CHUN-HUA GUO. Key words. matrix equations, minimal nonnegative solution, Markov chains, cyclic reduction, iterative methods, convergence rate
|
|
- Steven Davidson
- 6 years ago
- Views:
Transcription
1 CONVERGENCE ANALYSIS OF THE LATOUCHE-RAMASWAMI ALGORITHM FOR NULL RECURRENT QUASI-BIRTH-DEATH PROCESSES CHUN-HUA GUO Abstract The minimal nonnegative solution G of the matrix equation G = A 0 + A 1 G + A 2 G 2, where the matrices A i i = 0, 1, 2 are nonnegative and A 0 +A 1 +A 2 is stochastic, plays an important role in the study of quasi-birth-death processes QBDs The Latouche-Ramaswami algorithm is a highly efficient algorithm for finding the matrix G The convergence of the algorithm has been shown to be quadratic for positive recurrent QBDs and for transient QBDs In this paper, we show that the convergence of the algorithm is linear with rate 1/2 for null recurrent QBDs under mild assumptions This new result explains the experimental observation that the convergence of the algorithm is still quite fast for nearly null recurrent QBDs Key words matrix equations, minimal nonnegative solution, Markov chains, cyclic reduction, iterative methods, convergence rate AMS subject classifications 15A24, 15A51, 60J10, 60K25, 65U05 1 Introduction A discrete-time quasi-birth-death process QBD is a Markov chain with state space {i, j i 0, 1 j m}, which has a transition probability matrix of the form P = B 0 B A 0 A 1 A A 0 A 1 A A 0 A 1 where B 0, B 1, A 0, A 1, and A 2 are m m nonnegative matrices such that P is stochastic In particular, A 0 + A 1 + A 2 e = e, where e is the column vector with all components equal to one The matrix P is also assumed to be irreducible Thus, A 0 0 and A 2 0 The matrix equation, 11 G = A 0 + A 1 G + A 2 G 2 plays an important role in the study of the QBD see [12] and [16] It is known that 11 has at least one solution in the set {G 0 Ge e} ie, the set of substochastic matrices The desired solution G is the minimal nonnegative solution We assume that A = A 0 + A 1 + A 2 is irreducible Then, by the Perron-Frobenius Theorem see [17], there exists a unique vector α > 0 with α T e = 1 and α T A = α T The vector α is called the stationary probability vector of A By Theorem 723 in [12], the QBD is null recurrent if α T A 0 e = α T A 2 e; positive recurrent if α T A 0 e > α T A 2 e; and transient if α T A 0 e < α T A 2 e For our purpose, we may use this criterion as an alternative definition for the three classes of QBDs This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2, Canada chguo@mathureginaca 1
2 2 CHUN-HUA GUO The minimal nonnegative solution of 11 can be found by any of the following three fixed-point iterations see [3], [5], [9], [10], [14], [15], [18]: G n+1 = A 0 + A 1 G n + A 2 G 2 n, G 0 = 0, G n+1 = I A 1 1 A 0 + A 2 G 2 n, G 0 = 0, G n+1 = I A 1 A 2 G n 1 A 0, G 0 = 0 Among the three iterations, iteration 14 has the fastest rate of convergence An inversion free version of 14 has also been proposed in [1] and analysed in [1] and [5] These four iterations are adequate for most situations However, the convergence of all four iterations is sublinear when the QBD is null recurrent see [5] The convergence of these methods is also extremely slow if the QBD is nearly null recurrent The algorithm proposed by Latouche and Ramaswami [11] is a little more complicated However, it works very well even for nearly null recurrent QBDs The algorithm is as follows: Algorithm 11 Set For k = 0, 1,, compute H 0 = I A 1 1 A 2 ; L 0 = I A 1 1 A 0 ; G 0 = L 0 ; T 0 = H 0 U k = H k L k + L k H k ; H k+1 = I U k 1 H 2 k; L k+1 = I U k 1 L 2 k; G k+1 = G k + T k L k+1 ; T k+1 = T k H k+1 It is shown in [11] that the matrices H k and L k are well defined and nonnegative and that the sequence {G k } converges quadratically to the matrix G for positive recurrent QBDs and for transient QBDs The algorithm is called a logarithmic reduction algorithm in [11] We will call it the LR algorithm for Logarithmic Reduction or for Latouche-Ramaswami A similar method is proposed in [2] for positive recurrent QBDs Since the LR algorithm has the greatest advantage over the fixed-point iterations when the QBD is nearly null recurrent, it is important to know the convergence rate of the LR algorithm when the QBD is null recurrent Before we can determine the convergence rate, we will take a closer look into the LR algorithm and present some preliminary results 2 Preliminaries It was mentioned in [11] that G W Stewart pointed out that the LR algorithm is related to the cyclic reduction technique We will make this point more transparent and derive two equations relating H k and L k Let G and F be the minimal nonnegative solution of 11 and 21 F = A 2 + A 1 F + A 0 F 2, respectively We have the following fundamental result see [12], for example
3 CONVERGENCE OF THE LATOUCHE-RAMASWAMI ALGORITHM 3 Theorem 21 If the QBD is positive recurrent, then G is stochastic and F is substochastic with spectral radius ρf < 1 If the QBD is transient, then F is stochastic and G is substochastic with ρg < 1 If the QBD is null recurrent, then G and F are both stochastic It is clear that the matrix G is also the minimal nonnegative solution of G = L 0 + H 0 G 2 Thus, we have the infinite system 22 W 0 H 0 0 L 0 I H 0 L 0 I 0 I G G 2 = K for appropriate K 0 and W 0 As in [2], we apply the cyclic reduction algorithm to 22 and get a reduced system Multiplying both sides of the reduced system by a proper block diagonal matrix, we get an infinite system with the same structure as 22, but with G replaced by G 2 After repeated application of the cyclic reduction algorithm and the block diagonal scaling, we obtain for each n 0, 23 W n H n 0 L n I H n L n I 0 I G 2n G 2 2n = K n 0 0, where H n and L n are as in the LR algorithm From equation 23, we have 24 L n + G 2n H n G 2 2n = 0 for each n 0 Therefore, G = L 0 + H 0 G 2 = L 0 + H 0 L 1 + H 1 G 4 = L 0 + H 0 L 1 + H 0 H 1 L 2 + H 2 G 8 = In general, 25 G = G k + 0 i k H i G 2 2k, where G k is as in the LR algorithm It is clear that the matrix F is also the minimal nonnegative solution of F = H 0 + L 0 F 2 By repeating the whole process leading to the equation 24, we get for each n 0, 26 H n + F 2n L n F 2 2n = 0 From 26, we can see that H n F 2n for each n 0 Thus, we have by G G k F 2 2k 1 G 2 2k
