Communication-efficient Distributed Solutions to a System of Linear Equations with Laplacian Sparse Structure

Transcription

1 Communcaton-effcent Dstrbuted Solutons to a System of Lnear Equatons wth Laplacan Sparse Structure Peng Wang, Yuanq Gao, Nanpeng Yu, We Ren, Janmng Lan, and D Wu Abstract Two communcaton-effcent dstrbuted algorthms are proposed to solve a system of lnear equatons Ax = b wth Laplacan sparse A. A system of lnear equatons wth Laplacan sparse A can be found n many applcatons, e.g., the power flow problems and other network flow problems. The frst algorthm s based on the gradent descent method n optmzaton and the agents only share two parts of the system state nstead of that of the whole system state, whch saves sgnfcant communcaton. The two parts shared by every agent through a communcaton lnk are the state nformaton of ts own and ts neghbor connected by the communcaton lnk. The second method s obtaned from an approxmaton to the Newton method, whch converges faster. It requres twce as much communcaton as the frst one but s stll communcatoneffcent due to the low dmenson of each part shared between agents. The convergence at a lnear rate of both methods s proved. A comprehensve comparson of the convergence rate, communcaton burden, and computaton costs between the methods s made. Fnally, smulatons are conducted to show the effectveness of both methods. I. INTRODUCTION Solvng a system of lnear equatons Ax = b s one of the most fundamental problems n many research felds. Wth the emergence of Internet of Thngs, an ncreasng amount of sensors and actuators are beng ntegrated nto the networked systems around us. Hence, dstrbuted methods to solve a system of lnear equatons are attractng more attentons from the researchers. Many dstrbuted algorthms to solve Ax = b were proposed n the lterature, e.g. [1] [17]. These algorthms assume that each agent knows some rows of the augmented matrx ( A b ). The algorthms n [12] [17] are contnuoustme ones whle those n [1] [11] are dscrete-tme ones. In ths paper we focus on dscrete-tme dstrbuted algorthms to solve Ax = b. In [1], a geometrcally convergent dstrbuted algorthm s proposed for both synchronous and asynchronous updates under repeatedly ontly strongly connected graphs, whch requres locally feasble ntalzatons. Ref. [10] extends the results n [1] by consderng the nfluence of communcaton and computaton delays and arbtrary ntalzatons. In [2], dstrbuted algorthms are proposed to fnd the mnmum norm soluton of a system of lnear Peng Wang, Janmng Lan, and D Wu are wth the Pacfc Northwest Natonal Laboratory, 902 Battelle Boulevard, Rchland, WA Correspondng emals: peng.wang@pnnl.gov, anmng.lan@pnnl.gov, d.wu@pnnl.gov. Yuanq Gao, Nanpeng Yu, and We Ren are wth the Department of Electrcal and Computer Engneerng at the Unversty of Calforna, Rversde, 900 Unversty Ave., Rversde, CA Correspondng emals: ygao024@ucr.edu, nyu@ece.ucr.edu, and ren@ece.ucr.edu equatons assocated wth weghted nner products. Ref. [9] then broadens the results n [1] and [2], allowng arbtrary ntalzatons for convergence to a general soluton and specal ntalzaton for convergence to a soluton closest to a gven pont. When Ax = b has a unque soluton, a dstrbuted algorthm s desgned n [3] to allow arbtrary ntalzatons wth feedback of the devaton from local systems of lnear equatons and the geometrc convergence rate s proved. Also, when Ax = b has a unque soluton, a dstrbuted algorthm s proposed n [8] usng the subgradent method and the lnear convergence rate s proved. In [5] [7], a dstrbuted algorthm that converges n fnte tme s also proposed to solve Ax = b. The algorthm requres agents to share the nformaton of kernels of local equatons wth ther neghbors, whch may lead to non-robustness. A dstrbuted method to solve Ax = b s proposed n [4] usng M-Feer mappngs and the convergence rates for two specal cases are also specfed. In the lterature mentoned above, the agents need to share ther estmates of the state of the whole system. However, n many applcatons where the matrx A s sparse and the system s large-scale, the dstrbuted methods n [1] [10] wll lead to sgnfcant communcaton overhead. In ths case, communcaton-effcent dstrbuted algorthms to solve Ax = b are necessary. A communcaton-effcent dstrbuted method s proposed n [11] for general sparse matrces, but the method n [11] requres the nformaton of common nonzero parts of agents rows and ther neghbors columns. When the matrx A s Laplacan sparse, the common nonzero parts of dfferent rows and columns actually requre sharng nformaton of agents common neghbors, whch mght not be avalable to the agents. In ths paper, we develop communcaton-effcent dstrbuted algorthms to solve Ax = b wth Laplacan sparse A, whch requres less communcaton than that n [11]. Matrces wth Laplacan sparse structure can be found n many problems such as network flow problems. In partcular, the power flow problem n the smart grd doman nvolves network matrces wth Laplacan sparse property. In our proposed communcaton-effcent dstrbuted algorthms, the agents only transmt the nformaton of ther own and one of ther neghbors connected by the communcaton lnk, nstead of the state of the whole system as n [1] [10] or the state of ther common neghbors as n [11]. In the frst proposed method, only two parts of the state vector of the system are transmtted through each communcaton lnk whle n the second proposed method, two parts of the state vector and the gradent vector, respectvely, of the system are transmtted.

