| Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations 1996 Edition Contributor(s): Ahlbrandt, Calvin (Author), Peterson, A. C. (Author) |
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ISBN: 0792342771 ISBN-13: 9780792342779 Publisher: Springer
Binding Type: Hardcover - See All Available Formats & Editions Published: October 1996 Annotation: This book explores several aspects of discrete Hamiltonian systems. It is unique in that it provides interconnections between symplectic systems, recessive and dominant solutions, various discrete Riccati equations, and continued fraction representations of solutions of Riccati equations. It also presents variable step size discrete variational theory, a discrete Legendre transformation from discrete Euler-Lagrange equations to discrete Hamiltonian systems. Novel use is made of the implicit function theorem to show the importance of step size in numerical solutions of Hamiltonian systems. An "a priori" step size criterion shows how one can avoid parasitic numerical solutions. This book is accessible to students of mathematics, engineering, physics, chemistry and economics at the senior or beginning graduate level who have completed a course in matrix theory. It provides foundation work for engineering students studying optimal control and estimation as well as the variational problems arising in physics, chemistry, and economics. Click for more in this series: Texts in the Mathematical Sciences |
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| Additional Information |
| BISAC Categories: - Mathematics | Finite Mathematics - Mathematics | Calculus |
| Dewey: 515.39 |
| LCCN: 96045991 |
| Series: Texts in the Mathematical Sciences |
| Physical Information: 1.17" H x 6.44" W x 9.56" L (1.82 lbs) 376 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: This book should be accessible to students who have had a first course in matrix theory. The existence and uniqueness theorem of Chapter 4 requires the implicit function theorem, but we give a self-contained constructive proof ofthat theorem. The reader willing to accept the implicit function theorem can read the book without an advanced calculus background. Chapter 8 uses the Moore-Penrose pseudo-inverse, but is accessible to students who have facility with matrices. Exercises are placed at those points in the text where they are relevant. For U. S. universities, we intend for the book to be used at the senior undergraduate level or beginning graduate level. Chapter 2, which is on continued fractions, is not essential to the material of the remaining chapters, but is intimately related to the remaining material. Continued fractions provide closed form representations of the extreme solutions of some discrete matrix Riccati equations. Continued fractions solution methods for Riccati difference equations provide an approach analogous to series solution methods for linear differential equations. The book develops several topics which have not been available at this level. In particular, the material of the chapters on continued fractions (Chapter 2), symplectic systems (Chapter 3), and discrete variational theory (Chapter 4) summarize recent literature. Similarly, the material on transforming Riccati equations presented in Chapter 3 gives a self-contained unification of various forms of Riccati equations. Motivation for our approach to difference equations came from the work of Harris, Vaughan, Hartman, Reid, Patula, Hooker, Erbe & Van, and Bohner. |
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