efficient barycentric formula for representing the Lagrange basis. In the case of rational interpolation, we show that this barycentric formula leads to the definition of the Loewner matrix - a Cauchy-like matrix constructed directly from the given data. The main result states that the rank of this Loewner matrix is equal to the order (complexity) of the rational function that interpolates the data. Crucially, the SVD of the Loewner matrix leads to a state-space form of the Lagrange rational interplant.

From a model reduction and system identification perspective, the Loewner matrix has deep system-theoretic significance: it is the product of generalized controllability and observability matrices. This insight shows how one can use the singular value decay of the Loewner matrix to perform model reduction of large-scale dynamical systems. In fact, the Loewner matrix is closely related to the Krylov approach for model reduction.

Our talk is structured as a short tutorial and, along the way, we provide several numerical examples of the different applications of Lagrange rational interpolation. We show how to perform function approximation via rational interpolation and compare the results with the classical polynomial approach. The results show that the convergence curves of rational approximation have desirable properties that are not found in the polynomial case. We also show several model reduction examples and briefly discuss how to extend this approach to systems with multiple inputs and outputs, and to systems that depend on parameters, i.e., how to perform multi-variate Lagrange rational interpolation.

Cosmin graduated in Fall 2013 from Rice University with a PhD in Electrical and Computer Engineering. Cosmin was advised by Dr. Thanos Antoulas and his thesis was titled "Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems" (available at  http://aci.rice.edu ). His main research interests are model reduction of large-scale dynamical systems, system identification, rational interpolation and approximation.

While in graduate school, most courses attended by Cosmin were offered in the Computational And Applied Math department with only two exceptions: the model reduction and compressed sensing courses offered in Electrical Engineering. After graduation, Cosmin continued pursuing his Math passion in an industry job; he joined The Mathworks Inc. as a software developer with the MATLAB Math team. He is always interested to learn about your experience with the fundamental Math functionality included in MATLAB, such as fft, backslash, eig, svd etc.

• Seminar Schedule
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Antonio Cosmin Ionita

• Model Reduction
• System Identification
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Update – October 2013

This has been my last month here at Rice. I have now successfully completed my PhD.

My thesis is titled  Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems  ( download pdf ).

(And here is the customary toy duck that I received from the Office of Graduate and Postdoctoral Studies for a successful PhD submission.)

Welcome to Antonio Cosmin’s personal website!

I am a Ph.D. student with a passion for algorithms that lead to efficient simulations, and an extensive background in Electrical and Computer Engineering ( ECE )  and Computational and Applied Math ( CAAM ).

My Ph.D. advisor is Dr. A. C. Antoulas , and my research interests include

• modeling and simulation of dynamical systems
• model reduction of large-scale dynamical systems
• system identification from frequency domain measurements
• control theory of linear, bi-linear, non-linear, and parametrized systems
• rational interpolation and approximation
• numerical algorithms, and numerical linear algebra.
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Effective Computation of Rational Approximants and Interpolants

• Published: November 2000
• Volume 6 , pages 365–390, ( 2000 )

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• Bernhard Beckermann 1 &
• George Labahn 2  

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This paper considers the problem of effective algorithms for some problems having structured coefficient matrices. Examples of such problems include rational approximation and rational interpolation. The corresponding coefficient matrices include Hankel, Toeplitz and Vandermonde-like matrices. Effective implies that the algorithms studied are suitable for implementation in either a numeric environment or else a symbolic environment.

The paper includes two algorithms for the computation of rational interpolants which are both effective in symbolic environments. The algorithms use arithmetic that is free of fractions but at the same time control the growth of coefficients during intermediate computations. One algorithm is a look-around procedure which computes along a path of closest normal points to an offdiagonal path while the second computes along an arbitrary path using a look-ahead strategy. Along an antidiagonal path the look-ahead recurrence is closely related to the Subresultant PRS algorithm for polynomial GCD computation. Both algorithms are an order of magnitude faster than alternative methods which are effective in symbolic environments.

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Beckermann, B., Labahn, G. Effective Computation of Rational Approximants and Interpolants. Reliable Computing 6 , 365–390 (2000). https://doi.org/10.1023/A:1009942122633

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Yeung, R. Kacheong. "Stable rational interpolation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape7/PQDD_0021/NQ46952.pdf.

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