2 As each part of the state vector and the gradent vector s low dmensonal, both methods sgnfcantly reduce the communcaton burden. We propose the frst communcatoneffcent dstrbuted algorthm to solve Ax = b wth Laplacan sparse A based on a gradent descent method and prove ts geometrc convergence. Then we propose an accelerated communcaton-effcent dstrbuted algorthm based on an approxmaton to the Newton method. The algorthm based on the approxmated Newton method requres twce as much communcaton as the one based on the gradent method, but t converges faster. The rest of the paper s organzed as follows. In Secton II, some prelmnary knowledge on graph theory, Laplacan sparse matrx, and power flow problems s ntroduced. In Secton III, the two communcaton-effcent dstrbuted algorthms are proposed to solve Ax = b wth Laplacan sparse A and a comparson between them s made. Smulatons are conducted to llustrate the effectveness of the two methods n Secton IV. Fnally, the conclusons are stated n Secton V. II. PRELIMINARIES In ths part, we wll provde some prelmnary knowledge on graph theory, whch s necessary for dstrbuted algorthms, Laplacan sparse matrx, whch s our research focus n ths paper, and power flow problems, whch provde applcatons to our research focus. A. Graph Theory An mth order undrected graph, denoted by G(V, E), s composed of a vertex set V = {1,, m} and an edge set E V V. We use the par (, ) to denote the edge between vertex and vertex. We suppose that (, ) / E, V. We say that s a neghbor of f there s an edge between and. The neghbor set N of vertex s composed of the neghbors of vertex,.e., N = { : (, ) E}. The number of vertex s neghbors s denoted by N. The Laplacan matrx L = [l ] m m R m m assocated wth the graph G s defned such that 1, N l = N, = ;. 0, otherwse A path between and s a sequence of edges (, 1 ), ( 1, 2 ),, ( p, ). An undrected graph s connected f for every par of vertces and ( ), there s a path between them. B. Laplacan Sparse Matrx We consder the matrx A wth the followng specal sparse structure: Defnton 1 (Laplacan sparse matrx): A matrx A has the Laplacan sparse structure of a graph G f a 0 only f and are neghbors n G or =,.e., l 0, where a s the (, )th entry of matrx A. A block matrx A has the Laplacan sparse structure of a graph G f A s a nonzero matrx only f and are neghbors n G or =,.e., l 0, where A s the (, )th block of matrx A. Laplacan sparse matrces appear n many problems, e.g., power flow problems. In power flow problems, the Ybus of a power system has Laplacan sparse structure f the communcaton topology s the same as the physcal one. Also, the Jacoban of the power flow equatons, though more complex, can also be regarded as a Laplacan sparse matrx. See Secton II-C for detals. C. The Power Flower Problem In ths subsecton, we wll detal the relatonshp between power flow problem and the Laplacan sparse matrx. The power flow problem s very fundamental n the steadystate analyss of electrcal power systems. The power flow problem s typcally formulated as solvng a system of nonlnear equatons, known as the power flow equatons. Laplacan sparse matrces emerge n numercal solutons to power flow equatons, e.g., the Newton-Raphson method. We wll gve a bref ntroducton to sngle-phased power flow problems and the Newton-Raphson method. A comprehensve descrpton of power flow problems and the Newton- Raphson method can be found n [18]. Consder an electrcal network modeled by a weghted graph G = (V, E, W), where V s a set of nodes, E a set of lnks representng transmsson/dstrbuton lnes, and W a set of weghts assocated wth E. The value of w W depends on the electrcal characterstcs of the lnk, e.g., the mpedance of the conductor. Let v be the nodal voltage of node, s be the net complex power necton at node, and Y be the bus admttance matrx. Notce that Y has the same sparse structure as the Laplacan matrx of G and has Laplacan sparse structure of G. Let v = v e θ, s = p + q, and Y = G + B, where s the magnary unt, v s the nodal voltage magntude, θ s nodal voltage angle, p s net actve power necton, q s the net reactve