Discrete Hamiltonian systems: difference equations, continued fractions, and Riccati equations
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 pr...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1996
|
| Series: | Kluwer texts in the mathematical sciences
16 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | 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 |
| Physical Description: | XIV, 374 S. |
| ISBN: | 0792342771 |
Staff View
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| 245 | 1 | 0 | |a Discrete Hamiltonian systems |b difference equations, continued fractions, and Riccati equations |c by Calvin D. Ahlbrandt and Allan C. Peterson |
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| 520 | 3 | |a 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 | |
| 520 | |a It provides foundation work for engineering students studying optimal control and estimation as well as the variational problems arising in physics, chemistry, and economics | ||
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
PREFACE xi
1 SECOND ORDER SCALAR DIFFERENCE
EQUATIONS 1
1.1 Difference Equations and Recurrence Relations 1
1.2 Second Order Scalar Equations as Symplectic Systems 4
1.3 Wronskians of Solutions 9
1.4 Prepared Solutions 10
1.5 Generalized Zeros of Solutions 11
1.6 Disconjugacy and the Reid Roundabout Theorem 12
1.7 Disfocality 36
1.8 Comparison Theorems for Disfocality 39
1.9 Notes 43
2 CONTINUED FRACTIONS 45
2.1 The Scalar Case 45
2.2 Scalar Symplectic Continued Fractions 53
2.3 Companion Matrices 55
2.4 Symplectic C.F.s versus Companion Matrix C.F.s 58
2.5 Ratios of Bessel Functions 60
2.6 Matrix Continued Fractions 61
2.7 Continued Fractions in a Normed Ring 67
2.8 Notes 69
3 SYMPLECTIC SYSTEMS 71
3.1 Linear Systems and the Lagrange Identity 71
3.2 Symplectic Matrices 73
vii
viii Discrete Hamiltonian Systems
3.3 Solutions of Symplectic Systems 76
3.4 Discrete Linear Hamiltonian Systems 82
3.5 Even Order Difference Equations as Systems 85
3.6 Discrete Jacobi Equations 91
3.7 Discrete Wronskians Liouville s Theorem 93
3.8 Prepared Families Lagrangian Subspaces 96
3.9 Linear Independence and the Wronskian Test 101
3.10 Reduction of Order 105
3.11 Dominant and Recessive Solutions 113
3.12 The Normal Basis Theorem 119
3.13 The Connection Theorem 125
3.14 Essential Uniqueness of Recessive Solutions 130
3.15 Asymptotic Behavior of Solutions 132
3.16 The Olver Reid Construction 133
3.17 Associated Riccati Equations 135
3.18 Transformations of Symplectic Systems 144
3.19 Discrete Floquet Theory 145
3.20 Exponential Dichotomies 146
3.21 Periodic Symplectic Systems 149
3.22 Notes 150
4 DISCRETE VARIATIONAL THEORY 153
4.1 The Discrete Variational Problem 153
4.2 The Second Variation 160
4.3 Legendre s Necessary Condition 163
4.4 Discrete Hamiltonian Systems 165
4.5 Higher Order Difference Equations 168
4.6 Disconjugacy and Generalized Zeros 172
4.7 Variable Step Variational Problems 183
4.8 Discrete Hamiltonian Systems Yield Symplectic Integrators 189
4.9 Existence and Uniqueness of Local Solutions of Discrete
Hamiltonian Systems 191
4.10 Notes 197
Contents ix
5 SYMMETRIC THREE TERM
RECURRENCE RELATIONS 199
5.1 A Discrete Reid Roundabout Theorem 199
5.2 Discrete Legendre Conditions 213
5.3 A Sturmian Comparison Theorem 215
5.4 Prepared Bases 216
5.5 An Associated Bilinear Form 218
5.6 A Discrete Sturm Separation Theorem 221
5.7 Discrete Jacobi Conditions 226
5.8 Reduction of Order 228
5.9 Backwards Reduction of Order 231
5.10 Dominant and Recessive Solutions 233
6 DISCRETE RICCATI EQUATIONS FOR
THREE TERM RECURRENCES 263
6.1 A Riccati Equation for CU = 0. 263
6.2 Distinguished Solutions of Riccati Equations 269
6.3 Periodic Coefficient Riccati Equations 270
6.4 Constant Coefficient Riccati Equations 271
6.5 The Characteristic Equation 273
6.6 Minimality of the Distinguished Solution at oo 275
6.7 The Reverse Riccati Equation 278
6.8 Upper Solutions of the Reverse Riccati Equation 285
7 GREEN S FUNCTIONS FOR
NONHOMOGENEOUS SECOND ORDER
DIFFERENCE EQUATIONS 295
7.1 Introduction 295
7.2 Variation of Constants Formula 295
