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In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A ( k ) converge to zero. However, this fact alone does not necessarily guarantee that A ( k ) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q ( k ) . We prove that under certain assumptions A ( k ) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q ( k ) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q ( k ) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example.

This paper is concerned with the resilient dynamic output-feedback (DOF) distributed model predictive control (DMPC) problem for discrete-time polytopic uncertain systems under synchronous Round-Robin (RR) scheduling. In order to alleviate the computation burden and improve the system robustness against uncertainties, the global system is decomposed into several subsystems, where each subsystem under synchronous RR scheduling communicates with each other via a network. The RR scheduling is adopted to avoid data collisions, however the updating information at each time instant is unfortunately reduced, and the underlying RR scheduling of subsystems are deeply coupled. The main purpose of this paper is to design a set of resilient DOF-based DMPC controllers for systems under the consideration of polytopic uncertainties and synchronous RR scheduling, such that the desirable performance can be obtained at a low cost of computational time. A novel distributed performance index dependent of the synchronous RR scheduling is constructed, where the last iteration information from the neighbor subsystems is used to deal with various couplings. Then, by resorting to the distributed RR-dependent Lyapunov-like approach and inequality analysis technique, a certain upper bound of the objective is put forward to establish a solvable auxiliary optimization problem (AOP). Moreover, by using the Jacobi iteration algorithm to solve such a problem online, the distributed feedback gains are directly obtained to guarantee the convergence of system states. Finally, two examples including a distillation process example and a numerical example are employed to show the effectiveness of the proposed resilient DMPC strategy.

In this paper, we propose a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In our method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. We establish the quadratic convergence of our method specially. We also give some numerical examples to illustrate the effectiveness of the proposed method.

This paper deals with computational delay in distributed nonlinear model predictive control. A fast, cooperative distributed model predictive control algorithm is proposed based on parametric sensitivity. The implementation strategy is divided into two different stages. In the background stage, the future state is estimated one step forward with the current state and input. The local MPC controllers perform distributed optimization based on the predicted state and iterate to obtain the nominal optimal solutions. In the online stage, all the controllers correct their nominal optimal inputs through parametric sensitivity. Specifically, each controller formulates its local sensitivity equation based on the state estimation error. In order to solve these linear equations in a distributed way, Jacobi iterative method is applied. The overall algorithm can provide fast control action. A case study is provided to show the promising performance of the proposed method.

We introduce three new classes JGD ( n ) , LUDD ( n ) and DWDD ( n ) of n × n complex matrices with variants of the diagonal dominance property. Each of the three classes contains singular matrices. We investigate in depth the relations between the three classes and their relations with the classes of diagonally dominant and generalized diagonally dominant matrices. We also study the problem of providing necessary and sufficient conditions for the singularity of a matrix in each of the three classes JGD ( n ) , LUDD ( n ) and DWDD ( n ) . We present an application of the theory of the classes JGD ( n ) , LUDD ( n ) and DWDD ( n ) to the iterative Jacobi method for the solution of the linear system A x = b .

We apply the approach of Eisenhart lift to the classical Euler’s bicentric system, yielding a three-dimensional geodesic system. An extra potential is added to the geodesic Hamiltonian to preserve the separability of the system. We also make a generalization of the system by Jacobi method. The Stackel separability of all of these systems are verified. Their Hamilton–Jacobi equations are integrated in principle. The explicit forms of the Killing tensors on the associated Riemannian manifolds are also shown.

Asynchronous iterative methods for solving linear systems are gaining renewed interest due to the high cost of synchronization points in massively parallel codes. Historically, theory on asynchronous iterative methods has focused on asymptotic behavior, while the transient behavior remains poorly understood. In this paper, we study a model of the asynchronous Jacobi method without communication delays, which we call simplified asynchronous Jacobi. Simplified asynchronous Jacobi can be used to model asynchronous Jacobi implemented in shared memory or distributed memory with fast communication networks. Our analysis uses the idea of a propagation matrix, which is similar in concept to an iteration matrix. We show that simplified asynchronous Jacobi can continue to reduce the residual when some processes are slower than other processes. We also show that simplified asynchronous Jacobi can converge when synchronous Jacobi does not. We verify our analysis of simplified asynchronous Jacobi using results from asynchronous Jacobi implemented in shared and distributed memory.

An iterative Jacobi-like algorithm is described for transforming a skew-symmetric complex matrix A of even dimension into Murnaghan’s normal form. The decomposition allows to determine the singular values of A and to solve the system of linear equations A x = b in a least square sense without accidentally destroying the skew-symmetry. Complex skew-symmetric matrices arise in the context of the time-dependent variational principle (TDVP), that maps a quantum mechanical system to a Hamiltonian system in a high-dimensional curved phase space. When the skew-symmetric phase space metric becomes singular because of parameter redundancies, the equations of motion have to be solved in a least square sense. The presented algorithm ensures that the symplectic structure is retained also in the least square solution. As a test case it is applied to studying the deflection of a photoelectron by a hydrogen atom using the TDVP. A Fortran implementation of the skew-Jacobi algorithm is provided as supplementary material. Program summary Program title: skew-Jacobi Program Files doi: http://dx.doi.org/10.17632/sgsyn672nf.1 Licensing provisions: MIT license Programming language: python and Fortran 90 Nature of problem: Computing the singular values and associated vectors for a complex skew-symmetric matrix of even dimension Solution method: A sequence of block Jacobi rotations is constructed that iteratively reduces the matrix to normal form. The skew-symmetry is exploited so that singular values are produced in pairs.