4 4 CHUN-HUA GUO Therefore, if the QBD is positive recurrent or transient, the quadratic convergence of {G k } is an immediate consequence of Theorem 21 In this situation, it is also very easy to determine the limits of the sequences {H k } and {L k } The following result is necessary Theorem 22 Let Q be a stochastic matrix If Q r has a positive column for some integer r 1, then there is a unique vector q 0 such that q T Q = q T and q T e = 1 the vector q is called the stationary probability vector of Q Moreover, there are constants K > 0 and β 0, 1 such that Q n eq T Kβ n for all n 0 In particular, lim n Q n = eq T For a proof of this result, see [6] See also [7] for the special case when Q r is positive for some integer r 1 Obviously, the condition that Q r has a positive column for some r 1 is necessary for lim n Q n = eq T If the QBD is positive recurrent, then G is stochastic and ρf < 1 Assuming that G p has a positive column for some integer p 1, we get from 24 and 26 that lim n H n = 0 and lim n L n = eg T, where g is the stationary probability vector of G If the QBD is transient, then ρg < 1 and F is stochastic Assuming that F p has a positive column for some integer p 1, we have lim n L n = 0 and lim n H n = ef T, where f is the stationary probability vector of F The limits of {H n } and {L n } were determined in [11] in a different way If the QBD is null recurrent, then ρg = 1 and ρf = 1 In this case, 27 tells us nothing about the convergence rate of the LR algorithm It is also much more difficult to determine the limits of the sequences {H n } and {L n } These issues will be resolved in the next section 3 Convergence rate of the LR algorithm for the null recurrent case We start with an algebraic proof of a basic result about the LR algorithm An probabilistic proof was given in [11] Lemma 31 For each k 0, H k + L k e = e Proof First, H 0 +L 0 e = I A 1 1 A 0 +A 2 e = e Assuming that H k +L k e = e k 0, we have H k +L k 2 e = e So, I H k L k L k H k e = Hk 2 +L2 k e Therefore, H k+1 + L k+1 e = I H k L k L k H k 1 Hk 2 + L2 k e = e We have thus proved the result by induction In the above proof, we have used the fact that the sequences {H k } and {L k } are well defined ie, the matrices I H k L k L k H k are nonsingular It is noted in [11] that, when the QBD is null recurrent, it is not true in general that one of the two sequences {H k } and {L k } converges to 0 Our next result shows that neither of the two sequences can converge to 0 for null recurrent QBDs Lemma 32 For the null recurrent QBD, there is a sequence {α k } such that for all k 0, α k 0, αk T e = 1, αt k H k + L k = αk T, and αt k H ke = αk T L ke = 1 2 Proof Recall that α is the stationary probability vector of A 0 + A 1 + A 2 So, α T I A 1 = α T A 0 + A 2 = α T I A 1 H 0 + L 0 Let ˆα T = α T I A 1 Since α > 0 and A 0 0, ˆα T = α T A 0 + A 2 0 and c 0 = ˆα T e > 0 Since the QBD is null recurrent, we have α T A 2 e = α T A 0 e and thus ˆα T H 0 e = ˆα T L 0 e Let α 0 = ˆα/c 0 It is clear that α 0 has all the properties in the lemma, noting that α T 0 H 0 e + α T 0 L 0 e = α T 0 e = 1 Assuming that an α i i 0, with all the properties in the lemma, has been found, we are going to find an α i+1 satisfying these properties
5 CONVERGENCE OF THE LATOUCHE-RAMASWAMI ALGORITHM 5 Since we have α T i = α T i H i + L i = α T i H i + L i 2 = α T i H 2 i + L 2 i + H i L i + L i H i, α T i I H i L i L i H i = α T i H 2 i + L 2 i = α T i I H i L i L i H i H i+1 + L i+1 Since α i 0 and I H i L i L i H i is nonsingular, αi T I H il i L i H i = αi T H2 i +L2 i 0 Thus, αi T H2 i + L2 i e > 0 and we can define α T i+1 = αt i H2 i + L2 i α T i H2 i + L2 i e = αt i I H il i L i H i α T i H2 i + L2 i e It remains to prove α T i+1 H i+1e = α T i+1 L i+1e, which is equivalent to α T i H2 i e = αt i L2 i e Note that α T i H 2 i e α T i L 2 i e = α T i H i e L i e α T i L i e H i e = α T i H i L i e + α T i L i H i e = α T i I L i L i e + α T i I H i H i e = α T i L 2 i e α T i H 2 i e Thus, αi T H2 i e = αt i L2 i e Remark 31 The result in the above lemma has also been obtained independently by Ye [19] In [19] it is assumed that αi T H2 i + L2 i e 0 for each i In the first version of this paper, the author used the assumption that Hi 2 + L2 i e > 0 for each i Without this assumption, the short argument in the proof of the lemma showing αi T H2 i + L2 i e > 0 for each i was pointed out by two referees Our further analysis will rely on Theorem 22 In order to apply Theorem 22, we make the following assumption: 31 deta 0 + za 1 + z 2 A 2 zi has no zeros on the unit circle other than z = 1 This assumption may be verified easily when the matrices A 0, A 1, A 2 have special structures see [13], for example From [4] we know that, in the null recurrent case, assumption 31 is equivalent to the assumption that λ = 1 is a simple eigenvalue of G and F and there are no other eigenvalues of G or F on the unit circle It is easy to show that the latter assumption is in turn equivalent to the next assumption 32 G p and F q have each a positive column for some p 1 and some q 1 Note that assumption 32 for G is satisfied if G k in the LR algorithm has a positive column for some k 0, since G G k In particular, assumption 32 for G is satisfied if L 0 has a positive column Similar comments can be made on assumption 32 for F Since assumptions 31 and 32 are equivalent, Theorem 22 can be applied to G and F under assumption 31 We let f and g be the unique stationary probability vector of F and G, respectively Since H k + L k e = e for all k 0, the sequences {H k } and {L k } are bounded and hence have convergent subsequences Let {H nk } and {L nk } be convergent with lim H n k = H, lim L n k = L
6 6 CHUN-HUA GUO Then, by equations 24 and 26 and Theorem 22, L + eg T Heg T = 0, H + ef T Lef T = 0 Therefore, H = af T with a = e Le, and L = bg T with b = e He Note that a + b = 2e H + Le = e We have thus proved the following result Lemma 33 For the null recurrent QBD with assumption 31, if H, L is a limit point of {H k, L k }, then H = af T and L = bg T with a 0, b 0, and a + b = e To prove that the convergence of the LR algorithm is linear with rate 1/2, we will need to show that lim H k = 1 2 ef T, lim L k = 1 2 egt Lemma 33 is only one small step towards this goal Many other auxiliary results will be needed Although we are unable to show the convergence of the sequences {H k } and {L k } at the moment, it is fairly easy to show that the sequence {α k } in Lemma 32 converges Lemma 34 For the null recurrent QBD with assumption 31, lim α k = 1 f + g 2 Proof Let α be any limit point of {α k } and lim α nk = α We will prove that α = 1 2 f + g We may assume without loss of generality that lim H n k = af T, lim L n k = bg T for some a, b 0 with a + b = e By taking limits in we get α T n k = α T n k H nk + L nk, α T n k H nk e = α T n k L nk e, α T = α T af T + bg T, α T a = α T b Thus, α T a = α T e a = 1 α T a So, α T a = α T b = 1/2 and α T = 1 2 f T + g T, or α = 1 2 f + g As we have already seen, in the null recurrent case, the two equations 24 and 26 are not sufficient to determine the convergence of the sequences {H n } and {L n } We have to seek additional information from the recursions for the sequences {H n } and {L n } The next result is one such finding Lemma 35 For the null recurrent QBD with assumption 31, if and g T a 1, then lim H n k = af T, lim L n k = bg T, lim H n k +1 = âf T, lim L n k +1 = ˆbg T,
7 CONVERGENCE OF THE LATOUCHE-RAMASWAMI ALGORITHM 7 with â = 1 + gt a 1 + 2g T a a + gt a 1 + 2g T a b, ˆb = g T a 1 + 2g T a a gt a 1 + 2g T a b Proof Let ãf T, bg T be any limit point of {H nk +1, L nk +1} and let lim H n sk +1 = ãf T, lim L n sk +1 = bg T Since we get by letting k, I H nsk L nsk L nsk H nsk H nsk +1 = H nsk 2, I af T bg T bg T af T ãf T = af T af T Post-multiplying the above equality by e gives By Lemma 32, ã = f T a + f T bg T ãa + g T af T ãb λa + µb α T n k H nk e = α T n k L nk e = 1 2, α n sk +1 T H nsk +1e = α nsk +1 T L nsk +1e = 1 2 By taking limits in the above identities and using Lemma 34, we have f T + g T a = f T + g T b = f T + g T ã = f T + g T b = 1 So, f T a = 1 g T a = g T e g T a = g T b Similarly, f T b = g T a, f T ã = g T b, f T b = g T ã Thus, λ + µ = f T a + f T bg T ã + g T af T ã = f T a + f T bg T ã + f T ã = f T a + f T b = f T e = 1 Now, Thus, µ = g T af T ã = g T af T λa + µb = 1 µg T af T a + µg T af T b = 1 µg T a1 g T a + µg T a g T a1 g T aµ = g T a1 g T a Since g T a 1, we have µ = g T a/1 + 2g T a and λ = 1 µ = 1 + g T a/1 + 2g T a So, and ã = 1 + gt a 1 + 2g T a a + gt a 1 + 2g T a b, b = e ã = a + b ã = g T a 1 + 2g T a a gt a 1 + 2g T a b
