the availability of data about large real - world networked systems commonly known as complex networks has demanded the development of several graph - theoretic and algebraic methods to study the structure and dynamical properties of such usually giant graphs . in a seminal paper ,watts and strogatz introduced one of such graph - theoretic indices to characterize the transitivity of relations in complex networks .the so - called clustering coefficient represents the ratio of the number of triangles in which the corresponding node takes place to the the number of potential triangles involving that node ( see further for a formal definition ) .the clustering coefficient of a node is bounded between zero and one , with values close to zero indicating that the relative number of transitive relations involving that node is low . on the other hand, a clustering coefficient close to one indicates that this node is involved in as many transitive relations as possible . when studying complex real - world networks it is very common to report the average watts - strogatz ( ws ) clustering coefficient as a characterization of how globally clustered a network is idea , however , was not new as reflected by the fact that luce and perry had proposed 50 years earlier an index to account for the network transitivity , given by the total number of triangles in the graph divided by the total number of triads existing in the graph .such index , hereafter called the graph transitivity , was then rediscovered by newman in the context of complex networks . here again this index is bounded between zero and one , with small values indicating poor transitivity and values close to one indicating a large one ( see also ) .it was first noticed by bollobs and then by estrada that there are certain graphs for which the two clustering coefficients diverge .that is , there are classes of graphs for which the ws average clustering coefficient tends to one while the graph transitivity tends to zero as the size of the graphs grows to infinity .the two families indentified by bollobs and by estrada are illustrated in figure [ examples ] and they correspond to the so - called _ friendship _ ( or dutch windmill or -fan ) _ graphs _ and _ agave graphs _ , respectively .the friendship graphs are formed by glueing together copies of a triangle in a common vertex .the agave graphs are formed by connecting disjoint nodes to both nodes of a complete graph .the friendship graphs are members of a larger family of graphs known as the _ windmill graphs _a windmill graph consists of copies of the complete graph  with every node connected to a common one ( see figure [ windmill graphs ] ) . in a recent paper estrada has proved that the divergence of the two clustering coefficient indices also takes place for windmill graphs when the number of nodes tends to infinity .the agave graphs belong to the general class of complete split graphs , which are the graphs consisting of a central clique and a disjoint set of nodes which are connected to all the nodes of the clique .it is important to note that this divergence between the two clustering coefficients also takes place in certain real - world networks .this phenomenon has passed inadverted among the plethora of papers that deals with clustering in real - world networks . in table[ table1 ] we illustrate some results collected from the literature in which the divergence of the two clustering coefficients is observed .the dataset is selected to intentionally display the heterogeneity of classes of real - world networks which display such phenomenon .they correspond to the interchange of emails among the employees of the enron corporation , a caida version of the internet as autonomous system , a metabolic network , a social network of hollywood actors and a network of coauthors in biology .> p2cm > p1.7cm > p1.8cm > p1cm >p1 cm & & & & +  e - mail & 36 692 & 183 831 & 0.50 & 0.09 + internet & 26 475 & 106 762 & 0.21 & 0.007 + metabolic & 765 & 3 686 & 0.67 & 0.09 + actors & 449 913 & 25 516 482 & 0.78 & 0.20 + coauthors & 1 520 251 & 11 803 064 & 0.60 & 0.09 +  in this work we study a class of graphs which accounts for the clustering divergence phenomenon in networks with degree disassortativity . these graphs , which we call _ core - satellite graphs _ ,are formed by a central clique ( the core ) connected to several other cliques ( the satellites ) in such a way that they generalize both the complete split and the windmill graphs .we prove here that the clustering coefficients of these graphs diverge and that they are always disassortative .we also characterize completely the eigenstructure of the adjacency and the laplacian matrices of these graphs , and comment on the asymptotic behavior of such quantities as the infection threshold and synchronizability index of these graphs .finally , we provide evidence that supports the idea of using the core - satellite graphs as models of certain classes of real - world networks .here we always consider _ simple _ _ undirected and connected graphs _ formed by a set of nodes ( vertices ) and a set of edges between the nodes .let us now recall an important graph operation which will be helpful in this work .the _ join _ ( or _ complete product _ ) of graphs and is the graph obtained from by joining every vertex of with every vertex of .  let us now define some specific classes of graphs which will be considered in this work . as usual , we denote with the complement of a graph .  the complete split graph is .the special cases and are known as the star and agave graphs , respectively .  in figure [ complete split ] we show some examples of complete split graphs .the windmill graph is .these are the graphs consisting of copies of meeting in a common vertex .the special cases are known as the friendship graphs .  in figure [ windmill graphs ] we show some examples of windmill graphs .let us now define a few graph - theoretic invariants that will be studied in this work .the so - called _ watts - strogatz clustering coefficient _ of a node , which quantifies the degree of transitivity of local relations in a graph is defined as :    where is the the number of triangles in which the node participates . taking the mean of these valuesas varies among all the nodes in , one gets the _ average ws clustering coefficient _ of the network :    the so - called graph transitivity is defined as     where is the total number of triangles and .another important network parameter is the degree assortativity coefficient which measures the tendency of high degree nodes to be connected to other high degree nodes ( assortativity ) or to low degree ones ( disassortativity ) .the assortativity coefficient is mathematically expressed as :    let be the _ adjacency matrix _ of the graph .we denote by the eigenvalues of .the dominant eigenvalue is also referred to as the perron eigenvalue or spectral radius of , denoted .let be the diagonal matrix of the vertex degrees of the graph .then , the laplacian matrix of the graph is defined by .we denote by the eigenvalues of .the eigenvalue is known as the _ algebraic connectivity _ of the graph and the eigenvector associated to it is known as the _ fiedler vector _ . for reviews about spectral properties of graphs the readeris directed to the classic works .the main paradigm for defining the core - satellite graphs is the following .we consider a group of central nodes in a network which are connected among them .then , there are a few cliques of the same size which are connected to the central core but which are not connected among them .formally , the core - satellite graphs are defined below .let , and .the core - satellite graph is .that is , they are the graphs consisting of copies of ( the satellites ) meeting in a common clique ( the core ) , see fig .[ core - satellite ] .the core - satellite graph generalizes the following classes of graphs :  * star graphs : ; * agave graphs : ; * complete split graphs : ; * friendship graphs : ; * windmill graphs : .the core - satellite graphs were previously defined in the literature in the context of extremal graph theory . they are named _generalized friendship graphs _ in the works of erds et al . , chen et al . and faudree et al . . however , the same name appears in connection to at least another class of graphs .the generalized friendship graph is defined by ahmad et al . , arumugan and nalliah , shi et al . and by jahari and alikhani as the graph consisting of cycles of orders having a common vertex .these graphs are also known as flowers .thus , to avoid any confusion we propose here the more intuitive name of _ core - satellite _ for these graphs .let be a core - satellite graph and let us designate any node in the core as and any node in a satellite as . then , and , also and .we now state our first result .[ thm : clustering]let be a core - satellite graph . then , for given values of and the average watts - strogatz and the transitivity coefficients diverge when the number of cliques tends to infinity :      first , we obtain an expression for the watts - strogatz clustering coefficient of windmill graphs .the watts - strogatz clustering coefficient of any node in the core clique of is and the rest of the nodes have .thus , , which gives    now we consider the transitivity index of a windmill graph .the total number of triangles in is and the number of 2-paths is thus , after substitution we get obviously , and for any given values of and , which proves the result .next , we prove a result related to the way in wich nodes connect to each other in core - satellite graphs .let be a core - satellite graph .then , for any values of , , and the core - satellite graphs are disassortative , i.e. ,   we first need a result previously obtained by estrada that expresses the assortativity coefficient in terms of subgraphs of the corresponding graph ,    where represents the ratio of paths of length to paths of length , is the number of star subgraphs with a central node and 3 pendant nodes , and is the transitivity index .it has been previously proved in that the denominator of ( [ eq : assortativity ] ) is positive .thus , the sign of the assortativity coefficient depends on the sign of .it is straighforward to realize that in a core - satellite graph the number of paths of length 3 is zero .consequently , the sign of depends only on the sign of .we remind that . thus , let us consider the difference     because then ( notice that if the resulting graph is just . if the core - satellite graph corresponds to the star graph , which is a tree , and consequently has .thus , the graph is diassortative .let , then because we have that , which implies that , i.e. , and consequently , which proves the result .in this section we give a full description of the spectral properties ( eigenvalues and eigenvectors ) of core - satellite graphs , and we investigate extensions to more general types of graphs .we begin by proving the following result .[ thm : spectrum ] the spectrum of the core - satellite graph consists of :  1 .  the eigenvalue with multiplicity ; 2 .  the eigenvalue with multiplicity ; 3 .  