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General linear methods for the parallel solution of ordinary differential equations
... Butcher recently introduced a class of methods called DIMSIMs [11, 12, 13, 14] which contains methods with each of the above structures. Because of their particular relevance to our work, these methods will be discussed separately later. ...
... Although our introduction to the GLMs known as DIMSIMs was through an early seminar by Butcher, they have now been described in published work [11, 12, 13, 14]. The methods are classified into four types, depending on the structure of B 1 . ...
... Types 1 and 3 are explicit methods, intended for non-stiff problems, while type 2 is singly diagonally implicit, and type 4 is singly strictly diagonally implicit. The DIMSIMs described in [11, 12, 13] are constructed with r = s, A 1 = I, A 2 e = e, where e is an r-vector of all 1's, and B 2 is determined from the simplified order conditions A(r), B(r), and C(r) in terms of the c i 's, B 1 , and A 2 . This gives an order and a stage order of r. ...
We describe a new class of General Linear Methods (GLMs) with the following properties: the methods are strictly diagonally implicit, allowing efficient parallel implementation; the stability matrix has no spurious eigenvalues (i.e., only one eigenvalue of the stability matrix is non-zero); the methods have high stage order, allowing them to retain their order on problems for which order reduction might otherwise be a problem; the methods are L-stable, making them suitable for the solution of stiff problems. As well as presenting some order barriers, we derive and test methods of orders 2--6. Key words. Parallel, Numerical Solution, Initial Value Problem, IVP, Ordinary Differential Equation, ODE, General Linear Method, GLM, Order, Stability. AMS(MOS) subject classification. 65L05. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Information Technology Research Centre of Ontario. Contents 1 Introduction and Background 1 1.1 ...
... Butcher recently introduced a class of methods called DIMSIMs [11,12,13,14] which contains methods with each of the above structures. Because of their particular relevance to our work, these methods will be discussed separately later. ...
... Although our introduction to the GLMs known as DIMSIMs was through an early seminar by Butcher, they have now been described in published work [11,12,13,14]. The methods are classified into four types, depending on the structure of B 1 . ...
... The DIMSIMs described in [11,12,13] are constructed with r = s, A 1 = I, A 2 e = e, where e is an r-vector of all 1's, and B 2 is determined from the simplified order conditions A(r), B(r), and C(r) in terms of the c i 's, B 1 , and A 2 . This gives an order and a stage order of r. ...
This thesis derives a new class of general linear methods (GLMs) intended for the solution of stiff ordinary differential equations (ODEs) on parallel computers. Although GLMs were introduced by Butcher in the 1960s, the task of deriving formulas from the class with properties suitable for specific applications is far from complete. This thesis is a contribution to that work. Our new methods have several properties suited for the solution of stiff ODEs on parallel computers. They are strictly diagonally implicit, allowing parallelism in the Newton iteration used to solve the nonlinear equations arising from the implicitness of the formula. The stability matrix has no spurious eigenvalues (that is, only one eigenvalue of the stability matrix is non-zero), resulting in a solution free from contamination from spurious solutions corresponding to non-dominant, non-zero eigenvalues. From these two properties arises the name DIMSEM, for Diagonally IMplicit Single-Eigenvalue Method. The method...
... One important advantage of the continuous over the discrete approach is the ability to provide discrete schemes for simultaneous integration. These discrete schemes can as well be reformulated as general linear methods (GLM) [6]. The block methods are self-starting and can directly be applied to both initial and boundary value problems [7]. ...
... We get four discrete schemes. Hence, the hybrid block methods are as follows (6) Interpolating Equation (2) ...
In this research work,the construction of two-step hybrid block Simpson's methods with two, threeand four off-grid points for the solutions of first order stiff systems of ordinary differential equations (ODEs) in studied.In the derivation of the method, power series is adopted as basis function to obtain the main scheme through collocation and interpolations approach. Taylor series was adopted alongside, the method to generate non-overlapping numerical results.This is achieved by transforming a k-step multi-step method into continuous form and evaluating at various grid points to obtain the discrete schemes. The performance of the methods is demonstrated on some numerical experiments. The results revealed that the hybrid block Simpson's method is efficient, accurate and convergent on mildly stiff problems.
... Parallelism for IMEX schemes is scarcely explored [16,18], but it is well studied for traditional, unpartitioned GLMs [8][9][10]23]. One step of a GLM is Methods are frequently categorized into one of four types to characterize the suitability for stiff problems and parallelism [7]. ...
... In [8,9], Butcher developed a systematic approach to construct DIMSIMs of type 4 with "perfect damping at infinity". One of his primary results is presented in Theorem 4.1. ...