8 8 CHUN-HUA GUO The proof is completed since the limit point is uniquely determined by a and b We can now move a little closer to our goal Lemma 36 For the null recurrent QBD with assumptions 31 and 33 Each limit point af T of the sequence {H n } is such that 0 < g T a < 1, the sequence {H n, L n } has a limit point 1 2 ef T, 1 2 egt Proof Take any subsequence {H nk, L nk } such that lim H n k, L nk = a 0 f T, b 0 g T By the previous lemma, for each integer r 1, where 34 lim H n k +r, L nk +r = a r f T, b r g T, a k+1 = 1 + gt a k 1 + 2g T a k a k + gt a k 1 + 2g T a k b k, and b k+1 = e a k+1 for each integer k 0 Let p k = g T a k We have by 34 p k+1 = 1 + p kp k 1 + 2p k + p k1 p k 1 + 2p k = 2p k 1 + 2p k Since p 0 = g T a 0 > 0 by assumption, it is easy to show that lim p k = 1 2 By 34 we have which can be rewritten as a k+1 = Since lim g T a k = 1 2, we have for k large enough Thus gt a k g T e + a k 1 + 2g T a k, a k a k e = g T a k ak 1 2 e a k e 2 3 a k 1 2 e lim a r = lim b r = 1 r r 2 e Therefore, we can find a subsequence {H mk, L mk } such that lim H m k, L mk = 1 2 ef T, 1 2 egt This completes the proof The next result is quite straightforward Lemma 37 Let the relation between H k+1, L k+1 and H k, L k in the LR algorithm be denoted by H k+1, L k+1 = T H k, L k
9 CONVERGENCE OF THE LATOUCHE-RAMASWAMI ALGORITHM 9 Then 1 2 ef T, 1 2 egt is a fixed point of T Proof It is easy to verify that The result follows since 1 I 2 ef T 1 2 egt egt 2 ef T 1 2 ef T = 1 2 ef T 2, 1 I 2 ef T 1 2 egt egt 2 ef T 1 2 egt = 1 2 egt 2 M I 1 2 ef T 1 2 egt 1 2 egt 1 2 ef T = I 1 4 ef T + g T is a nonsingular M-matrix note that Me = e/2 Thus, we have shown that the sequence {H n, L n } defined by H k+1, L k+1 = T H k, L k k 0 has a limit point 1 2 ef T, 1 2 egt that is a fixed point of T By a theorem on general fixed-point iterations see [8, p 21], for example, we can conclude that the whole sequence {H n, L n } converges to this fixed point if the spectral radius of the Fréchet derivative of the operator T at the fixed point is less than 1 But, unfortunately, the spectral radius is not less than 1 in our case the spectral radius is equal to 4 when the matrices A 0, A 1, A 2 are 1 1 However, the sequence {H n, L n } can still converge since H n, L n may approach 1 2 ef T, 1 2 egt in a special way A delicate analysis for the error H n 1 2 ef T, L n 1 2 egt is necessary For notational convenience, let H = 1 2 ef T and L = 1 2 egt It is easy to see that H 2 = LH = 1 2 H, L2 = HL = 1 2 L We start with expressing H k+1 H in terms of H k H and L k L: H k+1 H = I H k L k L k H k 1 H 2 k I HL LH 1 H 2 = I H k L k L k H k 1 H 2 k H 2 + I H k L k L k H k 1 I HL LH 1 H 2 = I H k L k L k H k 1 H 2 k H 2 + I H k L k L k H k 1 I HL LH I Hk L k L k H k I HL LH 1 H 2 = I H k L k L k H k 1 H 2 k H 2 + H k L k HL + L k H k LHH = I H k L k L k H k 1 H k H k H + H k HH +H k L k L + H k HL + L k H k H + L k LHH To simplify the expression, observe that Thus, and H k H + L k LH = 1 2 Hk + L k e H + Le f T = 0 L k LH = H k HH Hk HL + L k LH H = 1 2 H k H + L k LH = 0
10 10 CHUN-HUA GUO Therefore, H k+1 H = I H k L k L k H k 1 H k H k H + H k HH Similarly, we can get H k H k HH + L k H k HH L k+1 L = I H k L k L k H k 1 L k L k L + L k LL L k L k LL + H k L k LL Now, for any ɛ 0, 1 4, we can find δ > 0 such that whenever H k H δ and L k L δ, 35 H k+1 H = I HL LH 1 HH k H + H k HH with W k ɛ H k H, and HH k HH + LH k HH + W k L k+1 L = I HL LH 1 LL k L + L k LL LL k LL + HL k LL + Z k with Z k ɛ L k L To get rid of the inverse in 35, we use 36 Since I HL LH 1 = I 1 2 H + L 1 = I H + L H + L2 + H + L HH k H + H k HH HH k HH + LH k HH = HH k H + H + LH k HH HH k HH + LH k HH = HH k H + 2LH k HH, and H + L i HH k H + 2LH k HH = HH k H + 2LH k HH for all i 1, we get by 35 and 36 that H k+1 H = HH k H + H k HH HH k HH + LH k HH Similarly, +HH k H + 2LH k HH + W k = H k HH + 2HH k H HH k HH + 3LH k HH + W k L k+1 L = L k LL + 2LL k L LL k LL + 3HL k LL + Z k For the scalar case, H = L = 1 2 So, the estimate for H k+1 H becomes H k+1 H 2H k H If we could replace 3LH k HH by 3HH k HH in the estimate, we would have H k+1 H 1 2 H k H instead Thus, we should try to show that 3H + LH k HH is small in the general case Let α = f + g/2 We have 3H + LH k HH = 3 2 eα T H k Hef T = 3 2 eα α k T H k Hef T,
11 CONVERGENCE OF THE LATOUCHE-RAMASWAMI ALGORITHM 11 since αk T H k He = 1 2 αt k 1 2e = 0 by Lemma 32 Similarly, 3H + LL k LL = 3 2 eα α k T L k Leg T Since lim α k = α by Lemma 34, we can find integer k 1 such that for all k k 1, 3H + LH k HH = P k, 3H + LL k LL = Q k with P k ɛ H k H and Q k ɛ L k L Now we have H k+1 H = H k HH + 2HH k H 4HH k HH + P k + W k 37 = 1 2 H k H 1 2 I 4HH k HI 2H + P k + W k, and L k+1 L = L k LL + 2LL k L 4LL k LL + Q k + Z k 38 = 1 2 L k L 1 2 I 4LL k LI 2L + Q k + Z k Next we will estimate the term H k HI 2H in 37 and the term L k LI 2L in 38 By the equations 24 and 26, we have H k I G 2 2k F 2 2k = F 2k G 2k F 2 2k, L k I F 2 2k G 2 2k = G 2k F 2k G 2 2k Now, Similarly, H k HI 2H = H k I ef T = F 2k G 2k F 2 2k + H k G 2 2k F 2 2k ef T = F 2k ef T G 2k eg T F 2 2k eg T F 2 2k ef T +H k G 2 2 k eg T F 2 2k + eg T F 2 2k ef T L k LI 2L = G 2k eg T F 2k ef T G 2 2k ef T G 2 2k eg T +L k F 2 2 k ef T G 2 2k + ef T G 2 2k eg T By Theorem 22, there are constants C 1 > 0 and β 0, 1 such that H k HI 2H C 1 β 2k, L k LI 2L C 1 β 2k for all k 0 Now, by 37 and 38, we have H k+1 H ɛ H k H + C 2 β 2k, L k+1 L ɛ L k L + C 2 β 2k for any k k 1 with H k H < δ and L k L < δ Let r = ɛ < 1 Since H, L is a limit point of {H k, L k } by Lemma 36, we can find l k 1 such that H l H < δ, L l L < δ, rδ + C 2 β 2l < δ, and β 2l r Now, it is clear that
12 12 CHUN-HUA GUO 39 and 310 are valid for all k l and that β 2l+j 1 r j for all j 0 Thus, we can obtain for any m 1 that H l+m H r m H l H + C 2 r m 1 β 2l + r m 2 β 2l β 2l+m 1 and that r m H l H + C 2 mr m, L l+m L r m L l L + C 2 mr m Therefore, lim H k = H and lim L k = L Moreover, since ɛ > 0 can be arbitrarily small, we also have lim sup k Hk H 1 2, k lim sup Lk L 1 2 In summary, we have proved the following result Theorem 38 For the null recurrent QBD with assumptions 31 and 33, we have lim H k = 1 2 ef T, lim L k = 1 2 egt It is clear that assumption 33 is necessary for the conclusion of the above theorem Since the assumption cannot be verified directly, we will give a sufficient condition that is easier to verify Proposition 39 Let the components of f and g be f i and g i i = 1, 2,, m, respectively, and let S f = {i 1 i m, f i = 0}, S g = {i 1 i m, g i = 0} If assumption 31 and the assumption that 311 S f S g or S g S f are satisfied, then assumption 33 is also satisfied Proof Let lim H n k = af T, lim L n k = bg T It is shown in the proof of Lemma 35 that f T + g T a = f T + g T b = 1, f T a = g T b, f T b = g T a If g T a = 1, then g T b = f T a = 0 By assumption 311, we would have f T +g T a = 0 or f T + g T b = 0, which is a contradiction Similarly, we get a contradiction if g T a = 0 Remark 32 Assumption 311 is certainly satisfied if one of F and G is irreducible in particular, if one of H 0 and L 0 is irreducible since one of S f and S g is an empty set in this case We are now ready to determine the convergence rate of the LR algorithm for the null recurrent case Recall that, for the sequence {G k } generated by the LR algorithm, 312 G G k = H 0 H 1 H k G 2k+1