the eigenvalues given by the roots of the quadratic equation   for any positive integer , we use to denote the column vector with all entries equal to 1 .let and denote , respectively , the adjacency matrices of the complete graphs with and nodes . note that , with and standing for the and identity matrices . with an obvious ordering of the nodes , the adjacency matrix of can be written as the block matrix \hline \\[-8pt ] { \bf 1}_{s}{\bf 1}_{c}^{t } & a(k_{s})\\ \vdots &  & \ddots\\[2pt ] { \bf 1}_{s}{\bf 1}_{c}^{t } &  &  & a(k_{s } ) \end{array}\right],\label{eq : adj}\ ] ] with copies of appearing on the block diagonal .now let the vector \hline \\[-8pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0 } \end{array}\right],\qquad{\bf x}_{1}\ne{\bf 0},\label{eq : ev1}\ ] ] be partitioned conformally to .then the equation becomes \hline \\[-8pt ] ( { \bf 1}_{c}^{t}{\bf x}_{1}){\bf 1}_{s}\\ \vdots\\[2pt ] ( { \bf 1}_{c}^{t}{\bf x}_{1}){\bf 1}_{s } \end{array}\right]=\left[\begin{array}{c } \lambda{\bf x}_{1}\\[2pt ] \hline \\[-8pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0 } \end{array}\right].\ ] ] clearly , this is equivalent to , and .hence , and there are linearly independent eigenvectors of the form ( [ eq : ev1 ] ) .next , consider vectors of the form \hline \\[-8pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0}\\ { \bf x}_{i}\\[2pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0 } \end{array}\right],\qquad{\bf x}_{i}\ne{\bf 0},\qquad i=1,\ldots,\eta,\label{eq : ev2}\ ] ] where corresponds to the diagonal block of .then the equation becomes \hline \\[-8pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0}\\ a(k_{s}){\bf x}_{i}\\[2pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0 } \end{array}\right]=\left[\begin{array}{c } { \bf 0}\\[2pt ] \hline \\[-8pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0}\\ \lambda{\bf x}_{i}\\[2pt ] { \bf 0}\\ \vdots\\[2pt ] { \bf 0 } \end{array}\right].\ ] ] clearly , this is equivalent to , and for .this yields another linearly independent eigenvectors associated with the eigenvalue .now , consider vectors of the form \hline \\[-8pt ] \alpha_{1}{\bf 1}_{s}\\ \vdots\\[2pt ] \alpha_{\eta}{\bf 1}_{s } \end{array}\right],\label{eq : ev3}\ ] ] with the not all zero .then the equation becomes \hline \\[-8pt ] \alpha_{1}a(k_{s}){\bf 1}_{s}\\ \vdots\\[2pt ] \alpha_{\eta}a(k_{s}){\bf 1}_{s } \end{array}\right]=\left[\begin{array}{c } { \bf 0}\\[2pt ] \hline \\[-8pt ] \lambda\alpha_{1}{\bf 1}_{s}\\ \vdots\\[2pt ] \lambda\alpha_{\eta}{\bf 1}_{s } \end{array}\right].\ ] ] since at least one of the must be nonzero , this is equivalent to the first of these two conditions implies that , and there are exactly linearly independent eigenvectors of the form ( [ eq : ev3 ] ) since the solution space of the homogeneous linear equation has dimension .to determine the two remaining eigenvectors and corresponding eigenvalues , let \hline \\[-8pt ] \beta{\bf 1}_{s}\\ \vdots\\[2pt ] \beta{\bf 1}_{s } \end{array}\right],\qquad\beta\ne0.\label{eq : ev4}\ ] ] then the equation becomes \hline \\[-8pt ] c{\bf 1}_{s}+\beta(s-1){\bf 1}_{s}\\ \vdots\\[2pt ] c{\bf 1}_{s}+\beta(s-1){\bf 1}_{s } \end{array}\right]=\left[\begin{array}{c } \lambda{\bf 1}_{c}\\[2pt ] \hline \\[-8pt ] \lambda\beta{\bf 1}_{s}\\ \vdots\\[2pt ] \lambda\beta{\bf 1}_{s } \end{array}\right],\ ] ] which reduces to the two conditions conditions ( [ eq : cond ] ) yield hence which is easily seen to become equation ( [ eq : quadratic ] ) upon multiplication of both sides by and rearranging terms .solving for yields the two remaining ( simple ) eigenvalues of : .\label{eq : roots}\ ] ] these two roots yield two distinct values of via ( [ eq : cond1 ] ) and thus two distinct eigenvectors of of the form ( [ eq : ev4 ] ) .it is obvious that these two eigenvectors are linearly independent ( indeed , mutually orthogonal ) .this theorem generalizes the spectral analysis of windmill graphs given in to core - satellite graphs .we emphasize that the proof of the previous theorem explicitly reveals the structure of the eigenvectors of ; see equations ( [ eq : ev1 ] ) , ( [ eq : ev2 ] ) , ( [ eq : ev3 ] ) and ( [ eq : ev4 ] ) , together with the fact that the eigenvectors associated with the complete graphs are known.  we also point out another consequence of theorem [ thm : spectrum ] .[ thm : extreme ] the two eigenvalues under item 3 in theorem [ thm : spectrum ] are the extreme eigenvalues of . in particular , the perron eigenvalue ( spectral radius ) of is given by \,,\ ] ] while the leftmost eigenvalue of is given by \,.\ ] ]  moreover , is strictly less than the other eigenvalues of .we have proving that exceeds the other eigenvalues of in magnitude .similarly , proving that is strictly less than all the other eigenvalues .it follows that the principal eigenvector of is of the form ( [ eq : ev4 ] ) .since this eigenvector is constant on each of the cliques forming , the eigenvector centrality of each node is the same for all the nodes in each clique , as one would expect .furthermore , for the nodes in the centrale core have higher centrality than the nodes in the satellite cliques , which is also to be expected .in other words , in ( [ eq : ev4 ] ) we have that .this can be shown using conditions ( [ eq : cond1 ] ) together with the computed expressions for .see also remark [ rm : centr ] below .[ rm : four ] it also follows that has exactly four distinct eigenvalues .moreover , it is clear from the expression given for that it must go to infinity as the graph size goes to infinity , i.e. , if at least one of the parameters , , or tends to infinity .it follows that the _ infection threshold _ ,defined as , vanishes as the graph size grows , showing that core - satellite graphs of even modest size offer almost no resistance to the spreading of infections , gossip , rumors , etc .this is of course not surprising since the diameter of a core - satellite graph is only 2 .we refer to for additional discussion of infection spreading in networks .see also for a discussion of the special case of windmill graphs .a more realistic scenario for modeling real - world complex networks is to consider graphs in which not all satellite cliques are identical .consequently , we consider the generalization of core - satellite graphs to the case where the satellites cliques are not restricted to have all the same number of nodes .  to this end , let and consider a graph consisting of a core clique with nodes and satellite cliques where cliques have nodes , cliques have nodes , ... , and cliques have nodes , where for . here and for all . as before, we assume that each node in every satellite clique is connected to each node in the core clique .we introduce the following notation : letting we denote with the _ generalized core - satellite graph _ just decribed .although this notation is very convenient for expressing our mathematical results , it can be very cumbersome for illustrative examples .for that last purpose we will denote the number of cliques of a given size by an integer , using a subindex for the size of the clique as illustrated in the figure [ generalized cs ] .note that such a graph has nodes .core - satellite graphs are obtained as special cases for . at the other end of the spectrum we have the generalized core - satellite graphs in which all the satellite cliques have a different number of nodes , corresponding to the case , for all .indicates that there is one clique of size 3 , one clique of size 4 and one clique of size 5.,scaledwidth=65.0% ]  we first investigate the phenomenon of clustering divergence and the degree assortativity of these graphs .unfortunately , it is difficult to extend the analytic approach used in section [ sec : gen ] to the case of generalized core - satellite graphs .therefore , we perform numerical experiments instead .our computations indicate that the clustering divergence phenomenon observed for the simple core - satellite graphs also occurs in the generalized case .these graphs also display degree disassortativity like their simpler analogues . in figure [ properties gcs ]we illustrate the divergence of the clustering coefficients of these graphs and the increase of their disassortativity as the number of satellite cliques in the graph increases . for the sake of brevity , we only report here results for three kinds of generalized core - satellite graphs having cores with 3 , 5 and 10 nodes ( we remark that in real - world networks it is rare to find cliques od size larger than 10 ) .then , we consider satellites of size 3 , 5 and 7 and generate graphs with and . as can be seen in fig .[ properties gcs ] , as the number of satellites increases the average watts - strogatz clustering coefficient approaches the asymptotic value 1 , while the transitivity index drops to zero .the increase of the watts - strogatz index is slower for the graphs with bigger core as expected by the fact that here more nodes with low clustering exists , namely , those in the core .however , the assortativity coefficient of these graphs decays much faster than that for graphs with smaller cores . in all cases ,the assortativity coefficient is negative , similarly to what happens with the simple core - periphery graphs .   with and .in the insets the plots for the global clustering ( transitivity ) and the assortativity coefficient are shown for the same graphs . , scaledwidth=65.0% ]  next , we study the spectral properties of the generalized core - satellite graphs .let us define the matrices }_{\eta_{i}{\rm \;times}}\,,\qquad i=1,\ldots , t\,,\ ] ] and the block diagonal matrices ,\qquad i=1,\ldots , t\ ] ] ( with identical diagonal blocks ) .then , for a suitable ordering of the nodes , the adjacency matrix of , can be written in block form as \hline \\[-5pt ] b_{1}^{t } & a_{1}\\[2pt ] b_{2}^{t } &  & a_{2}\\ \vdots &  &  & \ddots\\[2pt ] b_{t}^{t } &  &  &  & a_{t } \end{array}\right]\,.\label{eq : adj2}\ ] ] this matrix is with .we have the following generalization of theorem [ thm : spectrum ] .  [thm : spectrum2 ] the spectrum of the generalized core - satellite graph consists of  1 .  the eigenvalue with multiplicity ; 2 .  the eigenvalue with multiplicity , for all with ; 3 .the roots of the following algebraic equation of degree : each of multiplicity one .it is straightforward to verify that has linearly independent eigenvectors of the form ( [ eq : ev1 ] ) corresponding to the eigenvalue .likewise , it is easy to check that has linearly independent eigenvectors of the form ( [ eq : ev2 ] ) with , also corresponding to the eigenvalue .hence , has ( at least ) linearly independent eigenvectors corresponding to the eigenvalue ; since the total number of nodes is , there remain eigenvalues to account for .let us first assume that for some .consider a nonzero vector of the form { \bf 0}\\ \hline \alpha_{1}{\bf 1}_{s_{i}}\\ \alpha_{2}{\bf 1}_{s_{i}}\\ \vdots\\[2pt ] \alpha_{\eta_{i}}{\bf 1}_{s_{i}}\\ \hline { \bf 0}\\ \vdots\\[2pt ] { \bf 0 } \end{array}\right]\,,\qquad\alpha_{1}+\cdots+\alpha_{\eta_{i}}=0\,,\label{eq : ev5}\ ] ] where the ( possible ) nonzeros in occur in the positions that correspond to the diagonal block in .then one can easily check that is eigenvector of associated to the eigenvalue , and from the condition on the we immediately deduce that there are linear independent vectors of the form ( [ eq : ev5 ] ) .there remain eigenvalues to account for , where is between and .when all the satellite cliques are identical and we are back under the assumptions of theorem [ thm : spectrum ] , hence the remaining two eigenvalues are the roots of the quadratic equation ( [ eq : quadratic ] ) , to which ( [ eq : algebraic ] ) reduces for .assume and consider a vector of the form \hline \\[-8pt ] \beta_{1}{\bf 1}_{s_{1}\eta_{1}}\\ \vdots\\[2pt ] \beta_{t}{\bf 1}_{s_{t}\eta_{t } } \end{array}\right].\label{eq : ev7}\ ] ] we note that vectors of this form are automatically orthogonal to all eigenvectors of the form ( [ eq : ev1 ] ) ( with ) , to all eigenvectors of the form ( [ eq : ev2 ] ) ( with ) , and to all eigenvectors of the form ( [ eq : ev5 ] ) .also , the span of all vectors of the form ( [ eq : ev7 ] ) has dimension .hence , it must be the invariant subspace of spanned by the eigenvectors associated with the remaining eigenvalues of . finally , we observe that any vector in is also of the form ( [ eq : ev7 ] ) up to a scalar multiple .  