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The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization

Abstract

The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realization problem. We will review these connections pointing out the relationships between several results established independently in related contexts. Original proofs of the Mismatch Theorem and of the Matching Moment Property are given by using the properties of formal orthogonal polynomials and the Gauss quadrature for linear functionals.

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Acknowledgments

We would like to thank Zdeněk Strakoš for the helpful comments and improvements suggested.

Funding

This work has been supported by Charles University Research program No. UNCE/SCI/023 and by the Ministry for Scientific and Technological Development, Higher Education and Information Society of R. Srpska.

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Pozza, S., Pranić, M. The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization. Numer Algor 88, 647–678 (2021). https://doi.org/10.1007/s11075-020-01052-y

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Keywords

  • Linear functionals
  • Matching moments
  • Gauss quadrature
  • Formal orthogonal polynomials
  • Minimal realization
  • Look-ahead Lanczos algorithm
  • Mismatch theorem