High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multi-stage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge–Kutta methods.
... Parallelism for IMEX schemes is scarcely explored [16,18], but it is well-studied for traditional, unpartitioned GLMs [8,10,9,23]. One step of a GLM is: Methods are frequently categorized into one of four types to characterize suitability for stiff problems and parallelism [7]. ...
... In [8,9], Butcher developed a systematic approach to construct DIMSIMs of type 4 with "perfect damping at infinity." One of his primary results is presented in theorem 4.1. ...
High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multistage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.
... Let Z = 1 z z 2 · · · z p−1 z p T . Then, using Taylor series expansion [5] is: exp (cz) = zA exp (cz)+UZ+O(z p+1 ), exp (z)Z = zB exp (cz)+V Z+O(z p+1 ). ...
... The rejection criterion depends on error estimation (Section 3.2). Thus, to analyse this, it is necessary to first see the behaviour of the C i h i+1 y (i+1) (x) for i = 3, 4. Figure 4 shows the comparison of C 3 y (4) (x) and C 4 y (5) (x) for this problem, where C 3 and C 4 are error constants for the methods of order three and four, respectively, and y (4) (x) and y (5) (1) 10 -3 Fig. 3 Stepsize control, order control and the first two Nordsieck elements for problem A3 could be selected by the controller at each time step. As A3 is a periodic problem with period 2π, there are many times that the third-order error term C 3 y (4) (x) is less than the fourth-order term C 4 y (5) (x), though for very small intervals of time (Fig. 4). ...
This paper describes the implementation of a class of IRKS methods (Wright 2002). These GLM algorithms are practical with reliable error estimators (Butcher and Podhaisky, Appl. Numer. Math. 56, 345–357 2006). The current robust ODE solvers in variable stepsize as well as in variable-order mode are based upon heuristics. In this paper, we examine an optimisation approach, based on Euler-Lagrange theory (Butcher, IMA J. Numer. Anal. 6, 433–438 1986), (Butcher, Computing 44, 209–220 1990), to control the stepsize as well as the order and implement the GLMs in an efficient manner. A set of nonstiff to mildly stiff problems have been used to investigate this approach in fixed-order and variable-order modes.
... One important advantage of the continuous over discrete approach is the ability to provide discrete schemes for simultaneous integration. These discrete schemes can be reformulated as general linear methods (GLM) [5]. The block methods are self-starting and can be applied to both stiff and non-stiff initial value problem in differential equations. ...
... With the mathematical software, we obtain the continuous formulation of eqns. (5) and (6) of the form ...
In this paper, we consider the derivation of hybrid block method for the solution of general first order Initial Value Problem (IVP) in Ordinary Differential Equation. We adopted the method of Collocation and Interpolation of power series approximation to generate the continuous formula. The properties and feature of the method are analyzed and some numerical examples are also presented to illustrate the accuracy and effectiveness of the method.
... Diverse method(s) of obtaining approximate solution to general or special equations in the form of ) ,..., ' , , ( (1) have been derived for various classes of linear multistep methods. Amongst these are the methods of Fatunla (1988), Ayinde and Ibijola (2015), Onumanyi et al (1994), Butcher (1993), James et al (2012), , Mohammed and Yahaya (2010), Skwame et al (2012), Ajileye et al (2018). Ogunride et al (2020), Isah et al (2012) and Salawu et al (2020), where (1) is of first order. ...
In this paper, we adopt the general method of interpolation and collocation in Linear Multistep Methods in deriving some numerical schemes for solving second order ordinary differential equations. Different choices of the interpolating function in the form of shifted Legendre, shifted Chebyshev and Lucas polynomials with the same interpolation and collocation points are considered in order to establish uniformity or otherwise of the derived schemes for the various polynomials. Furthermore, probable disparities in the derived schemes for varied choices of interpolation and collocation points are also investigated. Results indicate that all the polynomials yield exactly the same schemes for the same choice of interpolation and collocation points but different schemes for different choices of interpolation and collocation points. However, numerical examples considered showed that all the derived schemes performed exactly in the same manner in terms of accuracy, regardless of the choices of interpolation or collocation points. Nevertheless, the derived schemes perform admirably better when compared with existing methods in literature
... with y(t) ∈ R m and f : R m → R m Lipschitz continuous. General linear methods (GLMs) [1] are numerical discretizations of (1) that generalize both Runge-Kutta methods (by computing multiple internal stages) and linear multistep methods (by transferring from one step to the next multiple pieces of information). One step of the GLM applied to (1) reads [2][3][4][5]: ...