for such vectors ,the equation becomes \hline \\[-8pt ] c{\bf 1}_{s_{1}\eta_{1}}+\beta_{1}(s_{1}-1){\bf 1}_{s_{1}\eta_{1}}\\ \vdots\\[2pt ] c{\bf 1}_{s_{t}\eta_{t}}+\beta_{t}(s_{t}-1){\bf 1}_{s_{t}\eta_{t } } \end{array}\right]=\left[\begin{array}{c } \lambda{\bf 1}_{c}\\[2pt ] \hline \\[-8pt ] \lambda\beta_{1}{\bf 1}_{s_{1}\eta_{1}}\\ \vdots\\[2pt ] \lambda\beta_{t}{\bf 1}_{s_{t}\eta_{t } } \end{array}\right].\label{cond:7}\ ] ] conditions ( [ cond:7 ] ) can be rewritten in the form together with rearranging conditions ( [ cond:9 ] ) we obtain substituting ( [ cond:10 ] ) into ( [ cond:8 ] ) we obtain or , equivalently , the left - hand side of the last equation can be rewritten as finally leading to the equation which is precisely ( [ eq : algebraic ] ) .next , we will show that the roots of this equation are all distinct , therefore each root yields a distinct value of via ( [ cond:10 ] ) and therefore a distinct eigenvector ( [ eq : ev7 ] ) , thus completing the proof . for suppose that there is a root of ( [ eq : algebraic ] ) of multiplicity at least two .then the vector \hline \\[-8pt ] \bar{\beta}_{1}{\bf 1}_{s_{1}\eta_{1}}\\ \vdots\\[2pt ] \bar{\beta}_{t}{\bf 1}_{s_{t}\eta_{t } } \end{array}\right],\qquad\bar{\beta}_{i}=\frac{c}{\bar{\lambda}-s_{i}+1}\,,\qquad i=1,\ldots , t,\ ] ] is an eigenvector of associated with .but then there must be another eigenvector of associated with the same eigenvalue and orthogonal to .as argued above , this eigenvector must lie in the subspace , and therefore be again of the form ( [ eq : ev5 ] ) .but when we impose to be of the form ( [ eq : ev5 ] ) and to satisfy we are forced to conclude that , contradicting the orthogonality requirement .hence , all roots of ( [ eq : algebraic ] ) must be simple .in particular , when all the cliques have different size ( for all ) there are eigenvalues given by the roots of ( [ eq : algebraic ] ) , and no eigenvalues of the form .this completes the proof .it is well known ( abel ruffini theorem ) that the solution by radicals of the general algebraic equation of degree 5 or higher is impossible .we do not know if the equation ( [ eq : algebraic ] ) is solvable by radicals for any values of , , and . for and equation ( [ eq : algebraic ] )can be solved by radicals , but the roots are given by rather complicated expressions ( cardano s and ferrari s formulas ) .hence , we do not attempt to explicitly write down the roots of ( [ eq : algebraic ] ) .the number of distinct eigenvalues of is given by where is the cardinality of the set .this quantity is equal to ( or , equivalently ) when all the satellite cliques have different sizes .[ thm : bounds ] the spectral radius ( perron eigenvalue ) of is given by the largest root of ( [ eq : algebraic ] ) and satisfies the bounds   we begin by recalling the well known inequalities which hold for the spectral radius of any nonnegative irreducible matrix with non - constant row sums ( * ? ? ?* lemma 2.5 ) . in our casethe upper bound is the maximum degree , which is attained by each node in the core clique and is equal to .the lower bound in ( [ eq : bound1 ] ) yields , which in general is not enough to conclude that is given by the largest root of ( [ eq : algebraic ] ) by theorem [ thm : spectrum2 ] . however , we can improve on this lower bound as follows .assume ( without loss of generality ) that and consider a vector of the form \hline \\[-8pt ] { \bf 1}_{s_{1}\eta_{1}}\\ { \bf 0}\\ \vdots\\[2pt ] { \bf 0 } \end{array}\right],\ ] ] then \hline \\[-8pt ] ( c-1+s_{1}){\bf 1}_{s_{1}\eta_{1}}\\ c{\bf 1}_{s_{2}\eta_{2}}\\ \vdots\\[2pt ] c{\bf 1}_{s_{t}\eta_{t } } \end{array}\right]\ge(c-1+s_{i}){\bf x}\,,\qquad\forall i=1,\ldots , t.\ ] ] hence , we have found a nonnegative vector such that for . by a well known result ( see ) this implies that . finally , it is straightforward to check that is not a root of ( [ eq : algebraic ] ) , therefore it must be , proving the lower bound in ( [ eq : bounds ] ) .[ rm : centr ] it follows that the principal eigenvector of is of the form ( [ eq : ev7 ] ) .since this eigenvector is constant on each of the cliques forming , the eigenvector centrality of each node is the same for all the nodes within a given clique , as one would expect .furthermore , the nodes in the core clique have higher centrality than the nodes in any of the satellite cliques ( assuming of course there is more than one of them ) , which is also to be expected . in other words , in ( [ eq : ev5 ] ) we have that for all . to see this , note that the are given by see ( [ cond:10 ] ) . by theorem [ thm : bounds ]we have for all , hence and therefore for all .the spectral radius of a generalized core - satellite graph goes to infinity as the graph size goes to infinity .let be the adjacency matrix of the star graph with nodes , with the central node numbered first .the spectrum of consists of the eigenvalue 0 of multiplicity , plus the eigenvalues .in particular , . if is the adjacency matrix of a generalized core - satellite graph with nodes , it is straightforward that , where the inequality is component - wise. it is well known that this implies , hence we conclude that the spectral radius of tends to infinity as the graph size tends to infinity .this result implies that in any generalized core - satellite graph , the infection threshold vanishes as the graph size grows , see remark [ rm : four ] .again , since the diameter of a generalized core - satellite graph is constant and equal to 2 , this is to be expected .we conclude this section with the description of the spectrum of the graph laplacian of . letting and the graph laplacian of can be written as \hline \\[-5pt ] -b_{1}^{t } & l_{1}+ci_{\eta_{1}s_{1}}\\[2pt ] -b_{2}^{t } &  & l_{2}+ci_{\eta_{2}s_{2}}\\ \vdots &  &  & \ddots\\[2pt ] -b_{t}^{t } &  &  &  & l_{k}+ci_{\eta_{t}s_{t } } \end{array}\right]\,.\label{eq : lap2}\ ] ]  we have the following result .[ thm : lspectrum2 ] the spectrum of the graph laplacian associated with the generalized core - satellite graph consists of  1 .  the eigenvalue with multiplicity ; 2 .  the eigenvalue with multiplicity , for ; 3 .  the eigenvalue with multiplicity ; 4 .the eigenvalue with multiplicity one .it is straightforward to verify that has linearly independent eigenvectors of the form ( [ eq : ev1 ] ) with associated to the eigenvalue .in addition , the vector \hline \\[-8pt ] \beta{\bf 1}_{\eta_{1}s_{1}}\\ \vdots\\[2pt ] \beta{\bf 1}_{s_{t}\eta_{t } } \end{array}\right],\qquad\beta=-\frac{c}{\sum_{i=1}^{t}\eta_{i}s_{i}}\,,\ ] ] is also eigenvector of associated with the eigenvalue .next , it is easily checked that every nonzero vector of the form ( [ eq : ev2 ] ) with is an eigenvector of associated with the eigenvalue , and there are exactly linearly independent eigenvectors of that form .consider now vectors of the form \hline \\[-8pt ] { \bf x}_{1}\\ \vdots\\[2pt ] { \bf x}_{t } \end{array}\right]\ , , \qquad{\rm where}\qquad{\bf x}_{i}=\left[\begin{array}{c } \alpha_{1i}{\bf 1}_{s_{i}}\\ \alpha_{2i}{\bf 1}_{s_{i}}\\ \vdots\\[2pt ] \alpha_{\eta_{i}i}{\bf 1}_{s_{i } } \end{array}\right ] \qquad(i=1,\ldots , t).\label{eq : ev8}\ ] ] then the equality is satisfied for if and only if equation ( [ eq : cond10 ] ) is a homogeneous linear equation in the unknowns , with and .since the number of unknowns is , equation ( [ eq : cond10 ] ) admits linearly independent solutions , each leading to a distinct eigenvector ( [ eq : ev8 ] ) of associated with .  finally , since is connected , has the simple eigenvalue zero associated with the eigenvector with constant entries .the laplacian eigenvalues of can also be obtained from general results about the laplacian eigenvalues of the join of two ( or more ) graphs and the fact that the laplacian eigenvalues of a complete graph with nodes are known ( they are with multiplicity and with multiplicity 1 ) ; see , e.g. , ( * ? ? ?* theorem 2.20 ) .the above proof has the advantage of being self - contained and of explicitly exhibiting the eigenvectors of .it follows from theorem [ thm : lspectrum2 ] that the laplacian of a generalized core - satellite graph , with , has exactly distinct eigenvalues . for a core - satellite graph ( where and ) , the number of distinct eigenvalues is four : , , , and ( see also remark [ rm : four ] ) .we observe that ( generalized ) core - satellite graphs are _ laplacian - integral _ ,i.e. , the laplacian eigenvalues are all integers . moreover , the _ algebraic connectivity _( smallest nonzero laplacian eigenvalue ) of a generalized core - satellite graph is equal to , the size of the core clique , and is independent of the number or size of the satellite cliques .this is to be expected , since generalized core - satellite graphs can be disconnected by disconnecting the core clique .  the ratio between the smallest nonzero eigenvalue of and the largest eigenvalue of is known as the _ synchronization index _ of the network : .theorem [ thm : lspectrum2 ] implies that in particular , for windmill graphs we recover the fact that , as already observed in .therefore , _ any two core - satellite graphs of the same size and with the same core clique have exactly the same synchronization index _ , a somewhat surprising result. moreover , if is fixed and grows , the synchronization index decreases .this implies that unless the core size grows and the number and size of satellite cliques remains constant ( or bounded ) , large ( generalized ) core - satellite graphs are bad synchronizers .we refer to for detailed discussions of network synchronizability .real - world graphs have a great variety of structures and complexities . hence , the existence of different mathematical models erds - rnyi , barabsi - albert , watts - strogatz , random geometric graphs , etc .however , not all of the structural properties of real - world graphs are captured by these models .one simple example is the local - global clustering coefficient divergence .this structural effect is simply due to the fact that in some networks the local clustering tends to the maximum while the global one tends to the minimum when the size of the graphs grows to infinity .here we have proposed a general class of graphs for which this clustering divergence is observed and can be studied analytically .these graphs here proposed to be called core - satellite graphs are characterized by a central core of nodes which are connected to a few satellites which may be of the same or different sizes . in this workwe have investigated some general properties of these graphs , e.g. , clustering coefficients , assortativity coefficients , as well as the eigenstructure of both the adjacency and the laplacian matrices .core - satellite graphs can also be easily modified so as to model other properties of complex networks , such as hierarchical structure .all of these make core - satellite graphs a flexible model for certain classes of real - world networks , opening some new possibilities for the analytic modeling of these systems .+ * acknowledgements *     l. f. costa , o. n. oliveira jr , g. travieso , f. a. rodrigues , p. r. villas boas , l. antiqueira , m. p. viana , l. e. correa rocha . _ analyzing and modeling real - world phenomena with complex networks : a survey of applications _ , adv .( 2011 ) 329412 .b. bollobs , mathematical results on scale - free random graphs , in : bornholdt , s. , schuster , h. g. ( eds . ) , handbook of graph and networks : from the genome to the internet , wiley - vch , weinheim , ( 2003 ) 1 - 32 .