power necton, G s the conductance matrx, and B s the susceptance matrx. Then the power flow equaton s as follows: V k=1 v v k (G k cos θ k + B k sn θ k ) p = 0, V k=1 v (1) v k (G k sn θ k B k cos θ k ) q = 0, where G k and B k are the (, k)th entry of matrces G and B, respectvely, and θ k s the nodal voltage angle dfference between nodes and k. As the matrx Y has Laplacan sparse structure of G, so do G and B. So when nodes and k are not connected, G k and B k are both zero. The Newton-Raphson method to solve (1) s ntroduced below. Defne the vector of unknowns x = [θ 2,, θ V, v 2,, v V ] T. x does not contan v 1, θ 1 snce they are known. Let p (x) = V k=1 v v k (G k cos θ k + B k sn θ k ), q (x) = V k=1 v v k (G k sn θ k B k cos θ k ), (2)

3 and then (1) s converted to p (x) p = 0, = 2, 3,, V, q (x) q = 0, = 2, 3,, V. In the Newton-Raphson method, Eq. (3) s teratvely solved. In each teratve step, we need to solve a system of lnear equatons as follows: (3) J x = y (4) where J s the Jacoban matrx of the functons n (3), y s a constant related to (2) n each step. ( The Jacoban ) matrx can be parttoned as J = Jpθ J p v, where J pθ = p(x) θ, J p v = p(x) v, J qθ = J qθ J q v q(x) θ, and J q v = q(x) v. When G k and B k are both zero, the (, k)th entres of J pθ, J p v, J qθ, J q v are all zeros. So each block of the Jacoban matrx has the same sparse structure as the matrx G or B. As G and B have Laplacan sparse structure of G, J pθ, J p v, J qθ, J q v also have Laplacan sparse structure of G. III. COMMUNICATION-EFFICIENT DISTRIBUTED SOLUTIONS TO LINEAR EQUATIONS In ths secton, we wll develop two communcatoneffcent dstrbuted algorthms to solve Ax = b. The frst algorthm s based on the gradent descent method whle the second one on the approxmated Newton method, whch converges faster than the frst one. The frst algorthm only requres communcaton of two parts of the state vector of the whole system through each communcaton lnk,.e., the two parts owned by the two agents connected by the lnk, whle the second algorthm also requres the communcaton of the two correspondng parts of the gradent vector of the whole system. A. Problem Formulaton Suppose that we have a group of m agents whch form the vertex set V and communcaton lnks between the agents whch form the edges between the vertces. Then we have a graph G(V, E) to represent the agent network. For the graph G, we have the followng assumpton: Assumpton 1: The communcaton topology of the agent network s fxed, undrected, and connected. Each agent knows some rows A, V of A and the correspondng entres b n b. All the agents work together to obtan the soluton of Ax = b. We further assume that the matrx A has a Laplacan sparse structure of G as n Defnton 1. Remark 1: If the agent network G, nstead of beng connected, conssts of several connected components, a system of lnear equatons Ax = b wth Laplcan sparse A of G s ndeed composed of several ndependent subsystems of lnear equatons. These subsystems of lnear equatons can be ndependently solved. Let A () parts n x, and L () be the nonzero part n A, x () the correspondng the nonzero parts of L, where L s the th row of the Laplacan matrx L. Then the local equatons A x = b s equvalent to A () x () = b. Example 1: We consder a network G composed of four agents wth the Laplacan matrx L = Then A = s a matrx wth Laplacan sparse structure of G. If each agent owns one row of A, then A (1) ( ) 1 = 1 3.4, A (2) 2 = ( ), A (3) 3 = ( ), and A (4) 4 = ( ). Correspondngly, x (1) = ( ) T x 1 x 2, x (2) = ( ) T x 1 x 2 x 3, x (3) = ( ) T x 2 x 3 x 4, and x (4) = ( ) T (1) x 3 x 4. L 1 = ( 1 1 ), L (2) 2 = ( ), L (3) 3 = ( ), and L (4) 4 = ( 1 1 ). If we formulate solvng Ax = b as a dstrbuted optmzaton problem as follows 1 mn A x b 2 2 V subect to x = x,, V, we then have the followng lemma: Lemma 1: Under Assumpton 1 and the Laplacan sparse structure of A, solvng Ax = b s equvalent to solvng subect to mn f = 1 2 x () A () x () b 2 V (5) = x (), x () = x (), (, ) E, where x () s agent s local estmate on x, N {}. Proof: If x s a soluton of Ax = b, t s obvous that x () = ( ) x, V s also an optmal soluton to N {} (5), where x () = ( ) x s a collecton of the entres N {} n x corresponds to the neghbors of agent and tself. ( If x (1), x (2),, x ( m) form ) a soluton of (5), denote x = T x (1)T 1 x (2)T 2 x ( m)t m. For any agent, we have x = x() = x (), N. Notce that A has Laplacan sparse structure, so x () = ( ) x and thus A N {} x = A () x () = b. We can then transfer the constraned optmzaton problem n (5) to an unconstraned optmzaton problem wth penalty functons as follows: Lemma 2: Let f p = 1 2 [ V ( x () (,) E A () x () b 2 x () 2 + x () x () 2 )]. If Ax = b has solutons, (5) s equvalent to (6) mn f p. (7)

4 Proof: If Ax = b has solutons, we can see that the solutons of (5) and (7) are those satsfyng that A () x () = b, V and x () = x (), x () = x (), (, ) E. So (5) s equvalent to (7). Next, we wll propose a communcaton-effcent dstrbuted algorthm to solve Ax = b by solvng (7) based on the gradent descent method. B. Communcaton-effcent Algorthm Based on Gradent Descent Method In ths subsecton, we propose a communcaton-effcent dstrbuted algorthm to solve Ax = b based on the gradent descent method wth a constant step sze and prove ts convergence at a geometrc rate. For smplcty, we suppose that dm(x () ) = 1, V,.e., the state owned by every agent s a scalar. It s not dffcult to extend the results to the cases when dfferent agents own states of dfferent dmensons. We frst apply the gradent descent method wth a constant step sze α to (7). For the gradent of f p, we have that f p x =A()T () (A () x () b ) [(x () N x () ) e () + (x () x () ) e () (8) ], where e () = ( ) T wth the poston of 1 located at agent s local ndex of. Also, dm(e () ) = and Let f(x) = N {} dm(x () ) = dm(x() ). x = ( x (1)T x (2)T x ( m)t ) T ( ( fp x (1) ) T From the gradent descent method we obtan that for agent, (9) ( fp x (2) ) T ( fp x ( m) ) T ) T. (10) x(k + 1) = x(k) α f(x(k)), x () (k + 1) =x () (k) αa ()T (A () x () (k) b ) α (k) x () (k)) e () α N (x () (x () N (k) x() (k)) e (). (11) Remark 2: The communcaton from agent to agent s only the estmate of the states of agent, e.g. x (), and agent, x (), by agent rather than the states of all agents. So (11) reduces sgnfcant communcaton compared wth the algorthms n [1] [10]. Also, compared wth [11], (11) does not requre sharng estmates of the states of ther common neghbors under Assumpton 1 whle the method n [11] requres the nformaton of the agents common neghbors, whch may not be avalable. As a result, (11) needs less communcaton between agents than the method n [11]. For the performance of (11), we have the followng result: Theorem 1: If the system of lnear equatons Ax = b has solutons and A s Laplacan sparse of the agent network, x () (k), V n (11) converges n fnte tme or at a lnear 2 rate to the optmal pont of (7) f 0 < α < λ max( 2 (f, p)) where 2 (f p ) s the Hessan of f p n (6) and λ max ( ) represents the maxmal egenvalue. Let x (k) = ( x (1)T 1 (k) x (2)T 2 (k) x ( m)t m (k)) T. (12) Then x (k), k = 1, 2, 3, converges to a soluton of Ax = b. When Ax = b has a unque soluton, f p s strongly convex and the convergence at a lnear rate of (11) s a drect result of Theorem n [19]. But when Ax = b has multple solutons, f p n (6) s not strongly convex and thus the results n [19] cannot be used to prove Theorem 1. But the proof of the convergence of (11) when Ax = b has more than one soluton s omtted due to space lmtaton. C. Communcaton-effcent Algorthm Based on Approxmated Newton Method A communcaton-effcent dstrbuted algorthm was proposed n the prevous subsecton, but we fnd n some smulaton examples that t s very slow. So n ths subsecton, we wll propose an approxmated Newton method to accelerate the process to solve Ax = b,.e., mnmze (6). For smplcty, we suppose that dm(x () ) = 1, V,.e., the state owned by every agent s a scalar. It s not dffcult to extend the results to the cases where dfferent agents own states of dfferent dmensons. Remark 3: We can fnd an approxmated Newton method to solve dstrbuted optmzaton problems n [20], whch requres the local obectve