13 CONVERGENCE OF THE LATOUCHE-RAMASWAMI ALGORITHM 13 Proposition 310 For each k 0, H 0 H 1 H k 0 Proof Equation 847 in [12] states that H 0 H 1 H k ij is the probability of first passage to the state 2 k+1, j before any of the states 0, starting from 1, i If all of these entries were equal to 0, it would be impossible to reach the states 2 k+1, and the transition probability matrix P would not be irreducible Remark 33 The above proof is provided by a referee In the first version of the paper, the author gave the statement in Proposition 310 as an assumption The next theorem is our main result It shows that the sequence {G k } converges to the minimal nonnegative solution of 11 at precisely the rate of 1/2 Theorem 311 For the null recurrent QBD with assumptions 31 and 33, k lim Gk G = 1 2 Proof Since lim H k = 1 2 ef T, we have lim H k e = 1 2e Therefore, for any ɛ 0, 1 2, we can find an integer k 0 such that 1 2 ɛe H ke 1 2 +ɛe for all k > k 0 Note that, by 312, for k > k 0 Thus, G G k = G G k e = H 0 H k0 H k0 +1 H k e 1 2 ɛk k 0 H 0 H k0 e G G k ɛk k 0 H 0 H k0 e In view of Proposition 310, it follows readily that lim sup k Gk G ɛ, k lim inf Gk G 1 2 ɛ Since ɛ can be arbitrarily small, we have lim k G k G = Improvement of the approximate solution in the null recurrent case By 312 and Theorem 38, it is easy to get a much better approximation to the matrix G from the sequence {G k } generated by the LR algorithm In fact, we have by 312 G G k 2G G k+1 = H 0 H k G 2k+1 G 2k+2 +H 0 H k 1 H k 2H k H k+1 G 2k+2 The first term converges to zero quadratically by Theorem 22 since G 2k+1 G 2k+2 = G 2k+1 eg T G 2k+2 eg T The second term is also much smaller than G G k+1 since lim H k 2H k H k+1 = 0 and lim H k H k+1 = 1 4 ef T by Theorem 38 Therefore, Gk+1 = 2G k+1 G k = G k+1 + G k+1 G k can be a much better approximation to G in the null recurrent case Of course, improvements may also be achieved for nearly null recurrent QBDs by using the above strategy 5 Examples In this section, we will present a few examples to illustrate the theoretical results in Section 3 and the simple strategy described in Section 4 for the improvement of the approximate solution For all examples, assumption 31 is checked through the equivalent assumption 32 Example 51 Consider the equation 11 with A 0 = , A 1 = , A 2 =
14 14 CHUN-HUA GUO It is easy to verify that the corresponding QBD is null recurrent We also find that G 1 = L 0 + H 0 L 1 is irreducible and has a positive column and that F 1 = H 0 + L 0 H 1 has a positive column Since G G 1 and F F 1, assumptions 31 and 311 are satisfied By Proposition 39, assumption 33 is also satisfied For this example, the exact minimal nonnegative solutions of 11 and 21 can be found to be G = Accordingly, we have , F = g T = 1431, 874, 120/2425, f T = 1, 0, 4/5 For the matrices H 18 and L 18, found by the LR algorithm using double precision, we have H ef T = , L egt = Note that H 18 and L 18 are already very close to 1 2 ef T and 1 2 egt, respectively We also find that, for the matrices G k computed by the LR algorithm, G G 17 = and G G 18 = Note that G G G G 17 For G 18 = 2G 18 G 17, we have G G 18 = So, G18 is a much better approximation for G For the next example, assumption 311 is satisfied although neither of S f and S g is empty Example 52 Consider the equation 11 with A 0 = , A 1 = , A 2 =
15 CONVERGENCE OF THE LATOUCHE-RAMASWAMI ALGORITHM 15 The corresponding QBD is clearly null recurrent Since 0 05 H 0 = L 0 = 0 05 for this example, assumptions 31 and 311 are satisfied It is easy to find that 0 1 G = F = 0 1 So, we actually have S f = S g = {1} For this example, we have H k = 1 2 ef T and L k = 1 2 egt for all k 0 We also have for each k /2 k+1 G k = 0 1 1/2 k+1 So, {G k } converges to G linearly with rate 1/2 and G k = 2G k G k 1 = G for all k 1 We can also find examples for which 311 is not satisfied Example 53 Consider the equation 11 with A 0 = , A 1 = The corresponding QBD is clearly null recurrent Since H 0 =, L = , A 2 = for this example, assumption 31 is satisfied It is easy to find that G =, F = So, we have S f = {1} and S g = {2} Therefore, assumption 311 is not satisfied However, the conclusions in our main results in Section 3 still hold In fact, we have H k = 1 2 ef T and L k = 1 2 egt for all k 0 We also have for each k 0 G k = 1 1/2 k /2 k+1 0 So, {G k } converges to G linearly with rate 1/2 and G k = 2G k G k 1 = G for all k 1 There are also examples for which assumption 31 is not satisfied The next example is provided by a referee Example 54 Consider the equation 11 with A 0 = , A 1 = 0, A 2 = The corresponding QBD is clearly null recurrent In this case, 0 1 G = F = 1 0
16 16 CHUN-HUA GUO So, 311 is true, but 31 is not satisfied It is easy to find that H k = L k = 1 2 I for each k 1 and that 0 1 1/2 G k = k+1 1 1/2 k+1 0 for each k 0 Thus, {G k } converges to G linearly with rate 1/2 and G k = 2G k G k 1 = G for all k 1 We do not have any examples of null recurrent QBDs for which the convergence of the LR algorithm is not linear with rate 1/2 Acknowledgments The author is grateful to the three referees for their very helpful comments REFERENCES [1] Z-Z Bai, A class of iteration methods based on the Moser formula for nonlinear equations in Markov chains, Linear Algebra Appl, , pp [2] D Bini and B Meini, On the solution of a nonlinear matrix equation arising in queueing problems, SIAM J Matrix Anal Appl, , pp [3] P Favati and B Meini, On functional iteration methods for solving nonlinear matrix equations arising in queueing problems, IMA J Numer Anal, , pp [4] H R Gail, S L Hantler, and B A Taylor, Spectral analysis of M/G/1 and G/M/1 type Markov chains, Adv Appl Probab, , pp [5] C-H Guo, On the numerical solution of a nonlinear matrix equation in Markov chains, Linear Algebra Appl, , pp [6] D J Hartfiel, Markov Set-Chains, Lecture Notes in Math 1695, Springer, Berlin, 1998 [7] R A Horn and C R Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985 [8] M A Krasnoselskii, G M Vainikko, P P Zabreiko, Ya B Rutitskii, and V Ya Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen, 1972 [9] G Latouche, Newton s iteration for non-linear equations in Markov chains, IMA J Numer Anal, , pp [10] G Latouche, Algorithms for evaluating the matrix G in Markov chains of P H/G/1 type, Cahiers Centre Études Rech Opér, , pp [11] G Latouche and V Ramaswami, A logarithmic reduction algorithm for quasi-birth-death processes, J Appl Probab, , pp [12] G Latouche and V Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, PA, 1999 [13] G Latouche and P G Taylor, Level-phase independence for GI/M/1-type Markov chains, J Appl Probab, , pp [14] B Meini, New convergence results on functional iteration techniques for the numerical solution of M/G/1 type Markov chains, Numer Math, , pp [15] M F Neuts, Moment formulas for the Markov renewal branching process, Adv in Appl Probab, , pp [16] M F Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press, Baltimore, MD, 1981 [17] R S Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962 [18] Q Ye, High accuracy algorithms for solving nonlinear matrix equations in queueing models, in Advances in Algorithmic Methods for Stochastic Models Proceedings of the 3rd International Conference on Matrix Analytic Methods, G Latouche and P G Taylor, eds, Notable Publications Inc, NJ, 2000, pp [19] Q Ye, On Latouche-Ramaswami s logarithmic reduction algorithm for quasi-birth-and-death processes, Research Report , Department of Mathematics, University of Kentucky, 2001