Abstract: core - satellite graphs ( sometimes referred to as generalized friendship graphs ) are an interesting class of graphs that generalize many well known types of graphs . in this paper we show that two popular clustering measures , the average watts - strogatz clustering coefficient and the transitivity index , diverge when the graph size increases . we also show that these graphs are disassortative . in addition , we completely describe the spectrum of the adjacency and laplacian matrices associated with core - satellite graphs . finally , we introduce the class of generalized core - satellite graphs and analyze their clustering , assortativity , and spectral properties .


this paper introduces quanfou v1.1 , quanglue v1.1 , quanoracle v1.1 , quanshi v1.1 , four java applications available for free .( source code included in the distribution . )  in a previous paper , we introduced the java applications quantree and quanlin .these two applications plus the four applications introduced in this paper , are part of a suite of java applications called quansuite .quansuite applications all depend heavily on a common class library called qwalk .each quansuite application compiles a different kind of input quantum evolution operator .the applications output a quantum circuit that equals the input evolution operator .before reading this paper , the reader should read ref. .many explanations in ref. still apply to this paper .rather than repeating such explanations in this paper , the reader will be frequently referred to ref. .the input evolution operator for a quansuite application can be specified either directly ( e.g. in quanfou , quanshi ) , or by giving a hamiltonian such that ( e.g. in quanglue and quanoracle ) .the standard definition of the evolution operator in quantum mechanics is , where is time and is a hamiltonian . throughout this paper, we will set  .if is proportional to a coupling constant , reference to time can be restored easily by replacing the symbol by , and the symbol by .the input evolution operator for quanfou is , the discrete fourier transform matrix , defined by :  ( u_dft)_p , q = ^pq = e^i , with .see ref. for a review of how to compile exactly .since we use an exact ( to numerical precision ) compilation of , the order of the suzuki(or other ) approximant and the number of trots are two parameters which do not arise in quanfou ( unlike quantree and quanlin ) .fig.[fig - qfou - main ] shows the * control panel * for quanfou .this is the main and only window of the application .    as to the input and output fields in the control panel for quanfou , we ve seen and explained these before in ref. .  as to the output files ( log , english , picture ) generated when we press the *write files * button , we ve seen and explained these before in ref. .for example , figs.[fig - qfou - log ] , [ fig - qfou - eng ] , [ fig - qfou - pic ] show an instance of these output files that was generated by quanfou .the input evolution operator for quanglue is , where :  h_glue = g ( + h.c . ) , for some .ref. explains our method for compiling exactly .since we use an exact ( to numerical precision ) compilation of , the order of the suzuki(or other ) approximant and the number of trots are two parameters which do not arise in quanglue ( unlike quantree and quanlin ) .fig.[fig - qglue - main ] shows the * control panel * for quanglue .this is the main and only window of the application .    as to the input and output fields in the control panel for quanglue, we ve seen and explained these before in ref. , except for the input fields * row 1 * and * row 2*.  row 1 , row 2 : : :  row 1 = and row 2 = or vice versa ,  where are the parameters defined above , the two  states being glued .  as to the output files ( log , english , picture ) generated when we press the * write files * button , we ve seen and explained these before in ref. .consider a tree with states , and leaves , with leaf inputs for .the input evolution operator for quanoracle is , where  h_oracle&= & g .ref. , in the appendix for banded oracles " , explains our method for compiling exactly .since we use an exact ( to numerical precision ) compilation of , the order of the suzuki(or other ) approximant and the number of trots are two parameters which do not arise in quanoracle ( unlike quantree and quanlin ) .fig.[fig - qora - main ] shows the * control panel * for quanoracle .this is the main and only window of the application .    as to the input and output fields in the control panel for quanoracle , we ve seen and explained these before in ref. , except for the input field *bands : : :  you must enter here an even number of integers separated by any  non - integer , non - white space symbols .say you enter   . if   for are as  defined above , then iff .each set   is a band " . if , the  band has a single element .quanoracle checks that   , , and   for all .it also checks  that .( if   , bands and   can be merged . if   , bands   and overlap . )  as to the output files ( log , english , picture ) generated when we press the * write files * button , we ve seen and explained these before in ref. .the input evolution operator for quanshi is the unitary operation that takes :  , where , with for some positive integer .we call the * state shift*.   can be easily expressed in matrix form .for example , for and ,  u_shift =  c||c|c|c|c|c|c|c|c| & & & & & & & & + & & & & 1 & & & & + & & & & & 1 & & & + & & & & & & 1 & & + & & & & & & & 1 & + & & & & & & & & 1 + & 1 & & & & & & & + & & 1 & & & & & & + & & & 1 & & & & & +  .appendix [ app - shift ] explains our method for compiling exactly .since we use an exact ( to numerical precision ) compilation of , the order of the suzuki(or other ) approximant and the number of trots are two parameters which do not arise in quanshi ( unlike quantree and quanlin ) .fig.[fig - qshi - main ] shows the * control panel * for quanshi .this is the main and only window of the application .as to the input and output fields in the control panel for quanshi , we ve seen and explained these before in ref. , except for the input field * shift*.  shift : : :  the parameter defined above .quanshi allows   and interprets a shift by   as the inverse of a shift by .  as to the output files ( log , english , picture ) generatedwhen we press the * write files * button , we ve seen and explained these before in ref. .in this appendix , we will show how to compile the unitary operation that takes                _ bool^3 ( 2 ^ 2 m_2 + 2 m_1 + m_0)p_&= & _ bool^3 ( 2 ^ 2 m_2p _ ) + 2n(1 ) + n(0 ) + & = & 2 ^ 2p_1 .. + 2n(1 ) + n(0 ) + & = & 2 ^ 2n(2 ) + 2n(1 ) + n(0 ) .this result can be easily generalized using induction to an arbitrary number of qubits .an exact compilation of is now readily apparent from eq.([eq - ushi - decomp ] ) .the matrices and are dfts matrices so we know how to compile them .the diagonal matrix is also easy to compile .for example , for ,

Abstract: this paper introduces quanfou v1.1 , quanglue v1.1 , quanoracle v1.1 , quanshi v1.1 , four java applications available for free . ( source code included in the distribution . ) each application compiles a different kind of input quantum evolution operator . the applications output a quantum circuit that equals the input evolution operator .