functons to be strongly convex. But n our problem, the local obectve functons 1 2 A() x () b are not strongly convex. Thus, the analyss n [20] does not apply to the problem n ths paper. The centralzed Newton method to mnmze f p s as x(k + 1) = x(k) ( 2 f p (x(k))) 1 f p (x(k)). For smplcty, we denote H = 2 f p (x(k)) and g(k) = f p (x(k)). Notce that H s a constant matrx for f p n (6). As n (8), the gradent of f p s f p x =A()T () (A () x () b ) [(x () N x () ) e () + (x () For the Hessan of f p, we have that and 2 f p x 2 f p x () x () = = A()T ()2 A () e () e ()T N = A ()T A () + dag( L () ) { x () ) e () ]. + e () e()t e () e ()T e () e()t, N, 0, and / N

5 where dag( L () ) s a dagonal matrx whole dagonal entres are the absolute values of L (), the nonzero entres of the th row of Laplacan matrx L. As we do not fnd any method to compute the nverse of the Hessan of f p n a dstrbuted way, we wll next approxmate t. Let D = γ 2 f p, where γ > 1 s a constant, and D = x ()2 dag(d 1,, D m ). Also, let F = H D. Then we have that (1 γ)(a ()T A () + dag( L () )), =, F = e () e ()T e () e()t, N, (13) 0, otherwse and we can see that F s Laplacan sparse. Then, H = D + F and H 1 = (D + F ) 1 = D 1 2 (I + D 1 2 F D 1 2 ) 1 D 1 2 D 1 2 (I D 1 2 F D 1 2 )D 1 2 = D 1 D 1 F D 1 Then the dstrbuted approxmated Newton method to solve Ax = b s x(k + 1) = x(k) (D 1 D 1 F D 1 )g(k), (14) where x s defned n (9). As matrx F s Laplacan sparse from (13), then for agent, (14) becomes x (k + 1) =x (k) (D 1 g (k) D 1 F D 1 g (k)). (15) N {} Then we have the followng result on the performance of (14) and (15): Theorem 2: Suppose that the system of lnear equatons Ax = b has a unque soluton and A s Laplacan sparse. Then under Assumpton 1, x () (k), V n (15) converges at a lnear rate to the optmal soluton of (7). x (k), k = 1, 2, 3, n (12) converges to the soluton of Ax = b. The proof of Theorem 2 s omtted due to space lmtaton. D. Comparson Between the Two Algorthms We make a bref comparson between the gradent descent based method (11) and the approxmated Newton based method (15) on the convergence rate, communcaton burdens, and computaton costs n ths subsecton. In terms of the convergence rate, although a quanttatve convergence rate s not avalable for ether method, we fnd through smulatons that the approxmated Newton based method (15) converges much faster than the gradent descent based method (11). The communcaton burden of the approxmated Newton based algorthm doubles that of the gradent descent based algorthm n each teraton. We already know that agent n (11) only requres two parts,.e., x () and x () the communcaton lnk (, ). In (15), let v () = D 1 When N, as F = e () e ()T e () e()t through g (k)., agent only requres v () and v () besdes x () and x (). So the algorthm based on approxmated Newton method requres twce as much communcaton as that based on the gradent descent method. But as the dmensons of x () and x () are usually very low n spte of the scale of the whole system, the algorthm based on approxmated Newton method s stll communcaton-effcent. For example, n three-phase power flow problems, dm(x () dm(v () ), dm(x () ), dm(v () ), and ) are all at most three, so what s transmtted from agent to s at most twelve scalars no matter how large the whole system s n the approxmated Newton based method. The approxmated network method (15) has a hgher computaton cost than the gradent descent based method (11). Frst, both algorthms need to calculate g (k) and then do a subtracton of two dm(x () )th order vectors. In addton, agent n (15) needs to do more computatons. These computatons nclude 1) computaton of a dm(x () )th order symmetrc matrx, D = A ()T A () ntalzaton step, + dag( L () ), whch can be done at the 2) the nverse of a dm(x () )th order symmetrc matrx, D 1, whch can be done at the ntalzaton step, 3) the multplcaton of D 1 and g (k) per teraton, 4) summaton of N vectors, v (), wth dmenson dm(x () ) after recevng the nformaton from ts neghbors per teraton. 