Approximating the eigenvalues and eigenvectors of birth and death matrices
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Approximating the eigenvalues and eigenvectors of birth and death matrices Natália Bebiano1 · Susana Furtado2 Received: 13 February 2020 / Revised: 31 August 2020 / Accepted: 7 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The objective of this note is to approximate a birth and death matrix B by a close Toeplitztype one for which explicit formulas for the eigenpairs are known. Numerical evidence of the approximation behavior of the eigenvalues and eigenvectors of B by those of such Toeplitztype matrices is provided. Keywords Birth and death matrix · Eigenvalue · Eigenvector · Toeplitz matrix Mathematics Subject Classification 15B05 · 65F15
1 Introduction Birth and death processes are special cases of continuous time Markov processes with only two types of state transitions: birth and death (see, e.g. Karlin 2014; Nowak 2006). Such processes occur in a wide variety of situations in science, engineering and business and have applications to several research fields, such as epidemics, queuing models, population dynamics, genetic models (see, e.g. Horne and Magagna 1970; Parthasarathy and Lenin 1999 and the references therein). The evolution in time of a birth and death process is characterized by a set of linear differential equations of the form d p(t) = Bp(t), dt
(1)
Communicated by Jinyun Yuan. This work was partially supported by project UID/MAT/00324/2019. This work was partially supported by FCT- Fundação para a Ciência e Tecnologia, under project UID/MAT/04721/2020.
B
Susana Furtado [email protected] Natália Bebiano [email protected]
1
CMUC and Departamento de Matemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal
2
CEAFEL and Faculdade de Economia do Porto, Universidade do Porto, 4200-464 Porto, Portugal 0123456789().: V,-vol
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N. Bebiano, S. Furtado
where p(t) = ( p1 (t), . . . pn (t))T , in which pi (t) is the probability of the system being in state i, i = 1, . . . , n, at instant t, and B is a so-called birth and death matrix (BD-matrix), that is, a tridiagonal matrix of the particular form ⎡ ⎤ −a0 b1 0 0 0 ⎢ a0 −a1 − b1 ⎥ b2 0 ⎢ ⎥ ⎢ ⎥ a1 −a2 − b2 b3 ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . a −a − b 2 3 3 (2) B=⎢ ⎥ ⎢ ⎥ . . . .. .. .. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎣ . −an−2 − bn−2 bn−1 ⎦ 0 an−2 −bn−1 , where the superdiagonal and subdiagonal elements are real positive numbers. We are focusing on homogeneous birth and death processes in which the associated matrix does not depend on time (see, e.g. Castillo and Zaballa 2020; Parthasarathy and Lenin 1999) for the nonhomogeneous case). The initial probabilities pi (0) are known and satisfy 0 ≤ pi (0) ≤ 1,
n
pi (0) = 1,
i = 1, . . . , n.
i=0
The solution of (1) is of the form p(t) =
n
ci wi eλi t ,
i=1
where the constant ci ’s are determined by the given initial conditions and wi = (wi1 , . . . , win )T is the eigenvector of B associated with the eigenvalue λi whose components sum is 1. The eigenvalues of B are real and distinct, one of them is 0 and the other ones are negative. It can be easil
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