Matrix analytic methods. Lecture 1: Structured Markov chains and their stationary distribution
1/29 Matrix analytic methods Lecture 1: Structured Markov chains and their stationary distribution Sophie Hautphenne and David Stanford (with thanks to Guy Latouche, U. Brussels and Peter Taylor, U. Melbourne
More informationIterative Solution of a Matrix Riccati Equation Arising in Stochastic Control
Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Chun-Hua Guo Dedicated to Peter Lancaster on the occasion of his 70th birthday We consider iterative methods for finding the
More informationA CONVERGENCE RESULT FOR MATRIX RICCATI DIFFERENTIAL EQUATIONS ASSOCIATED WITH M-MATRICES
A CONVERGENCE RESULT FOR MATRX RCCAT DFFERENTAL EQUATONS ASSOCATED WTH M-MATRCES CHUN-HUA GUO AND BO YU Abstract. The initial value problem for a matrix Riccati differential equation associated with an
More informationCensoring Technique in Studying Block-Structured Markov Chains
Censoring Technique in Studying Block-Structured Markov Chains Yiqiang Q. Zhao 1 Abstract: Markov chains with block-structured transition matrices find many applications in various areas. Such Markov chains
More informationStructured Markov chains solver: tool extension
Structured Markov chains solver: tool extension D. A. Bini, B. Meini, S. Steffé Dipartimento di Matematica Università di Pisa, Pisa, Italy bini, meini, steffe @dm.unipi.it B. Van Houdt Department of Mathematics
More informationN.G.Bean, D.A.Green and P.G.Taylor. University of Adelaide. Adelaide. Abstract. process of an MMPP/M/1 queue is not a MAP unless the queue is a
WHEN IS A MAP POISSON N.G.Bean, D.A.Green and P.G.Taylor Department of Applied Mathematics University of Adelaide Adelaide 55 Abstract In a recent paper, Olivier and Walrand (994) claimed that the departure
More informationOverload Analysis of the PH/PH/1/K Queue and the Queue of M/G/1/K Type with Very Large K
Overload Analysis of the PH/PH/1/K Queue and the Queue of M/G/1/K Type with Very Large K Attahiru Sule Alfa Department of Mechanical and Industrial Engineering University of Manitoba Winnipeg, Manitoba
More informationIEOR 6711: Stochastic Models I Professor Whitt, Thursday, November 29, Weirdness in CTMC s
IEOR 6711: Stochastic Models I Professor Whitt, Thursday, November 29, 2012 Weirdness in CTMC s Where s your will to be weird? Jim Morrison, The Doors We are all a little weird. And life is a little weird.
More informationOPTIMAL SCALING FOR P -NORMS AND COMPONENTWISE DISTANCE TO SINGULARITY
published in IMA Journal of Numerical Analysis (IMAJNA), Vol. 23, 1-9, 23. OPTIMAL SCALING FOR P -NORMS AND COMPONENTWISE DISTANCE TO SINGULARITY SIEGFRIED M. RUMP Abstract. In this note we give lower
More informationOn a quadratic matrix equation associated with an M-matrix
Article Submitted to IMA Journal of Numerical Analysis On a quadratic matrix equation associated with an M-matrix Chun-Hua Guo Department of Mathematics and Statistics, University of Regina, Regina, SK
More informationMatrix Analytic Methods for Stochastic Fluid Flows
Matrix Analytic Methods for Stochastic Fluid Flows V. Ramaswami, AT&T Labs, 2 Laurel Avenue D5-3B22, Middletown, NJ 7748 We present an analysis of stochastic fluid flow models along the lines of matrix-analytic
More informationMarkov Chains, Stochastic Processes, and Matrix Decompositions
Markov Chains, Stochastic Processes, and Matrix Decompositions 5 May 2014 Outline 1 Markov Chains Outline 1 Markov Chains 2 Introduction Perron-Frobenius Matrix Decompositions and Markov Chains Spectral
More informationIn particular, if A is a square matrix and λ is one of its eigenvalues, then we can find a non-zero column vector X with
Appendix: Matrix Estimates and the Perron-Frobenius Theorem. This Appendix will first present some well known estimates. For any m n matrix A = [a ij ] over the real or complex numbers, it will be convenient
More informationKernels of Directed Graph Laplacians. J. S. Caughman and J.J.P. Veerman
Kernels of Directed Graph Laplacians J. S. Caughman and J.J.P. Veerman Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207. caughman@pdx.edu, veerman@pdx.edu
More informationWeighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 401 412 (2013) http://campus.mst.edu/adsa Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes
More informationarxiv: v1 [math.na] 9 Apr 2010
Quadratic Vector Equations Federico Poloni arxiv:1004.1500v1 [math.na] 9 Apr 2010 1 Introduction In this paper, we aim to study in an unified fashion several quadratic vector and matrix equations with
More informationMulti Stage Queuing Model in Level Dependent Quasi Birth Death Process
International Journal of Statistics and Systems ISSN 973-2675 Volume 12, Number 2 (217, pp. 293-31 Research India Publications http://www.ripublication.com Multi Stage Queuing Model in Level Dependent
More informationBOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION
K Y BERNETIKA VOLUM E 46 ( 2010), NUMBER 4, P AGES 655 664 BOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION Guang-Da Hu and Qiao Zhu This paper is concerned with bounds of eigenvalues of a complex
More informationA probabilistic proof of Perron s theorem arxiv: v1 [math.pr] 16 Jan 2018
A probabilistic proof of Perron s theorem arxiv:80.05252v [math.pr] 6 Jan 208 Raphaël Cerf DMA, École Normale Supérieure January 7, 208 Abstract Joseba Dalmau CMAP, Ecole Polytechnique We present an alternative
More informationModified Gauss Seidel type methods and Jacobi type methods for Z-matrices
Linear Algebra and its Applications 7 (2) 227 24 www.elsevier.com/locate/laa Modified Gauss Seidel type methods and Jacobi type methods for Z-matrices Wen Li a,, Weiwei Sun b a Department of Mathematics,
More informationExamples of Countable State Markov Chains Thursday, October 16, :12 PM
stochnotes101608 Page 1 Examples of Countable State Markov Chains Thursday, October 16, 2008 12:12 PM Homework 2 solutions will be posted later today. A couple of quick examples. Queueing model (without
More informationModelling Complex Queuing Situations with Markov Processes
Modelling Complex Queuing Situations with Markov Processes Jason Randal Thorne, School of IT, Charles Sturt Uni, NSW 2795, Australia Abstract This article comments upon some new developments in the field