images are produced to record or display useful information .due to the visibility of images and the rapid development of science and technology , images play an increasingly important role in our lives .however , because of imperfections in the imaging and capturing process , the recorded image invariably represents a degraded version of the original scene ( ) .the undoing of these imperfections is crucial to many of the subsequent image processing tasks .there exists a wide range of different degradations .a very important example is the existence of noise .noise may be introduced by the medium through which the image is created and transmitted . in this paper , we concentrate on removing impulse noise and its mixture with gaussian noise .we present a numerical image by a matrix , where is the image domain , and represents the gray value at the pixel for 8-bit gray images .the additive gaussian noise model is : where is the original image , is the noisy one , and is the gaussian noise : are independent and identically distributed gaussian random variables with mean and standard deviation . in the sequelwe always denote by the original image , and the noisy one .the random impulse noise model is :   where is the impulse probability ( the proportion of the occurrence of the impulse noise ) , and are independent random variables uniformly distributed on  ] , as , [th1 ] where denotes the cardinality of .notice that when , ( ) means that which is the similarity principle in .recall that .theorem shows that is a good estimator of the original image if the number of similar patches is sufficiently large .here we use the weight instead of , as has the nice property that it is independent of if .this property is used in the proof , and makes the estimator to be `` almost '' non - biased : in fact , if the family is independent of the family ( e.g. this is the case when the similar patches are disjoint ) , then it is evident that . we can consider that this non - biased property holds approximately as for each there are few such that is dependent of . a closely related explanation about the biased estimation of nl - means can be found in .  notice that when is not similar to , the weight is small and negligible .therefore it is also reasonable to take all patches for the calculation . but selecting only similar patches can improve the restoration result , and can also speed up the algorithm .the difference between and is also small , so that theorem shows that nlm is also a good estimator of .but very often gives better restoration result .the next result is a convergence theorem in distribution .it states that in distribution with a rate as .  under the condition of theorem ,assume additionally that is a stationary sequence of random vectors .then as , where means the convergence in distribution , and is a mixture of centered gaussian laws in the sense that it has a density of the form  being the law of and . [ th2 ]  by theorems and , the larger the value of , the better the approximation of to , that is to say , the more similar patches , the better restored result .this will be confirmed in section [ secsim ] where we shall introduce the notion of degree of similarity for images , showing that the larger the degree of similarity , the better the quality of restoration .the proofs of the theorems will be given in section [ proofthm ] .to prove the two theorems , we will show a marcinkiewicz law of large numbers ( theorem ) and a convergence theorem in distribution for random weighted means ( theorem ) , which are more general than theorems and , respectively .   before stating the theorems, we first recall the notion of -dependence .  for an integer , a sequence of random variables is called to be -dependent if each subsequence is independent whenever for all ( that is , random variables with distances greater than are independent of each other . )let be a sequence of -dependent identically distributed random variables , with and for some and .then [ genth1 ]  we need the following lemma to prove it . if are -dependent and identically distributed random variables with and for some , then [ lemm1 ]  this lemma is a direct consequence of marcinkiewicz law of large numbers for independent random variables (page 118 ) , since for all is a sequence of i.i.d .random variables , and for each positive integer , we have where are positive integers with .  _ proof of theorem ._ notice that where since , by lemma with , we have since , and , again by lemma , we get thus the conclusion follows .   let be a stationary sequence of -dependent and identically distributed random variables with and .then that is , where and is a constant .[ genth2 ]  we need the following lemma to prove the theorem . let be a stationary sequence of -dependent and identically distributed random variables with and .set , then , and as , [ lemm2 ]  _ proof of theorem ._ as in the proof of theorem , we have where notice that the -dependence of and the stationarity imply those of . therefore by lemma , we get where since , by lemma with , we obtain thus the conclusion follows with .   for simplicity , denote by with in the following proof .we need to prove , for any  ] .  _ proof of theorem . _similarly to the proof of theorem , fix , and set . then and are independent , and .for large enough , is a sequence of -dependent identically distributed random vectors , and . hence applying theorem , for fixed , we get , for any positive integer , where depends on .this means that for any , let be the positive integer such that is independent of for all . then by fubini s theorem and lebesgue s dominated convergence theorem , we have where with being the law of . in other words , where is the law with density .this together with ( ) give the conclusion of theorem .   in this section , we introduce the notion of degree of similarity of images to describe the influence of the number of similar patches on the restoration results .  if two patches and have the same distribution and are independent of each other , then are independent variables with the same law ( recall that is the translation mapping of onto ) . therefore where is the sum of squares of independent random variables with normal law , and has the law with degrees of liberty . for ,let be determined by so that the value of is chosen to be small enough .it represents the risk probability .when , we consider that and are similar with confidence level .this leads us to the following definition of the degree of similarity .  for let be the proportion of patches similar to in the search window , andlet be their mean over the whole image .define the degree of similarity of the image with confidence level .we compute the ds values for different images corrupted by gaussian noise with .note that the ds values depend on . in our experiments ,we fix .the ds values of noisy images are shown in table .it can be seen that , for non - local means based methods , for example , our method in section [ pwmf ] , generally , the larger the ds value , the larger the psnr value ( see section [ exp ] ) .  .ds values of noisy images ( ) [ cols="^,^,^,^,^,^,^,^",options="header " , ]   [ time ]two convergence theorems , one for the almost sure convergence and the other for the convergence in law , are established to show the rate of convergence of nl - means .the notion of degree of similarity is also introduced to describe the influence of the proportion of similar patches in the application of nl - means .based on the convergence theorems , a new filter called patch - based weighted means filter ( pwmf ) is proposed to remove mixed noise , leading to an extension of nl - means .the choice of parameters has been carefully discussed .simulation results show that the new proposed filter is competitive compared to recently developed known algorithms .as the detection of impulse noise is crucial for removing the noise , which is done by the statistics road in this paper , we could further improve our results by improving road . in the future, we will consider a semi - local statistics to make use of redundancies of images , which has the possibility to well recover the textured regions .delon , j. , desolneux , a. : a patch - based approach for random - valued impulse noise removal . in : 2012 ieee international conference on acoustics , speech and signal processing ( icassp ) , pp .ieee ( 2012 )                        xu , h. , xu , j. , wu , f. : on the biased estimation of nonlocal means filter . in : international conference on multimedia computing and systems / international conference on multimedia and expo , pp .11491152 ( 2008 )  yang , j.x ., wu , h.r . :mixed gaussian and uniform impulse noise analysis using robust estimation for digital images .proceedings of the 16th international conference on digital signal processing pp . 468472 ( 2009 )

Abstract: we first establish a law of large numbers and a convergence theorem in distribution to show the rate of convergence of the non - local means filter for removing gaussian noise . we then introduce the notion of degree of similarity to measure the role of similarity for the non - local means filter . based on the convergence theorems , we propose a patch - based weighted means filter for removing impulse noise and its mixture with gaussian noise by combining the essential idea of the trilateral filter and that of the non - local means filter . our experiments show that our filter is competitive compared to recently proposed methods .  gaussian noise , impulse noise , trilateral filter , non - local means filter , convergence theorem , degree of similarity


linear systems of equations with a matrix with a toeplitz - type structure arise in many applications , such as in signal and image processing . consider the computation of an approximate solution of the linear system of equations where is a symmetric bttb matrix , i.e., is a symmetric block toeplitz matrix with each block being an symmetric toeplitz matrix .the eigenvalues of are assumed to decay smoothly to zero in magnitude without a significant gap . in particular, may be singular .linear systems of equations ( [ linsys ] ) with a matrix of this kind arise , for example , from the discretization of a linear ill - posed problem , such as a fredholm integral equation of the first kind in two space - dimensions with a displacement kernel .the right - hand side of ( [ linsys ] ) is assumed to be contaminated by an ( unknown ) error .we will refer to this error as `` noise '' .it may stem from measurement or discretization errors .let denote the ( unknown ) error - free vector associated with , i.e. , the ( unknown ) linear system of equations with error - free right - hand side , is assumed to be consistent ; however , we do not require the available system ( [ linsys ] ) to be consistent .we will assume that a fairly sharp bound for the norm of is known , i.e. , here and throughout this paper denotes the euclidean vector norm or the spectral matrix norm .this bound will help us determine a suitable number of iterations to carry out with a minimal residual iterative method and to construct a preconditioner for the solution of .let denote the moore penrose pseudoinverse of .we are interested in computing an approximation of the solution of minimal euclidean norm of the unavailable error - free linear system ( [ linsysnf ] ) .note that the solution of ( [ linsys ] ) , typically is dominated by the propagated error and , therefore , is useless .therefore all solution methods for seek to determine a suitable approximate solution that is not severely contaminated by propagated error .the computed approximate solution is the exact solution of an appropriately chosen nearby problem , whose solution is less sensitive to the error in than the solution of .the replacement of the given problem by a nearby problem is commonly referred to as regularization . among the most popular regularization methodsis tikhonov regularization , which replaces by a penalized least - squares problem , and truncated iteration , which is based on solving by an iterative method and terminating the iterations suitably early , see , e.g. , for discussions on these regularization methods . in this paperwe regularize by truncated iteration , and by choosing a suitable preconditioner .the evaluation of matrix - vector products with a bttb matrix of order can be carried out in only arithmetic floating point operations ( flops ) by using the fast fourier transform ( fft ) ; see , e.g. , .this makes it attractive to solve by an iterative method .we will use a preconditioner to increase the rate of convergence of the iterative method .are commonly preconditioned by block circulant with circulant block ( bccb ) matrices ; see for discussions , illustrations , and further references .the use of bccb preconditioners for bttb matrices is attractive both due to the spectral properties of the preconditioned matrix and because of the possibility to evaluate a matrix - vector product with a preconditioned matrix of order in only flops with the aid of the fft .a matlab software package for fast matrix - vector product evaluation is provided by redivo zaglia and rodriguez .  when preconditioning a bttb matrix that stems from the discretization of a linear ill - posed problem , it is desirable that an invariant subspace associated with the eigenvalues of of smallest magnitude is not affected by preconditioning to avoid severe propagation of the error in the right - hand side of ( [ linsys ] ) into the computed iterates .this is due to the fact that the eigenvectors associated with these eigenvalues are highly oscillatory ( have many sign changes ) and model noise rather than the desired solution .typically , we do not want these eigenvectors to be part of our computed approximation of .a nice introduction to bccb preconditioners for the solution of discretized linear ill - posed problems with a toeplitz - type matrix is presented by hanke et al . .the number of iterations have to be few enough to avoid severe propagation of the error in into the computed approximation of .the availability of the bound and the consistency of allow us to apply the discrepancy principle to determine a suitable number of iterations as well as to define the dimension of the invariant subspace that should not be affected by the preconditioner .roughly , the larger the error in , the larger should the dimension of the subspace that is not ( or only minimally ) affected by preconditioning be chosen .various other approaches to define bccb preconditioners for the iterative solution of discretized linear ill - posed problems with an error - contaminated right - hand side are described in the literature .for instance , hanke and nagy apply the l - curve criterion to determine a subspace that should not be affected by the preconditioner .the l - curve criterion implicitly estimates the norm of the error in .this criterion is able to estimate the norm of the error fairly accurately in some situations , but it is not a reliable error estimator ; see for discussions and illustrations .we therefore are interested in developing an approach for constructing bccb preconditioners that is not based on the l - curve criterion .hanke et al . apply a discrete picard condition to determine the dimension of the subspace that the bccb preconditioner should leave invariant .this approach typically works quite well in an interactive computing environment that allows the determination of whether the discrete picard condition holds by visual inspection , however , it is not straightforward to automatize . di benedetto et al . the application of a so - called superoptimal bccb preconditioner and do not explicitly choose the dimension of the subspace that should be unaffected by the preconditioner .this approach works well for some image restoration problems , but not for others ; see the discussion in .preconditioning is most useful when the error in is of small relative norm , because then many steps of an iterative method may be required to determine an accurate approximation of . when the error is large , only few steps can be carried out before the propagated error destroys the computed solution .preconditioning then does not reduce the computational effort by much .  