This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.
... One important advantage of the continuous over discrete approach is the ability to provide discrete schemes for simultaneous integration. These discrete schemes can be reformulated to general linear methods (GLM) [5]. ...
... These discrete schemes can as well be reformulated as general linear method (GLM) [2]. Many researchers have worked on the development of continuous linear multi-step method in finding solution to (1). ...
... The absolute stability regions of the HOBIM methods are constructed by reformulating the block integrators whose coefficients are in Tables 1-5 as General linear Methods of Butchers [4] using the notations introduced in Burage and Butchers [5] .The General linear method (GLM) is represented by a partitioned (s+r) × (s+r) characterized by the four matrices A, B, U and V expressed in ...
The search for higher order A-stable linear multi-step methods has been the interest of many numerical analyst and has been realized through either higher derivatives of the solution or by inserting additional off step points,supper future points and the likes.These methods are suitable for the solution of stiff differential equations which exhibit characteristics that place severe restriction on the choice of step size. It becomes necessary that only methods with large regions of absolute stability remain suitable for such equations. In this paper, high order block implicit multi-step methods of the hybrid form up to order twelve have been constructed using the multi-step collocation approach by inserting one or more off step points in the multi-step method. The accuracy and stability properties of the new methods are investigated and are shown to yield A-stable methods, a property desirable of methods suitable for the solution of stiff ODEs. The new High Order Block Implicit Multistep methods used as block integrators are tested on stiff differential systems and the results reveal that the new methods are efficient and compete favorably with the state of the art Matlab ode23 code.
... 1. dividing the computational cost of the right-hand side of the problem by using different processors for different components, 2. decoupling the system into independent sub-systems with fewer coordinates via an iterative process : all sub-systems can then be integrated simultaneously provided there is a sufficient number of processors [38], 3. designing new methods with some intrinsic parallelism, such as block Runge-Kutta methods [28,29], Parallel Diagonally Iterated Runge-Kutta methods (PDIRKs) [21,22,31,34,36,37], Multiply Implicit Runge-Kutta methods (MIRKs) [30], General linear methods [18,19,32] or DIMSIMs [4,5,7,9,10,11,12,13,14,15]. ...
Potential advantages and drawbacks of multi-value methods are discussed in detail. This presentation leads in a natural way to the definition and the construction of DIMSIMs. In particular, it is shown that DIMSIMs should be soon efficiently implemented in a parallel environment.
... For these methods, the stability is determined by the (11.2) where Lk to the usual Laguerre polynomial of degree Ic. The proof of (11.1) and (11.2) will be given in a forthcoming paper [15]. ...
A number of questions and results concerning Runge-Kutta and general linear methods are surveyed. These include order conditions and order bounds for Runge-Kutta methods, the A-stability of implicit Runge-Kutta methods based on Gaussian quadrature and transformation methods of implementation which lead to singly-implicit methods. The sections dealing with general linear methods include a discussion of the order conditions and an algebraic structure for carrying out order analyses as well as an introduction to a special function associated with parallel methods for stiff problems.
... It will be shown in a forthcoming paper [2] that, with this choice, the stability polynomial ...
The special case of diagonally-implicit multistage integration methods is considered in which the order, the stage order, the number of values passed between steps and the number of stages in a step, all coincide. It is shown that a similarity transformation can be applied to the matrices characterizing the method so as to simplify the expression for the stability polynomial and thus aid in the search for methods with acceptable stability.
... Ainsi nous avons fait un survol de quelques travaux reliés aux problèmes génériques suivants : (i) Calcul parallèle Stone, 2000;Chen et Lin, 2000); (ii) Les simulations distribuées (Chandy et MISRA, 1979;Dado et al., 1993); (iii) Ordonnancement Carlier, 1989); (iv)Équilibrage des lignes d'assemblage (Urban, 1998;Muth, 1963); (v) Gestion de projet (Milon, 1999;Sprecher, 2000) Le problème d'ordonnancement est un sujet qui intéresse aussi bien les informati- Burrage (1995), Butcher (1993aet 1993b) et Chartier (1994. ...
The simulation and the analysis of advanced systems such as robots require an increased design power, which sequential processors are far from satisfying. Parallel computing is a solution to reduce the execution times. By using independent processors together, we can increase our capacity of calculation and answer such requirements thereafter. In this project, made in collaboration with Opal-rt technologies and the Canadian Space Agency, we propose a practical solution to automate the parallelization of simulations developed under the Matlab/Simulink environment. The solution was developed in two phases: generation of the task graph, then the task assignment to the processors. Our feasibility study is conclusive and a little research work remains to be performed before launching our software on the market.