More informationNote that in the example in Lecture 1, the state Home is recurrent (and even absorbing), but all other states are transient. f ii (n) f ii = n=1 < +
Random Walks: WEEK 2 Recurrence and transience Consider the event {X n = i for some n > 0} by which we mean {X = i}or{x 2 = i,x i}or{x 3 = i,x 2 i,x i},. Definition.. A state i S is recurrent if P(X n
More informationMATH36001 Perron Frobenius Theory 2015
MATH361 Perron Frobenius Theory 215 In addition to saying something useful, the Perron Frobenius theory is elegant. It is a testament to the fact that beautiful mathematics eventually tends to be useful,
More informationThe MATLAB toolbox SMCSolver for matrix-analytic methods
The MATLAB toolbox SMCSolver for matrix-analytic methods D. Bini, B. Meini, S. Steffe, J.F. Pérez, B. Van Houdt Dipartimento di Matematica, Universita di Pisa, Italy Department of Electrical and Electronics
More informationDetailed Proof of The PerronFrobenius Theorem
Detailed Proof of The PerronFrobenius Theorem Arseny M Shur Ural Federal University October 30, 2016 1 Introduction This famous theorem has numerous applications, but to apply it you should understand
More informationA NEW EFFECTIVE PRECONDITIONED METHOD FOR L-MATRICES
Journal of Mathematical Sciences: Advances and Applications Volume, Number 2, 2008, Pages 3-322 A NEW EFFECTIVE PRECONDITIONED METHOD FOR L-MATRICES Department of Mathematics Taiyuan Normal University
More informationCentral limit theorems for ergodic continuous-time Markov chains with applications to single birth processes
Front. Math. China 215, 1(4): 933 947 DOI 1.17/s11464-15-488-5 Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes Yuanyuan LIU 1, Yuhui ZHANG 2
More informationOn the Class of Quasi-Skip Free Processes: Stability & Explicit solutions when successively lumpable
On the Class of Quasi-Skip Free Processes: Stability & Explicit solutions when successively lumpable DRAFT 2012-Nov-29 - comments welcome, do not cite or distribute without permission Michael N Katehakis
More informationWe first repeat some well known facts about condition numbers for normwise and componentwise perturbations. Consider the matrix
BIT 39(1), pp. 143 151, 1999 ILL-CONDITIONEDNESS NEEDS NOT BE COMPONENTWISE NEAR TO ILL-POSEDNESS FOR LEAST SQUARES PROBLEMS SIEGFRIED M. RUMP Abstract. The condition number of a problem measures the sensitivity
More informationON SOME BASIC PROPERTIES OF THE INHOMOGENEOUS QUASI-BIRTH-AND-DEATH PROCESS
Comm. Korean Math. Soc. 12 (1997), No. 1, pp. 177 191 ON SOME BASIC PROPERTIES OF THE INHOMOGENEOUS QUASI-BIRTH-AND-DEATH PROCESS KYUNG HYUNE RHEE AND C. E. M. PEARCE ABSTRACT. The basic theory of the
More informationR, 1 i 1,i 2,...,i m n.
SIAM J. MATRIX ANAL. APPL. Vol. 31 No. 3 pp. 1090 1099 c 2009 Society for Industrial and Applied Mathematics FINDING THE LARGEST EIGENVALUE OF A NONNEGATIVE TENSOR MICHAEL NG LIQUN QI AND GUANGLU ZHOU
More informationBirth-death chain models (countable state)
Countable State Birth-Death Chains and Branching Processes Tuesday, March 25, 2014 1:59 PM Homework 3 posted, due Friday, April 18. Birth-death chain models (countable state) S = We'll characterize the
More informationAdvanced Queueing Theory
Advanced Queueing Theory 1 Networks of queues (reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity, BCMP networks, mean-value analysis, Norton's
More informationApplied Mathematics Letters. Comparison theorems for a subclass of proper splittings of matrices
Applied Mathematics Letters 25 (202) 2339 2343 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Comparison theorems for a subclass
More informationCover Page. The handle holds various files of this Leiden University dissertation
Cover Page The handle http://hdlhandlenet/1887/39637 holds various files of this Leiden University dissertation Author: Smit, Laurens Title: Steady-state analysis of large scale systems : the successive
More informationComputing the pth Roots of a Matrix. with Repeated Eigenvalues
Applied Mathematical Sciences, Vol. 5, 2011, no. 53, 2645-2661 Computing the pth Roots of a Matrix with Repeated Eigenvalues Amir Sadeghi 1, Ahmad Izani Md. Ismail and Azhana Ahmad School of Mathematical
More informationMarkov Chains and Stochastic Sampling
Part I Markov Chains and Stochastic Sampling 1 Markov Chains and Random Walks on Graphs 1.1 Structure of Finite Markov Chains We shall only consider Markov chains with a finite, but usually very large,
More informationMATH 56A: STOCHASTIC PROCESSES CHAPTER 1
MATH 56A: STOCHASTIC PROCESSES CHAPTER. Finite Markov chains For the sake of completeness of these notes I decided to write a summary of the basic concepts of finite Markov chains. The topics in this chapter
More informationThe Eigenvalue Shift Technique and Its Eigenstructure Analysis of a Matrix
The Eigenvalue Shift Technique and Its Eigenstructure Analysis of a Matrix Chun-Yueh Chiang Center for General Education, National Formosa University, Huwei 632, Taiwan. Matthew M. Lin 2, Department of
More informationMath Homework 5 Solutions
Math 45 - Homework 5 Solutions. Exercise.3., textbook. The stochastic matrix for the gambler problem has the following form, where the states are ordered as (,, 4, 6, 8, ): P = The corresponding diagram
More informationSimultaneous Transient Analysis of QBD Markov Chains for all Initial Configurations using a Level Based Recursion
Simultaneous Transient Analysis of QBD Markov Chains for all Initial Configurations using a Level Based Recursion J Van Velthoven, B Van Houdt and C Blondia University of Antwerp Middelheimlaan 1 B- Antwerp,
More informationImproved Newton s method with exact line searches to solve quadratic matrix equation
Journal of Computational and Applied Mathematics 222 (2008) 645 654 wwwelseviercom/locate/cam Improved Newton s method with exact line searches to solve quadratic matrix equation Jian-hui Long, Xi-yan
More informationA TANDEM QUEUE WITH SERVER SLOW-DOWN AND BLOCKING
Stochastic Models, 21:695 724, 2005 Copyright Taylor & Francis, Inc. ISSN: 1532-6349 print/1532-4214 online DOI: 10.1081/STM-200056037 A TANDEM QUEUE WITH SERVER SLOW-DOWN AND BLOCKING N. D. van Foreest
More informationACCURATE SOLUTION ESTIMATE AND ASYMPTOTIC BEHAVIOR OF NONLINEAR DISCRETE SYSTEM
Sutra: International Journal of Mathematical Science Education c Technomathematics Research Foundation Vol. 1, No. 1,9-15, 2008 ACCURATE SOLUTION ESTIMATE AND ASYMPTOTIC BEHAVIOR OF NONLINEAR DISCRETE
More information6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities
6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities 1 Outline Outline Dynamical systems. Linear and Non-linear. Convergence. Linear algebra and Lyapunov functions. Markov
More informationClassification of Countable State Markov Chains
Classification of Countable State Markov Chains Friday, March 21, 2014 2:01 PM How can we determine whether a communication class in a countable state Markov chain is: transient null recurrent positive
More informationKey words. Strongly eventually nonnegative matrix, eventually nonnegative matrix, eventually r-cyclic matrix, Perron-Frobenius.
