D. A. Bini, G. Latouche, and B. Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0003
- Subject:
- Mathematics, Numerical Analysis
This chapter is concerned with infinite block Toeplitz matrices, their relationships with matrix power series and matrix Laurent power series, and the fundamental problem of solving matrix equations ...
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This chapter is concerned with infinite block Toeplitz matrices, their relationships with matrix power series and matrix Laurent power series, and the fundamental problem of solving matrix equations and computing canonical factorizations. It introduces the concept of a Wiener algebra and provides a natural theoretical framework where the convergence properties of algorithms for solving Markov chains can easily be proved. Furthermore, the notion of canonical factorization provides a powerful tool for solving infinite linear systems, such as the fundamental system involving the stationary probability distribution of structured Markov chains.Less
This chapter is concerned with infinite block Toeplitz matrices, their relationships with matrix power series and matrix Laurent power series, and the fundamental problem of solving matrix equations and computing canonical factorizations. It introduces the concept of a Wiener algebra and provides a natural theoretical framework where the convergence properties of algorithms for solving Markov chains can easily be proved. Furthermore, the notion of canonical factorization provides a powerful tool for solving infinite linear systems, such as the fundamental system involving the stationary probability distribution of structured Markov chains.
David F. Hendry
- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780198293545
- eISBN:
- 9780191596391
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198293542.003.0006
- Subject:
- Economics and Finance, Econometrics
The ‘time‐series’ approach to econometrics is critically evaluated, and analytical test power response surfaces presented. Non‐stationarity, differencing and ‘error‐correction’ models (though not yet ...
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The ‘time‐series’ approach to econometrics is critically evaluated, and analytical test power response surfaces presented. Non‐stationarity, differencing and ‘error‐correction’ models (though not yet named) are discussed. Residual autocorrelation is reinterpreted using Sargan's common factor approach, and embryonic ideas presented on how to explain competing models’ findings to reduce the proliferation of conflicting results. Finally, the respective roles of criticism and construction are considered.Less
The ‘time‐series’ approach to econometrics is critically evaluated, and analytical test power response surfaces presented. Non‐stationarity, differencing and ‘error‐correction’ models (though not yet named) are discussed. Residual autocorrelation is reinterpreted using Sargan's common factor approach, and embryonic ideas presented on how to explain competing models’ findings to reduce the proliferation of conflicting results. Finally, the respective roles of criticism and construction are considered.
Jean-Marc Couveignes
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0005
- Subject:
- Mathematics, Number Theory
The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of ...
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The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of a deterministic polynomial time algorithm for these purposes plays an important role in this book. More important, it will also explain what it means to compute with real or complex data in polynomial time. The chapter first recalls basic definitions in computational complexity theory, it then deals with the problem of computing square roots. The more general problem of computing complex roots of polynomials is treated thereafter and, finally, the chapter studies the problem of finding zeros of a converging power series.Less
The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of a deterministic polynomial time algorithm for these purposes plays an important role in this book. More important, it will also explain what it means to compute with real or complex data in polynomial time. The chapter first recalls basic definitions in computational complexity theory, it then deals with the problem of computing square roots. The more general problem of computing complex roots of polynomials is treated thereafter and, finally, the chapter studies the problem of finding zeros of a converging power series.
Charles Fefferman and C. Robin Graham
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0005
- Subject:
- Mathematics, Geometry / Topology
As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic ...
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As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic self-dual Einstein metric in dimension 4 defined near the boundary with prescribed real-analytic conformal infinity.Less
As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic self-dual Einstein metric in dimension 4 defined near the boundary with prescribed real-analytic conformal infinity.