Under Matlab/Simulink, we can develop simulations using a functional bloc-diagram language. The logic of this kind of language already allows a well-structured code. However, to automate parallelization, it is necessary to formalize the model in the form of a graph to make sure that the tasks are well identified and that the constraints of precedence and communication delays are well defined. That is why we generate a directed acyclic graph (DAG). The generation of such a graph is a big challenge. Firstly, the real model contains cycles and the blocks are gathered in a hierarchical way with various levels. Secondly, it is necessary to make this procedure automatic for any kind of models. The solution that we developed is based on the Matlab tools. We created Simulink compiler which surfs the model to detect all its components. This compiler allows, for a given level of details or for a “ desired granularity ”, to determine in an automatic way the tasks and their links. Moreover a good characterization of the loops in the model enables us to
avoid the cycles in the generated graph.
Once the DAG is generated, we must assign the tasks to the different CPUs with respect to the communication delays and the precedence constraints while aiming the shortest possible execution time over all CPUs - which means the objective is to minimize the makespan. We performed a literature review on this subject. There are several methods of resolving this problem but, due to the nature of our project - a feasibility study - we began with what is simplest and rather sure, a priority-based heuristic and a “Clustering” algorithm. We do not allow task duplication but we explore it. We applied these types of heuristic to our problem. Adapting these algorithms to our situation was more difficult than expected. The implementation of the code gave, after many improvements, good results - as good as we had planned.
To validate our results we performed several tests. On the one hand, a group of tests relates to a realistic example of a robot simulation. This test aims to validate our method in a real context and to ensure the accuracy of the results after the automation of the whole process. In other words, these tests prove the feasibility of the project. On the other hand, another group of tests use a group of representative graphs to test the robustness, the stability and the performance of our heuristics in cases when there is no practical control of the scheduler. The results of these tests are conclusive. Our heuristics provides good solutions and the best schedules in a controlled context are still the best when the scheduler is not under control, which is the case of the Rt-lab environment.
This proves that the automation of the parallelization of a schematic model, the type of the Simulink/Matlab models, is possible. The heuristics give acceptable results. However, due to the fact that we obtain worse solutions when the DAG is generated with a finer granularity, we suspect that metaheuristics could increase the quality of the solutions. Also, while the report is conclusive on the practical level, it is necessary to continue the research tasks before delivering the product to the market.
Keywords: distributed simulation, parallel computing, Simulink, scheduling, priority-based heuristic, clustering, communication delays.
... For general linear methods (methods which use both information from previous steps and internal stages, i.e., they are a combination of one-step and multistep methods) parallelism is possible by requiring that the internal stages should be decoupled from each other. This makes it possible to construct some nice methods, [But93,EJ95] but there is still quite a few problems to solve for this class of methods regarding error estimation and variable step size. ...
v Resume vii Chapter 1. Classical Theory of Ordinary Differential Equations 1 1.1. Stiff Initial Value Problems 2 1.2. Numerical Methods 4 1.2.1. Runge-Kutta Methods 4 1.2.2. Method Order 4 1.3. Stability Properties 6 1.3.1. A-Stability 6 1.3.2. L-Stability and Stiff Accuracy 6 1.3.3. B-Stability and Algebraic Stability 7 Chapter 2. Parallelism 9 2.1. Parallel Computers 10 2.1.1. Communication 10 2.1.2. The IBM SP-2 11 2.2. Programming Parallel Machines 12 2.2.1. Programming Languages 13 2.2.1.1. Fortran and High Performance Fortran 14 2.2.1.2. MPI -- Message Passing Interface 14 2.3. Parallel Integration of Initial Value Problems 16 2.3.1. Parallelism Across Space 16 2.3.2. Parallelism Across Time 16 2.3.3. Parallelism Across the Method 16 Chapter 3. Construction of Runge-Kutta Methods 19 3.1. The W-Transformation 20 3.2. B-stable and Stiffly Accurate Methods 22 i 3.2.1. Forcing Stiff Accuracy 23 3.2.2. Computation of X 29 3.2.3. An Example 30 3.3. Stiffly Accurate Methods with Hi...
... Other possibilities for parallel Runge-Kutta and related methods have been discussed in e.g. 14,6] as well as in the recent book on the topic, 3]. ...