May 7, DETERMINING WHETHER A MATRIX IS STRONGLY EVENTUALLY NONNEGATIVE LESLIE HOGBEN 3 5 6 7 8 9 Abstract. A matrix A can be tested to determine whether it is eventually positive by examination of its
More informationFrame Diagonalization of Matrices
Frame Diagonalization of Matrices Fumiko Futamura Mathematics and Computer Science Department Southwestern University 00 E University Ave Georgetown, Texas 78626 U.S.A. Phone: + (52) 863-98 Fax: + (52)
More informationDiscrete time Markov chains. Discrete Time Markov Chains, Limiting. Limiting Distribution and Classification. Regular Transition Probability Matrices
Discrete time Markov chains Discrete Time Markov Chains, Limiting Distribution and Classification DTU Informatics 02407 Stochastic Processes 3, September 9 207 Today: Discrete time Markov chains - invariant
More informationHomework 3 posted, due Tuesday, November 29.
Classification of Birth-Death Chains Tuesday, November 08, 2011 2:02 PM Homework 3 posted, due Tuesday, November 29. Continuing with our classification of birth-death chains on nonnegative integers. Last
More informationLecture 10: Powers of Matrices, Difference Equations
Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each
More informationLecture 7. µ(x)f(x). When µ is a probability measure, we say µ is a stationary distribution.
Lecture 7 1 Stationary measures of a Markov chain We now study the long time behavior of a Markov Chain: in particular, the existence and uniqueness of stationary measures, and the convergence of the distribution
More informationAffine iterations on nonnegative vectors
Affine iterations on nonnegative vectors V. Blondel L. Ninove P. Van Dooren CESAME Université catholique de Louvain Av. G. Lemaître 4 B-348 Louvain-la-Neuve Belgium Introduction In this paper we consider
More informationMatrix functions that preserve the strong Perron- Frobenius property
Electronic Journal of Linear Algebra Volume 30 Volume 30 (2015) Article 18 2015 Matrix functions that preserve the strong Perron- Frobenius property Pietro Paparella University of Washington, pietrop@uw.edu
More informationComponentwise perturbation analysis for matrix inversion or the solution of linear systems leads to the Bauer-Skeel condition number ([2], [13])
SIAM Review 4():02 2, 999 ILL-CONDITIONED MATRICES ARE COMPONENTWISE NEAR TO SINGULARITY SIEGFRIED M. RUMP Abstract. For a square matrix normed to, the normwise distance to singularity is well known to
More informationTwo Results About The Matrix Exponential
Two Results About The Matrix Exponential Hongguo Xu Abstract Two results about the matrix exponential are given. One is to characterize the matrices A which satisfy e A e AH = e AH e A, another is about
More information2. Transience and Recurrence
Virtual Laboratories > 15. Markov Chains > 1 2 3 4 5 6 7 8 9 10 11 12 2. Transience and Recurrence The study of Markov chains, particularly the limiting behavior, depends critically on the random times
More information642:550, Summer 2004, Supplement 6 The Perron-Frobenius Theorem. Summer 2004
642:550, Summer 2004, Supplement 6 The Perron-Frobenius Theorem. Summer 2004 Introduction Square matrices whose entries are all nonnegative have special properties. This was mentioned briefly in Section
More informationMarkov Chains, Random Walks on Graphs, and the Laplacian
Markov Chains, Random Walks on Graphs, and the Laplacian CMPSCI 791BB: Advanced ML Sridhar Mahadevan Random Walks! There is significant interest in the problem of random walks! Markov chain analysis! Computer
More information= P{X 0. = i} (1) If the MC has stationary transition probabilities then, = i} = P{X n+1
Properties of Markov Chains and Evaluation of Steady State Transition Matrix P ss V. Krishnan - 3/9/2 Property 1 Let X be a Markov Chain (MC) where X {X n : n, 1, }. The state space is E {i, j, k, }. The
More informationProperties for the Perron complement of three known subclasses of H-matrices
Wang et al Journal of Inequalities and Applications 2015) 2015:9 DOI 101186/s13660-014-0531-1 R E S E A R C H Open Access Properties for the Perron complement of three known subclasses of H-matrices Leilei
More informationInterval solutions for interval algebraic equations
Mathematics and Computers in Simulation 66 (2004) 207 217 Interval solutions for interval algebraic equations B.T. Polyak, S.A. Nazin Institute of Control Sciences, Russian Academy of Sciences, 65 Profsoyuznaya
More informationTHE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR
THE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR WEN LI AND MICHAEL K. NG Abstract. In this paper, we study the perturbation bound for the spectral radius of an m th - order n-dimensional
More informationInterlacing Inequalities for Totally Nonnegative Matrices
Interlacing Inequalities for Totally Nonnegative Matrices Chi-Kwong Li and Roy Mathias October 26, 2004 Dedicated to Professor T. Ando on the occasion of his 70th birthday. Abstract Suppose λ 1 λ n 0 are
More informationA FAST MATRIX-ANALYTIC APPROXIMATION FOR THE TWO CLASS GI/G/1 NON-PREEMPTIVE PRIORITY QUEUE
A FAST MATRIX-ANAYTIC APPROXIMATION FOR TE TWO CASS GI/G/ NON-PREEMPTIVE PRIORITY QUEUE Gábor orváth Department of Telecommunication Budapest University of Technology and Economics. Budapest Pf. 9., ungary
More informationSolving Nonlinear Matrix Equation in Credit Risk. by Using Iterative Methods
Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 39, 1921-1930 Solving Nonlinear Matrix Equation in Credit Risk by Using Iterative Methods a,b Gholamreza Farsadamanollahi 1, b Joriah Binti Muhammad and
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationThe Kalman filter is arguably one of the most notable algorithms
LECTURE E NOTES «Kalman Filtering with Newton s Method JEFFREY HUMPHERYS and JEREMY WEST The Kalman filter is arguably one of the most notable algorithms of the 0th century [1]. In this article, we derive
More informationOn hitting times and fastest strong stationary times for skip-free and more general chains
On hitting times and fastest strong stationary times for skip-free and more general chains James Allen Fill 1 Department of Applied Mathematics and Statistics The Johns Hopkins University jimfill@jhu.edu
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationCAAM 454/554: Stationary Iterative Methods
CAAM 454/554: Stationary Iterative Methods Yin Zhang (draft) CAAM, Rice University, Houston, TX 77005 2007, Revised 2010 Abstract Stationary iterative methods for solving systems of linear equations are
More informationHessenberg Pairs of Linear Transformations
Hessenberg Pairs of Linear Transformations Ali Godjali November 21, 2008 arxiv:0812.0019v1 [math.ra] 28 Nov 2008 Abstract Let K denote a field and V denote a nonzero finite-dimensional vector space over
More informationLecture 15 Perron-Frobenius Theory