in this paper, we will use the bound ( [ errbd ] ) to determine both the bccb preconditioner and the number of iterations to be carried out .a perturbation bound guides our choice of preconditioner .this is described in section [ sec2 ] .a few computed examples are presented in section [ sec3 ] and concluding remarks can be found in section [ sec4 ] .we discuss the construction of the preconditioner , the stopping criterion for the iterative method , and outline the minimal residual iterative methods used .let for the moment be a symmetric positive definite toeplitz matrix and let be the closest circulant matrix to in the frobenius norm .t. chan proposed the use of as a preconditioner for ; see also .the eigenvalues of are given by the discrete fourier transform of the first column of ; their computation with the fft requires only flops .since we would like the preconditioner not to affect the invariant subspace of associated with the smallest eigenvalues , we set the eigenvalues of smallest magnitude of to unity for some suitable , analogously as in .we refer to this preconditioner as .subsection [ sub2.4 ] describes how to determine using the error bound .  the linear systems ( [ linsys ] ) of interest to us have a bttb matrix , i.e., is the kronecker product of two toeplitz matrices we will determine a preconditioner that is the kronecker product of two circulant matrices where is defined by first determining the closest circulant to in the frobenius norm and then setting its eigenvalues of smallest magnitude to one for . in this wayour preconditioner does not affect the invariant subspace of associated with the eigenvalues of smallest magnitude .the eigenvectors of this subspace are highly oscillatory and primarily model noise and not the desired solution .this construction of requires only flops .we refer to the bccb preconditioner so determined as .the determination of this preconditioner is somewhat faster than of the bccb preconditioner described in , because the latter requires that all its eigenvalues be formed .once the bccb preconditioner has been defined , we compute an approximate solution of the preconditioned linear system of equations , using one of the minimal residual iterative methods described in subsection [ sub2.6 ] . as for the initial approximation of , define the bccb matrix , where is obtained from by setting the eigenvalues of smallest magnitude to zero , for , and define the initial iterate then is chosen to be .let denote the computed iterates .the number of iterations to be carried out is determined with the aid of the discrepancy principle .this stopping criterion prescribes that the iterations be terminated as soon as an iterate satisfies where is a user - specified parameter independent of .typically , is chosen close to unity when is known to be a fairly sharp upper bound for ; cf . .we obtain the approximation of the desired vector .let for the moment be a toeplitz matrix and let be the closest circulant in the frobenius norm .order the eigenvalues of according to assume that a bound ( [ errbd ] ) is known .let be the number of eigenvalues of largest magnitude of that are not set to unity .we choose , where is the solution of the minimization problem here denotes the largest integer smaller than or equal to and .this choice of is suggested by the following perturbation result .[ prop1 ] given a rank- matrix , , a vector in the range of , and such that .let , , and satisfy then where is the condition number of the matrix , and denotes the ratio between the smallest singular values of the matrices and .  from and , one has taking norms on both sides , one gets hence , finally , dividing by and exploiting the inequality yield ( [ ineq ] ) .let be the circulant obtained by setting the eigenvalues of smallest magnitude of to zero . in this context ,one replaces by and by in proposition [ prop1 ] .now , letting and , we obtain by ( [ rhs ] ) that satisfies the hypothesis of proposition [ prop1 ] .inequality ( [ ineq ] ) reads in the final estimate , we assume that the eigenvalues of of smallest magnitude are close to , so that and . this discussion suggests the choice , where is determined by ( [ choice_p ] ) .however , since we do not know whether the eigenvalues of of smallest magnitude are close to , we will choose to secure that we do not `` over - precondition '' and thereby obtain a large propagated error in the computed approximation of .we remark that a `` standard '' circulant preconditioner generally over - preconditions and gives a large propagated error in the computed solution .we turn to bttb matrices and first consider the matrix .let be the closest circulant to in the frobenius norm .then is the closest bccb matrix to in the frobenius norm .let to be the solution of the minimization problem where .the choice is suggested by proposition [ prop1 ] , where one replaces by , the bccb matrix obtained setting the eigenvalues of smallest magnitude of each circulant matrix to zero , and replaces by , so that the vector in proposition [ prop1 ] is given by in ( [ init_guess ] ) with . in the computed examples, we will let to avoid to over - precondition .  finally , consider bttb matrices of the form ( [ tt ] ) . to determine a bccb preconditioner of the kind ( [ cc ] ) , we sort the eigenvalues of the circulant matrices , , according to and let be the number of eigenvalues of largest magnitude of that are not set to unity .let the index pair solve the minimization problem where .similarly as above , we let for .the preconditioned matrix , where and are given by ( [ tt ] ) and ( [ cc ] ) , respectively , is constructed as follows :  compute for :  1 .the closest circulant to in the frobenius norm .2 .  the fft of the first column of the matrix . this gives the eigenvalues of .the eigenvectors are the columns of the fourier matrix .permute the columns of the fourier matrix so that the eigenvalues are ordered according to decreasing magnitude , and denote the permuted fourier matrix by .the columns of generally become more oscillatory with increasing column number .3 .  the truncation index , , by using ( [ choice3_p ] ) . this defines the diagonal matrices ,\\\widetilde{\lambda}_{p_j}&=&{\rm diag}[\lambda^{(j)}_1,\lambda^{(j)}_2,\ldots , \lambda^{(j)}_{p_j } , 0,\ldots , 0].\end{aligned}\ ] ] neither the matrices nor and have to be explicitly formed .we are now in a position to define the preconditioner and related matrices , but hasten to point out that these matrices do not have to be explicitly formed . introduce where the superscript denotes transposition and complex conjugation .we compute the initial approximate solution in ( [ init_guess ] ) without explicitly forming the matrix .indeed , the spectral factorization can be applied to evaluate for any in flops with the fft , and the same holds for matrix - vector products with the matrices and .krylov subspace methods for the iterative solution of require matrix - vector product evaluations with the preconditioned matrix .it is well known that these matrix - vector product evaluations can be carried out quickly with the aid of the fft .we outline for completeness the evaluation of matrix - vector products with the matrix in the simplified situation when is a toeplitz matrix and is a circulant .we express as a sum of a circulant and a skew - circulant .this splitting and the spectral factorizations where , yield that the preconditioned linear system of equations with can be expressed in the form which is used in the computations .each iteration requires the evaluation of the fft of four -vectors .the computation of these ffts is the dominating computational work .we remark that the dominating computational effort to evaluate a matrix - vector product with the matrix , which is required when solving the unpreconditioned system by a krylov subspace method , also is the computation of the fft of four -vectors . therefore , the number of iterations required by the iterative method is the proper measure of the computational effort both for preconditioned and unpreconditioned linear systems of equations .the situation is analogous when is the tensor product of two toeplitz matrices .we omit the details .gmres is a popular iterative method for the solution of large linear systems of equations with a square nonsingular nonsymmetric matrix that arise from the discretization of well - posed problems , such as dirichlet boundary value problems for elliptic partial differential equations ; see , e.g. , saad .the iterate determined by this method solves the minimization problem where is an initial approximate solution , , and is a krylov subspace .it has been observed that a modification of gmres , which we refer to as the range restricted gmres method ( rrgmres ) , often yields a more accurate approximation of the desired solution than ( standard ) gmres when stems from the discretization of a linear ill - posed problem and the right - hand side is contaminated by error ; see .the rrgmres method determines iterates in shifted krylov subspaces , where is a small integer .we propose that an rrgmres method be used for the solution of the preconditioned problem when and are symmetric positive definite , rrgmres can be simplified to a range restricted minres method that only requires simultaneous storage of a few -vectors , the number of which is bounded independently of the number of iterations ; see for details .the calculations of this section were carried out in matlab with machine epsilon about .for all the examples , we chose in ( [ discrp1 ] ) .* example 1 .* this is an image deblurring test problem from the matlab package regularization tools .the original image and the symmetric bttb matrix are those defined by the matlab function blur.m .we choose the dimensions , half - bandwidth for each toeplitz block specified by the parameter and width of the gaussian point spread specified by the parameter ..example 1 : blur .a hyphen signifies that no preconditioner is used .[ cols="^,^,^,^",options="header " , ]this paper presents a novel method to determine bccb preconditioners to be used for iterative solution of discretized linear ill - posed problem with a bttb matrix .the computed examples show that the number of iterations is reduced by roughly a half when using the proposed preconditioner , whereas the quality of the computed solution is about the same as without preconditioning .also , we would like to mention that instead of using the circulant preconditioners described , one may use the generalized optimal circulant preconditioners described in at the same computational cost .this may be attractive for certain toeplitz and bttb matrices .99 r. h .- f .chan and x .- q .jin , _ an introduction to iterative toeplitz solvers _ , siam , philadelphia , 2007. t. chan , _ an optimal circulant preconditioner for toeplitz systems _ , siam j. sci . statist .comput . , 9 ( 1988 ) , pp .766771 . t. chan and j. a. olkin , _ circulant preconditioners for toeplitz - block matrices _ , numer .algorithms , 6 ( 1994 ) , pp . 89101 .f. di benedetto , c. estatico , and s. serra capizzano , _ superoptimal preconditioned conjugate gradient iteration for image deblurring _, siam j. sci .comput . , 26 ( 2005 ) , pp .. l. dykes , f. marcelln , and l. reichel , _ the structure of iterative methods for symmetric linear discrete ill - posed problems _ , bit , 54 ( 2014 ) , pp . 129145 .l. dykes and l. reichel , _ a family of range restricted iterative methods for linear discrete ill - posed problems _ , dolomites research notes on approximation , 6 ( 2013 ) , pp .. h. w. engl , m. hanke , and a. neubauer , _ regularization of inverse problems _ , kluwer , dordrecht , 1996 .m. hanke and j. g. nagy , _ restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques _ ,inverse problems , 12 ( 1996 ) , pp . 157173 .m. hanke , j. nagy , and r. plemmons , _ preconditioned iterative regularization for ill - posed problems _ , in : l. reichel , a. ruttan , and r. s. varga , eds ., numerical linear algebra , de gruyter , berlin , 1993 , pp .p. c. hansen , _ rank - deficient and discrete ill - posed problems _ , siam , philadelphia , 1998 .p. c. hansen , _ regularization tools version 4.0 for matlab 7.3 _ , numer .algorithms , 46 ( 2007 ) , pp . 189194 .m. k. ng , _ iterative methods for toeplitz systems _ , oxford university press , oxford , 2004 .s. kindermann , _ convergence analysis of minimization - based noise level - free parameter choice rules for linear ill - posed problems _ , electron ., 38 ( 2011 ) , pp . 233257 .a. neuman , l. reichel , and h. sadok , _ implementations of range restricted iterative methods for linear discrete ill - posed problems _ , linear algebra appl ., 436 ( 2012 ) , pp .39743990 . s. noschese and l. reichel , _ the structured distance to normality of toeplitz matrices with application to preconditioning _linear algebra appl . , 18 ( 2011 ) , pp .429447 . m. redivo zaglia and g. rodriguez , _ smt : a matlab toolbox for structured matrices_ , numer .algorithms , 59 ( 2002 ) , pp . 639659 .l. reichel and g. rodriguez , _ old and new parameter choice rules for discrete ill - posed problems _ , numer .algorithms , 63 ( 2013 ) , pp .y. saad , _ iterative methods for sparse linear systems _, 2nd ed . ,siam , philadelphia , 2003 . c. van der mee , g. rodriguez , and s. seatzu , _ fast computation of two - level circulant preconditioners _algorithms , 41 ( 2006 ) , pp . 275295 . c. van der mee , g. rodriguez , and s. seatzu , _ fast superoptimal preconditioning of multiindex toeplitz matrices _ , linear algebra appl . , 418 , ( 2006 ) , pp .

