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Index

A

A, A, AI, AI , see "admissiblevariations"

admissible pair, 343admissible functions,

fixed endpoints, F, 153, 184,199

one fixed endpoint , F I , 157admissible variations,

conjugate problem,A, 29, 200A, 154, 200, 343AI , 158,344

focal problem, AI , 36

B

Bessel functions , 60bilinear form , 218Bohner, Martin, 151, Chapter 9BVP, boundary value problem,

18

C

Cauchy function , 16Cauchy matrix function , 296characteristic equation, 273conjoined, 44, 202

family, 96conjugate solutions, 44connection theorem, 125

371

continued fraction , Chapter 2approximants, 56CCF, companion matrix, 55-

62generator, 46matrix, 62MKB, Matrizenkettenbriiche,

62reverse , 64-66SCF, symplectic, 53-58

series equivalence, 54-57transformations, 57

continuous case , 167, 188

D

DARE, discrete algebraic Riccatiequation, 139

differential equations, 167convergence to , 188-189

disconjugacy,even order, 172systems, 172three term, 172

disconjugate, 12, 207, 354discrete algebraic Riccatiequa-

tion, 140, 147, 271, 272discrete Hamiltonian system, 186discrete Jacobi condition, 225discrete Legendre condition, 213discrete linear Hamiltonian system,

187

372

discrete matrix Riccati equation,137

discrete momentum variable, 166,186

discrete Riccati equation, 137disfocal, 24, 37distinguished solution, 269, 275

at -00, 285dominant solution, 54, 113, 234­

236DRE, discrete Riccati equation,

137dual discrete matrix Riccati equa­

tion, 138DVRE, discrete variational Ric­

cati equation, 139

E

equation of motion, 331Euler equation, 331Euler-Lagrange equation, 156even order, 85, 169, 333exponential dichotomies, 147

F

:F, fixed endpoints, 153, 184:F1 , one fixed endpoint, 157Fibonacci recurrence relation, 3first integral, 167Floquet theory, discrete , 145flows, symplectic, 78focal point, 330formally self-adjoint, 74fundamental set of solutions, 120Fundamental Theorem for difference

calculus, 112

G

generalized self-adjoint equation,162

INDEX

generalized zero, 11, 170, 172,207,221, 354

generator of a continued fraction,46

Green's function, 26Green's matrix function, 301

H

Hamiltonian system, 166discrete, 186

Hamiltonian system, linear 82Hamiltonian, 165

I

"iff" == if and only if, 10image, Im == range, 339inverting symplectic matrices, 74isotropic, 44IVP, initial value problem, 100

J

Jacobi condition, 226Jacobi equation, 91

K,L

Kalman filtering, 66, 137, 151Lagrange identity, 201Lagrangian subspace, 97Lagrangian, 165left disfocal, 329left matrix product notation, 78Legendre's necessary condition, 163Legendre-Clebsch transformation,

346Levrie, Paul , 69linearly independence in a ring,

69linearly independent, 102Liouville's Theorem, 94

INDEX

M

Matlab, 149, 340matrix continued fraction, MCF,

62maximal dimension of a prepared

family, 97minimal solution of a Riccati

equation, 61MKB, Matrizenkettenbriiche, 62module, right unitary, 101Moore-Penrose pseudo-inverse, 340mutually prepared, 221Mobius function identity, 65

N

norm on a ring , 68normal pair of solutions, 120normalized prepared, 334

P

Parabola Theorem, 69Perron, 0 ., 69Picone Identity, 345pinv, pseudo-inverse, 340positive definite, 31positive definite, 213, 344positive semidefinite, 163, 213prepared basis , 114, 218, 334prepared family, 96

maximal dimension, 97prepared pair , 95, 202prepared solution, 10, 95, 203, 334principal solutions, 44product convention , 3product operator P;, 79

R

range , 339Rayleigh quotient, 274

373

recessive at 00 , 62, 115, 235-236essential uniqueness, 118examples, 116-117nonexistence, 117

recurrence relation, 92reduction of order, 13, 105, 112,

228reduction of order, backwards , 231-

232regular element of a ring, 67Reid Roundabout Theorem, 208reverse continued fraction, 64reverse discrete algebraic Riccati

equation, 289Reverse Riccati equation, 280reverse steady state equation, 289

Riccati equationcontinuous, 136discrete, 31, 137

algebraic , 140, 147, 271­272

applications, 137, 151convergence to continuous,

189filtering, 137Floquet theory, 150game, 137predictor, 151reverse algebraic, 289reverse, 280steady state, 280

Riccati transformations, 140-141

Riccati operator, 31right disfocal, 326right focal BVP, 303right unitary module, 101

ring, 67Cauchy sequence in a ring,

68

374

complete normed , 68convergence in a normed ring,

68linearly independent ring ele-

ments, 69norm on a ring, 68regular element of a ring, 67unit in a ring, 67

Runckel, Hans-J., 69

S

Schelling, A., 138second variation, 160self-conjoined, 95semi-group property, 79sesquilinear form, 218singular value decomposition (SVD),

337solution, 2, 31, 281solution independent from X o, 116spurious solution, 196steady state equation, 271-272strengthened Jacobi condition, 226Sturm Separation Theorems, dis-

cretecomparison, 215separation, 221

summation by parts formula, 156vector case, 185

summation conventions, 105, 111,296

summation operator, 112SVD, singular value decomposi­

tion, 337symplectic approximants, 56symplectic matrix, 56, 74

inverse, 74characterization, 74

symplectic continued fraction , SCF,53

INDEX

symplectic system, 7, 74, 80, 343

T

theoremsConnection, 63, 125Discrete Sturm Comparison,

215Discrete Sturm Separation, 221Existence and Uniqueness, 194Fundamental Theorem for

Difference Calculus, 112Implicit Function, 191Liouville, 94Normal Basis, 120Parabola, 69Reduction of Order, 13, 105,

112,228Reduction of Order, Reverse,

232Reid Roundabout, 208

three term recurrences, Chapters1 & 5

transformations,continued fractions, 57Legendre , 186Riccati equations, 140-141

U,V

unique two point property, 133,247

unit in a ring, 67variable stepsize,

continued fraction, 66variational theory, 183

w,z

Wronskian, 333Wronskian test, 103, 123

Znojil, M., 69

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