The construction of stiffly accurate and B-stable multi-implicit Runge-Kutta methods for parallel implementation is discussed. A fifth and a seventh order method is constructed and a promising numerical comparison with the efficient Radau5 code of E. Hairer and G. Wanner is conducted. AMS subject classification: 65L06, 65Y05 Keywords. Multi-implicit Runge-Kutta methods, parallel computation 1. Introduction This paper describes the construction of B-stable and stiffly accurate multiimplicit Runge-Kutta methods -- MIRK methods (not to be confused with monoimplicit RK-methods [7]). When solving stiff ordinary differential equations or differential algebraic equations, stiff accuracy and B-stability seems to be desirable properties, [11, 10]. Since parallel computation is becoming widely available it is natural to ask how to construct such methods which are suitable for implementation in a parallel environment. One possible way to do this--- as will be shown below---is by requiring tha...
This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.
This paper concerns the functions defined by the functional equations \begin{array}{*{20}{l}}
{\exp \left( z \right) = \phi \left( {\frac{z}{{\exp \left( z \right)}}} \right),} \\
{\left( {1 - z} \right)\exp \left( {tz} \right) = \psi \left( {\frac{z}{{\left( {1 - z} \right)\exp \left( {tz} \right)}}} \right)}
\end{array} and their relationship to the numerical solution of ordinary differential equations in a parallel environment. It will be shown that φ and its Padé approximations are related to stability questions for type 3 DIMSIM methods and that ψ and its Padé approximations have a similar relationship to type 4 DIMSIMs. An introduction to DIMSIM methods will be presented as a special class of general linear methods. The particular features which identify DIMSIMs from the much wider class of general linear methods are the natural meaning of the quantities being approximated, high stage-order and ease of implementation in a sequential or parallel environment. In particular type 3 and 4 DIMSIMs are intended for parallel implementation and for non-stiff and stiff differential equation systems respectively. Particular methods will be derived from approximations to φ and ψ selected for their stability properties.
We describe the construction of diagonally implicit multistage integration methods (DIMSIMs) of type 1 and 2 with the same stability properties as explicit Runge-Kutta methods or implicit SDIRK methods, respectively, of appropriate order. Such methods are intended for the numerical integration of nonstiff or stiff differential systems in a sequential computing environment. Examples of pqrst DIMSIMs are given with p, q, r, s ≤ 4 and t = 1 or 2, where p is the order, q is the stage order, r is the number of external stages, s is the number of internal stages, and t is the type of the method. Coefficients of the methods of order 4 were obtained numerically with the aid of continuation programs from PITCON, ALCON, and HOMPACK.
The identification of high order diagonally implicit multistage integration methods with appropriate stability properties requires the solution of high dimensional nonlinear equation systems. The approach to the solution of these equations, and hence the construction of suitable methods, that we will describe in this paper, is based on computation of the coefficients of the stability polynomial by a variant of the Fourier series method and solving the resulting systems of polynomial equations by least squares minimization. Examples of explicit and implicit methods of order 5 and 6 are given which are appropriate for nonstiff or stiff differential systems in a sequential computing environment. The coefficients of these methods were obtained numerically with the aid of lmdif. f and lmder. f from MINPACK. These programs minimize the sum of the squares of nonlinear functions by a modification of the Levenberg-Marquardt algorithm. The derived explicit and implicit methods have the same stability properties as explicit Runge-Kutta and SDIRK methods, respectively, of the same order.
General linear methods, as multistage multivalue methods, are the natural generalizations of linear multistep and Runge-Kutta methods. This survey contains a discussion of the traditional methods and a motivation for the general linear type of generalization. The new methods are introduced in terms of their formulation and the basic properties of consistency, stability and convergence. The order of general linear methods has to be looked at from a new point of view and it is shown how to use an algebraic structure (equivalent to B-series) to express conditions for a given order. Linear and non-linear stability for the new methods brings the theories for the classical methods into a comprehensive formulation and known results are outlined. Recently a number of subfamilies have been introduced and some of these are considered in detail. This applies in particular to methods with the property known as ‘inherent Runge-Kutta stability’. These seem to have prospects of yielding useful and efficient methods, and some progress towards their practical implementation is outlined. Finally, the relationship between stability functions and order of methods is discussed in a setting wide enough to include general linear methods as well as multiderivative methods, such as Obreshkov methods. The classical barriers due to Ehle, Daniel-Moore and Dahlquist (second barrier) all fit into a common pattern and these are explored in a general setting.
We investigate some classes of general linear methods withs internal andr external approximations, with stage orderq and orderp, adjacent to the class withs=r=q=p considered by Butcher. We demonstrate that interesting methods exist also ifs+1=r=q, p=q orq+1,s=r+1=q, p=q orq+1, ands=r=q, p=q+1. Examples of such methods are constructed with stability function matching theA-acceptable generalized Pad approximations to the exponential function.
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