5) a subtracton of two vectors wth dmenson dm(x () ) per teraton. We can see that the extra computaton burdens of (15) depends on dm(x () ) and dm(x () ). In many problems, dm(x () ) and dm(x () ) are usually very small n spte of the scale of the whole system. For example, n three-phase power flow problems, dm(x () ) s at most three and dm(x () ) s at most 3 N, whch s also small due to the sparse physcal connectons n the real world systems. But as (15) usually converges much faster than (11), the total communcaton and computaton costs of (15) may be lower than those of (11). IV. SIMULATION RESULTS In ths secton, we conduct smulaton studes to llustrate the effectveness of the communcaton-effcent dstrbuted algorthms (11) and (15). The frst smulaton set up s based on solvng one step of update equatons n the Newton-Raphson approach to solve the power flow problem n (1). The testng system s the IEEE 13-bus test feeder. We treat each bus as an agent and assume that the communcaton network and the physcal network share the same topology. The state vector of the whole system conssts of the magntudes and phase angles of the complex voltages of all buses. As the buses may be of one-phase, two-phase, or three-phase, dfferent agents have state vectors of dfferently dmensonal.

6 Fg. 1. (a) Norms of Errors between x () (k) and Accurate Solutons We can see that n Secton ( II-C the ) Jacoban matrx Jpθ J conssts of four blocks J = p v. Every block has J qθ J q v the same sparse structure as the Ybus of the power system and thus s Laplacan sparse. The Jacoban matrx s more complex than a Laplacan sparse matrx. However, we can stll solve (4) wth the methods n (11) and (15). The smulaton results are shown n Fg. 1(a). The step sze α n (11) s chosen as , and we observe that f we choose α = , (11) dverges. So α = mght be very close to the largest step sze that makes (11) converge n ths example. The coeffcent γ n (15) s selected as 2. The second smulaton set up s based on solvng a random generated system of lnear equatons wth the Laplacan sparse structure of ts communcaton topology. And the communcaton graph s a 30th-order undrected rng. The smulaton results are shown n Fg. 1(b). The step sze α n (11) s chosen as 0.1, and we observe that f we choose α = 0.15, (11) dverges. So α = 0.1 mght be very close to the largest step sze that makes (11) converge n ths example. The coeffcent γ n (15) s selected as 2. From the two smulaton examples, we can see that both algorthms converges to the soluton of Eq.(4), and the approxmated Newton based algorthm converges much faster than that of the gradent descent based algorthm (11). V. CONCLUSIONS Two communcaton-effcent dstrbuted algorthms have been proposed n ths paper. The frst algorthm was based on gradent descent method whle the second one was based on an approxmaton to the Newton method. The frst algorthm only requred two parts of the state vector to be communcated nstead of the whole state vector, whle the second algorthm requred twce as much communcaton as the frst one. In the frst algorthm, the two parts communcated through each communcaton lnk were the states owned by the two agents connected by the communcaton lnk. In the second algorthm, two parts of the gradent vector were also transmtted besdes the two parts of the state vector. The second algorthm converged faster than the frst one at the prce of heaver computaton and communcaton burdens. It was proven that both algorthms converged at a lnear rate. Future works may nclude convergence rate analyss and asynchronous updates. (b) REFERENCES [1] S. Mou, J. Lu, and A. S. Morse, A dstrbuted algorthm for solvng a lnear algebrac equaton, IEEE Transactons on Automatc Control, vol. 60, no. 11, pp , Nov [2] P. Wang, W. Ren, and Z. Duan, Dstrbuted mnmum weghted norm soluton to lnear equatons assocated wth weghted nner product, n 2016 IEEE 55th Conference on Decson and Control (CDC), Dec 2016, pp [3] L. Wang, D. Fullmer, and A. S. Morse, A dstrbuted algorthm wth an arbtrary ntalzaton for solvng a lnear algebrac equaton, n 2016 Amercan Control Conference (ACC), July 2016, pp [4] P. Wang, W. Ren, and Z. 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