7.3 The Green s Matrix Function for the Conjugate Problem 298
7.4 Green s matrix function for the right focal BVP 303
7.5 A Green s Matrix Function for a General Two Point BVP 308
7.6 Notes 317
x Discrete Hamiltonian Systems
8 DISCONJUGACY CRITERIA 319
8.1 Introduction 319
8.2 A Sufficient Condition for Disconjugacy 319
8.3 A Sufficient Condition for Right Disfocality 326
8.4 Notes 329
9 DISCRETE LINEAR HAMILTONIAN
SYSTEMS 331
9.1 Preliminaries 331
9.2 The Moore Penrose Inverse 337
9.3 The Quadratic Form 343
9.4 Notes 356
REFERENCES 357
INDEX 371
|
| any_adam_object | 1 |
| author | Ahlbrandt, Calvin D. Peterson, Allan C. |
| author_facet | Ahlbrandt, Calvin D. Peterson, Allan C. |
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| id | DE-604.BV011223653 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:06:05Z |
| institution | BVB |
| isbn | 0792342771 |
| language | English |
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| physical | XIV, 374 S. |
| publishDate | 1996 |
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| publisher | Kluwer |
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| series | Kluwer texts in the mathematical sciences |
| series2 | Kluwer texts in the mathematical sciences |
| spelling | Ahlbrandt, Calvin D. Verfasser aut Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations by Calvin D. Ahlbrandt and Allan C. Peterson Dordrecht [u.a.] Kluwer 1996 XIV, 374 S. txt rdacontent n rdamedia nc rdacarrier Kluwer texts in the mathematical sciences 16 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 Systèmes hamiltoniens ram Hamiltonian systems Diskretes System (DE-588)4401225-1 gnd rswk-swf Riccati-Differentialgleichung (DE-588)4230752-1 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Differenzengleichung (DE-588)4012264-5 gnd rswk-swf Riccati-Differentialgleichung (DE-588)4230752-1 s Hamiltonsches System (DE-588)4139943-2 s DE-604 Differenzengleichung (DE-588)4012264-5 s Diskretes System (DE-588)4401225-1 s Peterson, Allan C. Verfasser aut Kluwer texts in the mathematical sciences 16 (DE-604)BV005450041 16 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007530303&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ahlbrandt, Calvin D. Peterson, Allan C. Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations Kluwer texts in the mathematical sciences Systèmes hamiltoniens ram Hamiltonian systems Diskretes System (DE-588)4401225-1 gnd Riccati-Differentialgleichung (DE-588)4230752-1 gnd Hamiltonsches System (DE-588)4139943-2 gnd Differenzengleichung (DE-588)4012264-5 gnd |
| subject_GND | (DE-588)4401225-1 (DE-588)4230752-1 (DE-588)4139943-2 (DE-588)4012264-5 |
| title | Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations |
| title_auth | Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations |
| title_exact_search | Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations |
| title_full | Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations by Calvin D. Ahlbrandt and Allan C. Peterson |
| title_fullStr | Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations by Calvin D. Ahlbrandt and Allan C. Peterson |
| title_full_unstemmed | Discrete Hamiltonian systems difference equations, continued fractions, and Riccati equations by Calvin D. Ahlbrandt and Allan C. Peterson |
| title_short | Discrete Hamiltonian systems |
| title_sort | discrete hamiltonian systems difference equations continued fractions and riccati equations |
| title_sub | difference equations, continued fractions, and Riccati equations |
| topic | Systèmes hamiltoniens ram Hamiltonian systems Diskretes System (DE-588)4401225-1 gnd Riccati-Differentialgleichung (DE-588)4230752-1 gnd Hamiltonsches System (DE-588)4139943-2 gnd Differenzengleichung (DE-588)4012264-5 gnd |
| topic_facet | Systèmes hamiltoniens Hamiltonian systems Diskretes System Riccati-Differentialgleichung Hamiltonsches System Differenzengleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007530303&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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