EE363 Winter 2005-06 Lecture 15 Perron-Frobenius Theory Positive and nonnegative matrices and vectors Perron-Frobenius theorems Markov chains Economic growth Population dynamics Max-min and min-max characterization
More informationFor δa E, this motivates the definition of the Bauer-Skeel condition number ([2], [3], [14], [15])
LAA 278, pp.2-32, 998 STRUCTURED PERTURBATIONS AND SYMMETRIC MATRICES SIEGFRIED M. RUMP Abstract. For a given n by n matrix the ratio between the componentwise distance to the nearest singular matrix and
More informationMaximizing the numerical radii of matrices by permuting their entries
Maximizing the numerical radii of matrices by permuting their entries Wai-Shun Cheung and Chi-Kwong Li Dedicated to Professor Pei Yuan Wu. Abstract Let A be an n n complex matrix such that every row and
More informationRecursive Solutions of the Matrix Equations X + A T X 1 A = Q and X A T X 1 A = Q
Applied Mathematical Sciences, Vol. 2, 2008, no. 38, 1855-1872 Recursive Solutions of the Matrix Equations X + A T X 1 A = Q and X A T X 1 A = Q Nicholas Assimakis Department of Electronics, Technological
More informationCondition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models
J Sci Comput (2018) 76:759 776 https://doi.org/10.1007/s10915-018-0641-x Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models Jie Meng 1 Sang-Hyup Seo 1 Hyun-Min
More informationEXTINCTION TIMES FOR A GENERAL BIRTH, DEATH AND CATASTROPHE PROCESS
(February 25, 2004) EXTINCTION TIMES FOR A GENERAL BIRTH, DEATH AND CATASTROPHE PROCESS BEN CAIRNS, University of Queensland PHIL POLLETT, University of Queensland Abstract The birth, death and catastrophe
More informationThe Kemeny Constant For Finite Homogeneous Ergodic Markov Chains
The Kemeny Constant For Finite Homogeneous Ergodic Markov Chains M. Catral Department of Mathematics and Statistics University of Victoria Victoria, BC Canada V8W 3R4 S. J. Kirkland Hamilton Institute
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More information1 Matrices and Systems of Linear Equations. a 1n a 2n
March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real
More informationGeometric Mapping Properties of Semipositive Matrices
Geometric Mapping Properties of Semipositive Matrices M. J. Tsatsomeros Mathematics Department Washington State University Pullman, WA 99164 (tsat@wsu.edu) July 14, 2015 Abstract Semipositive matrices
More informationc 1995 Society for Industrial and Applied Mathematics Vol. 37, No. 1, pp , March
SIAM REVIEW. c 1995 Society for Industrial and Applied Mathematics Vol. 37, No. 1, pp. 93 97, March 1995 008 A UNIFIED PROOF FOR THE CONVERGENCE OF JACOBI AND GAUSS-SEIDEL METHODS * ROBERTO BAGNARA Abstract.
More informationRECENT RESULTS FOR SUPERCRITICAL CONTROLLED BRANCHING PROCESSES WITH CONTROL RANDOM FUNCTIONS
Pliska Stud. Math. Bulgar. 16 (2004), 43-54 STUDIA MATHEMATICA BULGARICA RECENT RESULTS FOR SUPERCRITICAL CONTROLLED BRANCHING PROCESSES WITH CONTROL RANDOM FUNCTIONS Miguel González, Manuel Molina, Inés
More informationNOTES ON THE PERRON-FROBENIUS THEORY OF NONNEGATIVE MATRICES
NOTES ON THE PERRON-FROBENIUS THEORY OF NONNEGATIVE MATRICES MIKE BOYLE. Introduction By a nonnegative matrix we mean a matrix whose entries are nonnegative real numbers. By positive matrix we mean a matrix
More informationInfinite-Horizon Average Reward Markov Decision Processes
Infinite-Horizon Average Reward Markov Decision Processes Dan Zhang Leeds School of Business University of Colorado at Boulder Dan Zhang, Spring 2012 Infinite Horizon Average Reward MDP 1 Outline The average
More informationIrregular Birth-Death process: stationarity and quasi-stationarity
Irregular Birth-Death process: stationarity and quasi-stationarity MAO Yong-Hua May 8-12, 2017 @ BNU orks with W-J Gao and C Zhang) CONTENTS 1 Stationarity and quasi-stationarity 2 birth-death process
More informationNecessary and sufficient conditions for strong R-positivity
Necessary and sufficient conditions for strong R-positivity Wednesday, November 29th, 2017 The Perron-Frobenius theorem Let A = (A(x, y)) x,y S be a nonnegative matrix indexed by a countable set S. We
More informationLinear-fractional branching processes with countably many types
Branching processes and and their applications Badajoz, April 11-13, 2012 Serik Sagitov Chalmers University and University of Gothenburg Linear-fractional branching processes with countably many types
More informationEigenvalue comparisons in graph theory
Eigenvalue comparisons in graph theory Gregory T. Quenell July 1994 1 Introduction A standard technique for estimating the eigenvalues of the Laplacian on a compact Riemannian manifold M with bounded curvature
More informationTHE RELATION BETWEEN THE QR AND LR ALGORITHMS
SIAM J. MATRIX ANAL. APPL. c 1998 Society for Industrial and Applied Mathematics Vol. 19, No. 2, pp. 551 555, April 1998 017 THE RELATION BETWEEN THE QR AND LR ALGORITHMS HONGGUO XU Abstract. For an Hermitian
More informationSpectral inequalities and equalities involving products of matrices
Spectral inequalities and equalities involving products of matrices Chi-Kwong Li 1 Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187 (ckli@math.wm.edu) Yiu-Tung Poon Department
More informationA NEW PROOF OF THE WIENER HOPF FACTORIZATION VIA BASU S THEOREM
J. Appl. Prob. 49, 876 882 (2012 Printed in England Applied Probability Trust 2012 A NEW PROOF OF THE WIENER HOPF FACTORIZATION VIA BASU S THEOREM BRIAN FRALIX and COLIN GALLAGHER, Clemson University Abstract
More informationAn estimation of the spectral radius of a product of block matrices
Linear Algebra and its Applications 379 (2004) 267 275 wwwelseviercom/locate/laa An estimation of the spectral radius of a product of block matrices Mei-Qin Chen a Xiezhang Li b a Department of Mathematics
More informationIntegers With Digits 0 or 1
MATHEMATICS OF COMPUTATION VOLUME 46, NUMBER 174 APRIL 1986, PAGES 683-689 Integers With Digits 0 or 1 By D. H. Lehmer, K. Mahler and A. J. van der Poorten Abstract. Let g > 2 be a given integer and Y
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 13, pp. 38-55, 2002. Copyright 2002,. ISSN 1068-9613. ETNA PERTURBATION OF PARALLEL ASYNCHRONOUS LINEAR ITERATIONS BY FLOATING POINT ERRORS PIERRE
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationAn improved convergence theorem for the Newton method under relaxed continuity assumptions
An improved convergence theorem for the Newton method under relaxed continuity assumptions Andrei Dubin ITEP, 117218, BCheremushinsaya 25, Moscow, Russia Abstract In the framewor of the majorization technique,
More informationZ-Pencils. November 20, Abstract
Z-Pencils J. J. McDonald D. D. Olesky H. Schneider M. J. Tsatsomeros P. van den Driessche November 20, 2006 Abstract The matrix pencil (A, B) = {tb A t C} is considered under the assumptions that A is
More information