Methods for determination the minimal generating vector of the homogeneous linear recurrent systems
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LAZARI, Alexandru. Methods for determination the minimal generating vector of the homogeneous linear recurrent systems. In: International Conference of Young Researchers , 11 noiembrie 2011, Chişinău. Chişinău: Tipogr. Simbol-NP SRL, 2011, Ediția 9, p. 78. ISBN 978-9975-4224-7-5.
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International Conference of Young Researchers
Ediția 9, 2011
Conferința "International Conference of Young Researchers "
Chişinău, Moldova, 11 noiembrie 2011

Methods for determination the minimal generating vector of the homogeneous linear recurrent systems


Pag. 78-78

Lazari Alexandru
 
Moldova State University
 
 
Disponibil în IBN: 21 mai 2021


Rezumat

Let a discrete dynamical system L with the set of states V  . The value v(t)V represents the state of the system at the moment of time t  0,1,2, . It is known the values v(t) , t  0,m1, and the following states v(t) , t  m,m1,m 2, are calculated using the recurrent relation 1 0 ( ) ( 1 ) m k k v t q v t k       . The dynamic of the system L is represented by the homogeneous linear m recurrence 0 ( ( ))t v v t    on . The vector [ ] 1 0 ( ( )) v m m t I v t    is called the initial state and the vector 1 0 ( )m k k q q    ─ the generating vector of the sequence v . The main notation and properties were presented in [1, 2]. Its are noted:  Rol[K] ( Rol[K][m] ) ─ the set of the homogeneous linear ( m ) recurrences on the set K ;  G[K](v) (G[K][m](v) ) ─ the set of the generating vectors (with length m ) of the sequence v ;  H[K](v) ( H[K][m](v) ) ─ the set of the characteristic polynomials (by degree m ) of the sequence v ;  [ ] 1 0, 1, 0, 1 0 (( ) ) k v j i m k k p j s i B i z        , where 1 [ ] 0 ( ) k ( ) [ ][ ]( ) p s q k m k z z H z H K m v       and 00 1 def  . It is defined the notion of “dimension” for the homogeneous linear recurrence v on the set K . The dimension is noted dim[K](v) and represents the minimum positive integer value d that satisfies vRol[K][d] . It is shown that the corresponding minimal generating vector qG[K][d](v) exists and is unique for each not null sequence v . The first minimalization method is based on elimination of the characteristic zeroes: Theorem 1. Let vRol[K][m] , [ ] [ ] 1 , 0, 1, 0, 1 (( ) ) ( ) k v v T m k j k p j s x I B A        , k z , k  0, p 1 − all distinct roots of the polynomial P(z)H[K][m](v) , k t − the count of zeroes from the queue of the vector , 0, 1 ( ) k k j j s A   , k  0, p 1 , : k k k z K t t    , : ( ) ( ) k k t k k z K T z z z     , : ( ) ( ) k k t k k z K U z z z     . Let R(z)K[z] the polynomial by maximum degree that verifies T(z) R(z) . If v not is a null sequence, then dim[K](v)  mt r and one from the minimal characteristic polynomials is ( ) ( ) [ ][ ]( ) ( ) ( ) P z S z H K m t r v R z U z     , where r  deg(R(z)) . The second minimalization method is based on the matrix rank definition. The following theorem shows how may be calculated the minimal generating vector and the dimension of the sequence 0 ( ( ))t v v t    : Theorem 2. If vRol[ ][m] not is a null sequence, then [ ] dim[ ]( ) ( ) v m v  R  rang A and 0 1 1 ( , , , ) [ ][ ]( ) R q q q q G R v    , where the vector 1 2 0 ( , , , ) R R x q q q    is the unique solution of the system [ ] [ ] ( ) v T v T R R A x  f with the given free terms [ ] ( ( ), ( 1), , (2 1)) v R f  v R v R  v R  and the matrix [ ] , 0, 1 ( ( )) v R i j R A v i j     . The second method has a greater applicability than the first method, because not requires the knowledge of complex roots of the characteristic polynomial. The demonstrations of the theorems 1 and 2 were published in [3].

Cuvinte-cheie
Homogeneous Linear Recurrence, minimal generating vector, characteristic polynomial, dimension