The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization
| Autor: | Miroslav S. Pranić, Stefano Pozza |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Generalization
Applied Mathematics Numerical analysis Lanczos algorithm 010103 numerical & computational mathematics Positive-definite matrix 01 natural sciences Hermitian matrix 010101 applied mathematics symbols.namesake Orthogonal polynomials symbols Applied mathematics Gaussian quadrature 0101 mathematics Realization (systems) Mathematics |
| Zdroj: | Numerical Algorithms. 88:647-678 |
| ISSN: | 1572-9265 1017-1398 |
| DOI: | 10.1007/s11075-020-01052-y |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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