Abstract: circulant preconditioners are commonly used to accelerate the rate of convergence of iterative methods when solving linear systems of equations with a toeplitz matrix . block extensions that can be applied when the system has a block toeplitz matrix with toeplitz blocks also have been developed . this paper is concerned with preconditioning of linear systems of equations with a symmetric block toeplitz matrix with symmetric toeplitz blocks that stem from the discretization of a linear ill - posed problem . the right - hand side of the linear systems represents available data and is assumed to be contaminated by error . these kinds of linear systems arise , e.g. , in image deblurring problems . it is important that the preconditioner does not affect the invariant subspace associated with the smallest eigenvalues of the block toeplitz matrix to avoid severe propagation of the error in the right - hand side . a perturbation result indicates how the dimension of the subspace associated with the smallest eigenvalues should be chosen and allows the determination of a suitable preconditioner when an estimate of the error in the right - hand side is available . this estimate also is used to decide how many iterations to carry out by a minimum residual iterative method . applications to image restoration are presented .  _ dedicated to ken hayami on the occasion of his 60th birthday . _  ill - posed problem , deconvolution , fft , image deblurring  * ams subject classifications . * 65f10 , 65f15 , 65f30 .


the ongoing computational revolution in physics relies on accurate solutions of fundamental equations , newton s ( or lagrange s or hamilton s ) laws of motion , in the case of classical mechanics .the determinism of these ordinary differential equations is illusory in many cases , as typically the equations are `` lyapunov unstable '' .such instabilities grow exponentially fast , , where is the largest lyapunov exponent of the solution .particle mechanics , our own research interest , provides many examples ranging from one - particle chaos to biomolecule simulations using models with many thousands of atomic degrees of freedom .we consider here the simplest particle model for chaos , a one - body `` cell model '' with the periodic four - body cell boundaries shown in * figure 1*. the resulting motion , approximated with the simplest possible `` leapfrog '' integrator , described below , is generally lyapunov unstable. we simplify the initial conditions by starting the particle trajectory in the field - free cell interior .we benchmark this problem with three quadruple - precision integrators using timesteps chosen to maximize accuracy .we compare the resulting benchmark trajectory to six other trajectories from self - starting double - precision algorithms typical of molecular dynamics simulations .five of these algorithms are `` symplectic '' , including the justifiably - popular leapfrog algorithm .the two others are runge - kutta algorithms .unit cell is shown at the left .the black regions , with potential energy greater than one half , are inacessible to the wanderer particle .initially the wanderer is at the origin with velocity . outside the central diamond - shaped regionthe fixed scatterers at the cell corners exert repulsive forces on the wanderer particle .a visually - accurate trajectory , calculated with a quadruple - precision fifth - order runge - kutta integrator , using five million timesteps and , is shown at the right with filled circles marking the configurations at times 10 , 20 , 30 , and 40 .the open circle corresponds to the maximum time ., width=384 ]  in the following sections we describe the specially - smooth differential equations governing the motion of the wandering cell - model particle , and then quantify the algorithmic accuracy with which leapfrog and the six more sophisticated integrators `` solve '' this same problem .our conclusions make up the final summary section .cell models played a role in models of the liquid state long before the development of molecular dynamics. the geometry treated here is shown at the left in * figure 1*. a mass point , the `` wanderer '' particle , moves in a periodic square cell with a motionless fixed particle at each of the four vertices .using periodic boundary conditions the equations of motion are : the force on the wanderer is the gradient of the potential function , a sum over the contributions of the four corner scatterers located at : ^ 4 { \rm \ for \ } | r - r_i| < 1 \ .\ ] ] after advancing the coordinates one timestep it is convenient to localize the motion to the cell centered on the origin .whenever the wanderer moves `` out '' , we replace it `` in '' the basic unit cell as follows :  we choose initial conditions and show an accurate benchmark solution of the motion equations for a time of 50 at the righthandside of * figure 1 * . at times of 10 , 20 , 30 , 40 , and 50 the benchmark values of are :  10 : + 0.321356333887505 , + 0.585921713605895 + 20 : + 0.81481797353866 , -0.572042192203162 + 30 : -0.040449409487 , + 0.38290501902 + 40 : + 0.3742439 , -0.842854 + 50 : + 0.4696 , -0.3568 +  * figure 1 * shows a unit cell of a periodic two - dimensional lattice in which a single particle moves in the field of scattering particles arranged in a fixed square lattice with nearest - neighbor spacing of 2 .the potential energy maximum of unity is twice the energy of the initial condition , shown at the center of the cell .the benchmark solution of the motion equations is shown at the right .this same accurate trajectory was obtained with both the candy - rozmus fourth - order and a runge - kutta fifth - order integrator using 50 million and 500 million timesteps , respectively .the two trajectories agree throughout within visual accuracy . at a time of 50 : } \ ; \ ] ] } \ .\ ] ]we consider seven solution algorithms for the wanderer particle trajectory , [ 1 ] leapfrog ( symplectic ) , [ 2 ] fourth - order candy - rozmus symplectic , [ 3 ] monte carlo symplectic , [ 4 ] sixth - order symplectic , [ 5 ] fourth - order runge - kutta , [ 6 ] fifth - order runge - kutta , and [ 7 ] eighth - order schlier - seiter - teloy symplectic .for the first six of these we use a fixed timestep typical of `` accurate '' molecular dynamics simulations .solutions for those six integrators appear in * figures 2 - 7 * .the particle mass is unity and the energy is one half . for the last integrator , which has a trajectory visually identical to that of * figure 1 * we have chosen timesteps as small as 0.00000001 in order to obtain ten - digit accuracy in the wanderer trajectory up to a time of 50 .let us consider the details of all the integrators next .`` symplectic '' integrators automatically obey liouville s theorem by advancing the solution of hamiltonian problems in time according to a series of phase - volume - conserving shears .symplectic algorithms alternate steps advancing the coordinates and momenta in time .the simplest example is equivalent to the strmer - verlet `` leapfrog algorithm '' :  this algorithm is said to be `` second order'' , with a fixed - time coordinate error of order for when applied to the simple harmonic oscillator .it is time reversible in that changing gives the same trajectory points either forward or backward in time .how does the simulation begin ?starting out at the origin , with the wanderer speed equal to unity and a fixed timestep , the first 420 steps leave the momenta unchanged and becomes 1.0004 . during the 421st step the upper right scattereris contacted and begins to repel the wandering particle with a force : where and are the wanderer coordinates .after an elapsed time we reverse the sign of the time so as to integrate _backward _ to see how closely the wanderer returns to its initial location .so long as we find that the trajectory reverses to within a distance 0.01 of the origin .we will see that this retracing of steps does not guarantee a match with the accurate trajectory shown at the right in * figure 1*. both the trajectory reversal and the conservation of energy are poor diagnostics for trajectory _ accuracy _ , where accuracy means reproducing correct values of the coordinates .higher - order algorithms , with fixed - time integration errors of order , ,  can be developed from taylor s series about giving small increments in the coordinates and momenta as three- , four- , five- term series in .candy and rozmus fourth - order integrator ( with an error of order at a fixed not - too - large , time ) is a simple example , cited in the very useful summary paper by gray , noid , and sumpter :  reference 7 gives the analytic forms of all of the coefficients .notice that the coefficients incrementing the coordinates sum to unity as do also those incrementing the momenta .each timestep requires three separate evaluations of the forces .although it is usual to provide coefficients in integration algorithms to many significant figures , in most cases an approximate rendition is sufficient .it is quite possible to develop algorithms with a monte carlo method , adjusting the coefficients to minimize the trajectory error for the simple harmonic oscillator problem .an integrator requiring five force evaluations per timestep was developed by monte carlo sampling adjusting the coefficients subject to the constraints of time reversibility and normalization so that the monte carlo trajectory optimization occurs in a four - dimensional space .the resulting integrator was successful in modelling many - body dynamics but is here applied to the cell - model problem of * figure 1 * :   the cell model trajectory using this monte carlo integrator is illustrated in * figure 4*.   yoshida developed and applied a general technique for finding a variety of higher - order symplectic integrators. his sixth - order time - reversible integrator advances the coordinates eight times per timestep , using the symmetric ( so as to guarantee time - reversibility ) set of eight coefficients which sum to unity :  between the successive coordinate updates there is a force calculation and an update of the momenta , using seven coefficients , which likewise sum to unity :    runge - kutta integrators ( _ circa _ 1900 , as described in wikipedia ) advance both coordinates and momenta _ simultaneously _ in a series of stages within each timestep .as in the symplectic case the variables at time are expressed as series in , putting conditions on the summed - up coefficients for each power of to be treated correctly by the algorithm .the main advantage of runge - kutta methods is that they can be applied to arbitrary sets of ordinary differential equations , not just those from hamiltonian mechanics .the fourth - order `` classic '' runge - kutta method has been a standard workhorse model for solving sets of coupled ordinary differential equations for 100 years. applied to the harmonic oscillator the fourth - order algorithm suffers a loss in energy proportional to the fifth power of the timestep .the fifth - order runge - kutta integrator behaves in the opposite manner with the energy increasing rather than decreasing .hybrid `` adaptive '' models , incorporating both fourth- and fifth - order algorithms , provide a simple means for the automatic control of integration errors .the harmonic oscillator is an excellent test case of integrator accuracy where lyapunov instability is absent. * figure 6 * illustrates the same cell - model orbit for the classic fourth - order runge - kutta integrator .* figure 7 * shows a fifth - order runge - kutta integrator : }\ ] ] }\ ] ] }\ ] ] }\ ] ] }\ ] ] }\ ] ] here yp [ ... ] represents the righthandside of the vector differential equation where the six force evaluations in each timestep are indicated by .both runge - kutta integrators return to the origin with errors no more than 0.01 with reversal at time 42 . forward in time their trajectories are accurate through times of 35 and 37 , the last being the best of the double - precision integrators .the energy errors for the two runge - kutta integerators are in the thirteenth and fourteenth decimal places .ernst teloy , christoph schlier , and ansgar seiter developed and implemented a useful eighth - order time - reversible symplectic integrator with 17 force evaluations per step . applied to the harmonic oscillator the rms coordinate error increases by about eight orders of magnitude when the timestep is increased by a factor of ten , consistent with an eighth - order method .  for the readers convenience we reproduce here the 18 coefficients required to implement the method .they can be found quoted to 35 decimal places at christoph schlier s freiburg website or in reference 12 .this precision is steadily reduced , digit by digit , through lyapunov instability , described in more detail in section iv . in the cell - model casethe rate of precision loss is 0.7 , one binary bit per unit time .accordingly , for the eighth - order integrator in quadruple precision at time 50 we would expect an exponentially amplified error of order .in fact , we find a trajectory error of order using a timestep of , as is shown below .even so the eighth - order integrator with loses only seven of the original 35 digits in energy along with twenty digits in position when run forward and backward for 50,000 steps to match the time illustrated in all the figures .as was illustrated and emphasized in references 12 and 13 energy conservation and trajectory reversibility are both of them misleading diagnostics of trajectory accuracy .it is only through a study of _ convergence _ that trajectories can be validated . for the eighth - order symplectic integrator the timestep dependence ofthe coordinates at time 50 is as follows :    for the convenience of the reader we reproduce the integrator coefficients here from reference 12 , together with a short harmonic - oscillator program to demonstrate their use .c ( 1 ) = + 0.04463 79505 23590 22755 9139996257 33590 d00 + c ( 2 ) = + 0.13593 25807 16909 59145 5432642134 95574 d00 + c ( 3 ) = + 0.21988 44042 71470 72254 4455350696 06167 d00 + c ( 4 ) = + 0.13024 94678 05238 28601 62119 37781 96846 d00 + c ( 5 ) = + 0.10250 36569 39750 69608 2612410077 79814 d00 + c ( 6 ) = + 0.43234 52186 93585 47487 98325 78848 77035 d00 + c ( 7 ) = -0.00477 48291 69168 81658 02248 90639 62934 d00 + c ( 8) = -0.58253 47690 40408 45493 1128379308 61212 d00 + c ( 9 ) = -0.03886 26428 21118 17697 73742 08751 89743 d00 + c(10 ) = + 0.31548 72853 79404 79698 2736037972 74199 d00 + c(11 ) = + 0.18681 58374 32971 55471 52615 35039 72746 d00 + c(12 ) = + 0.26500 27549 90620 83398 3460029630 79872 d00 + c(13 ) = -0.02405 08473 57473 61993 5735879824 07554 d00 + c(14 ) = -0.45040 49249 97722 51180 9228967121 51891 d00 + c(15 ) = -0.05897 43301 55923 86914 57532 39267 66330 d00 + c(16 ) = -0.02168 47617 18613 35324 93438 86847 07580 d00 + c(17 ) = + 0.07282 08003 35901 28173 7618926412 34244 d00 + c(18 ) = + 0.55121 42963 41970 67334 40560 13815 94315 d00 +  oscillator program with q , p , dt and 18 c(i ) : +  do i = 1,17,2 + q = q + c(i)*p*dt + p = p - c(i+1)*q*dt + enddo + do i = 17,1,-2 + q = q + c(i)*p*dt + if(i.gt.1 ) p = p - c(i-1)*q*dt + enddofor chaotic systems the algorithmic accuracy of numerical integrators deteriorates exponentially rather than linearly in the time .the underlying exponential lyapunov instability of dynamical systems is easily measured by following the motion of a `` reference '' trajectory in the usual way , for instance with any one of the seven algorithms discussed here .an additional `` satellite '' trajectory , separated from the reference by a small length , is also followed using the same algorithm . at the end of each timestepthe separation is rescaled , maintaining the length of the offset between the trajectories constant , but allowing the direction to vary : \ ; \\ ] ] \ .\ ] ] the largest lyapunov exponent is simply the average value of the growth rates measured at the ends of every timestep prior to rescaling : \ \rangle \ .\ ] ] previous studies of this cell model, with the same initial condition , have shown that the largest lyapunov exponent is about 0.7 .this means that an error of the order at the initiation of a run of length 50 will increase by a factor of .this exponential growth rate explains why it is that _ all _ of the double - precision integrators fail , from the standpoint of reproducing a reversible trajectory , at about the same time , at about _half _ the time where quadruple - precision trajectories fail .it is because these trajectories are just approximations that the most sophisticated biomolecule simulations are based on the rudimentary leapfrog algorithm rather than more sophisticated algorithms .of course , even the slightest difference in the error prior to amplification will yield a different history .just summing the particle interactions in a different order leads to qualitatively different histories once the lyapunov instability rises to the level of visibility , an increase of 16 digits for routine double - precision simulations .the phase - shift errors in all of the algorithms discussed here can be measured by choosing the initial velocity for which the roundoff errors in the and directions are identical .  if high accuracy is required , as in astronomical simulations , multiple precision can be employed , as demonstrated by lorenz attractor simulations using a precision of thousands of decimal digits .but , as joseph ford was fond of pointing out , lyapunov instability is incompatible with high accuracy .doubling the number of significant figures in the integration algorithm only doubles the time for which the simulation is accurate .recently hanno rein and david siegel developed and implemented a relatively complicated fifteenth - order integrator for gravitational problems with the provocative title `` a fast , adaptive , high - order integrator for gravitational dynamics , accurate to machine precision over a billion orbits '' .evidently this integrator is not at all intended for long - time applications to chaotic problems , where errors grow exponentially with time .conversations with ben leimkuhler and mark tuckerman , both of whom summarily dismiss the use of runge - kutta techniques , due to their monotonic energy drift , plus the appearance of rein and siegel s high - order long - time work led to the present article .to summarize , for simple chaotic simulations ( such as classical fluids ) symplectic integrators attain accuracies similar to those obtained with runge - kutta integration and are primarily limited by lyapunov instability .although energy conservation and trajectory reversibility characterize symplectic integrators , those properties do not ensure trajectory accuracy .the reversibility of the double - precision leapfrog integrator , to a time of 47 and back , exceeds that of all the more accurate double - precision integrators .  for usit was illuminating to find that the humble leapfrog integrator , presumably nearing its 330th anniversary , is nearly as useful as are its more complex relatives , and is certainly far more economical . for higher accuracythere is little distinction between the symplectic and the runge - kutta integrators for chaotic problems , because both types lose accuracy at the very same rate , determined by the maximum lyapunov exponent .it is significant that all of the integrators used here conserve energy almost perfectly for the benchmark problem .they also reverse back to the initial conditions _ even when their trajectories are inaccurate_. one takeaway message from these simulations is the one to which joseph ford devoted much thought and many thought - provoking words , among them these taken from reference 16 :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` newtonian determinism assures us that chaotic orbits exist and are unique , but they are nevertheless so complex that they are humanly indistinguishable from realisations of truly random processes . ''_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  liao has confronted the lyapunov instability problem headon for the lorenz attractor. by using 3500-term series expansions coupled with 4180-digit arithmetic he followed the evolution of the lorenz model to a time of 10,000 . like the continuing discovery of the digits of activity will last as long as mankind .lyapunov instability often shows up in peculiar places . simply changingthe order of operations in adding up forces or in computing the weights of contributions to differential equations righthandsides can provide the seeds from which macroscopic change develops .we learned this lesson in simulating the collisions of mirror - image manybody drops and crystals . to retain accurate mirror symmetry it was necessary to symmetrize the force calculations at every timestep. thank ben leimkuhler , hanno rein , and mark tuckerman for their useful and stimulating comments , clint sprott for his constant encouragement , and christoph schlier for several helpful emails .we are grateful to the gnu fortran project for furnishing their no - cost quadruple - precision compiler . to find it google `` gnu fortran project '' .s. k. gray , d. w. noid , and b. g. sumpter , `` symplectic integrators for large scale molecular dynamics simulations : a comparison of several explicit methods '' , the journal of chemical physics * 101 * , 4062 - 4072 ( 1994 ) .h. rein and d. s. spiegel , `` ias15 : a fast , adaptive , high - order integrator for gravitational dynamics , accurate to machine precision over a billion orbits '' , ar 1409.4779 , also available in the monthly notices of the royal astonomical society .w. g. hoover and c. g. hoover , `` what is liquid ?lyapunov instability reveals symmetry - breaking irreversibility hidden within hamilton s many - body equations of motion '' , condensed matter physics * 18 * , 13003 - 1 - 13 ( 2015 ) = ar 1405.2485 .

Abstract: time - reversible symplectic methods , which are precisely compatible with liouville s phase - volume - conservation theorem , are often recommended for computational simulations of hamiltonian mechanics . lack of energy drift is an apparent advantage of such methods . but _ all _ numerical methods are susceptible to lyapunov instability , which severely limits the maximum time for which chaotic solutions can be `` accurate '' . the `` advantages '' of higher - order methods are lost rapidly for typical chaotic hamiltonians . we illustrate these difficulties for a useful reproducible test case , the two - dimensional one - particle cell model with specially smooth forces . this hamiltonian problem is chaotic and occurs on a three - dimensional constant - energy shell , the minimum dimension for chaos . we benchmark the problem with quadruple - precision trajectories using the fourth - order candy - rozmus , fifth - order runge - kutta , and eighth - order schlier - seiter - teloy integrators . we compare the last , most - accurate particle trajectories to those from six double - precision algorithms , four symplectic and two runge - kutta .  0.5 cm