A fast parallelable Jacobi iteration type optimization method for non-smooth convex composite optimization is presented. Traditional gradient-based techniques cannot solve the problem. Smooth approximate functions are attempted to be used as a replacement of those non-smooth terms without compromising the accuracy. Recently, proximal mapping concept has been introduced into this field. Techniques which utilize proximal average based proximal gradient have been used to solve the problem. The state-of-art methods only utilize first-order information of the smooth approximate function. We integrate both first and second-order techniques to use both first and second-order information to boost the convergence speed. A convergence rate with a lower bound of O([Formula presented]) is achieved by the proposed method and a super-linear convergence is enjoyed when there is proper second-order information. In experiments, the proposed method converges significantly better than the state of art methods which enjoy O([Formula presented]) convergence. National Research Foundation (NRF) This work was conducted within the Rolls-Royce@NTU Corporate Lab with support from the National Research Foundation (NRF) Singapore un-der the Corp Lab@University Scheme and Energy Research Institute@NTU under Interdisciplinary Graduate School in Nanyang Technological University.

The current manuscript focuses on solving systems of multi-pantograph equations. The spectral tau method is applied for solving systems of multi-pantograph equations with shifted Jacobi polynomials as basis functions. The convergence analysis of the proposed technique is also investigated. We introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods.

Minimum mean square error (MMSE) linear detector can achieve the near-optimal bit error rate (BER) performance for uplink multi-user massive multiple input multiple output (MIMO) systems, but it involves matrix inversion with high complexity, especially when the number of users is high. In order to reduce the complexity, Jacobi iterative method has been applied for low-complexity MMSE detector without employing the computationally intensive matrix inversion. However, its convergence rate is low. In this paper, we propose a novel low-complexity MMSE detector based on the refinement of Jacobi iterative method. This refinement is given by the use of band matrix in order to accelerate the convergence rate, as well as guaranteeing the near-optimal BER performance. Analytical and simulation results show the efficiency of the proposed detector in comparison to the recently Jacobi-based detector.

A new real structure-preserving Jacobi algorithm is proposed for solving the eigenvalue problem of quaternion Hermitian matrix. By employing the generalized JRS-symplectic Jacobi rotations, the new quaternion Jacobi algorithm can preserve the symmetry and JRS-symmetry of the real counterpart of quaternion Hermitian matrix. Moreover, the proposed algorithm only includes real operations without dimension-expanding and is generally superior to the state-of-the-art algorithm. Numerical experiments are reported to indicate its efficiency and accuracy.

We study the traces of Hecke operators acting on the space of Jacobi forms and give the trace formula in the case of general odd squarefree level N by using results of Skoruppa and Zagier. As an application, we obtain some trace relations of them, and construct an isomorphism map from the space of all Jacobi cusp new forms of level N and index 1 having the eigenvalue +1 with respect to any Atkin–Lehner operator on the level to the space of all Jacobi cusp new forms of level 1 and index N whose eigenvalue with respect to any Atkin–Lehner operator on the index is equal to +1.

An approximate method using Jacobi polynomials is proposed for the approximate solutions of fractional vibration equation.Convergence and numerical stability of the proposed method are shown.Results are discussed for different cases of Jacobi polynomials.Results are compared with existing analytical methods. In this paper, we present a method based on the Jacobi polynomials for the approximate solution to fractional vibration equation (FVE) of large membranes. Proposed method converts the FVE into Sylvester form of algebraic equations, whose solution gives the approximate solution. Convergence analysis of the proposed method is given. It is also shown that our approximate method is numerically stable. Numerical results are discussed for different values of wave velocities and fractional order involved in the FVE. These numerical results are shown through figures for particular cases of Jacobi polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third kind, (4) Chebyshev polynomial of fourth kind, (5) Gegenbauer polynomial. The accuracy of the proposed method is proved by comparing results of our method and other exiting analytical methods. Comparison of results are presented in the form of tables for particular cases of FVE and Jacobi polynomials.

Based on the generalized Jacobi polynomials with indices ( − 1 , − 1 ) , a new upper bound for a posteriori error estimates is proposed, investigated and implemented for generalized Jacobi–Galerkin spectral element approximations. To simplify discussion for the error estimates, the second-order partial differential equation with homogeneous Dirichlet boundary conditions is considered on a unit interval.

Currently, community detection in complex networks has become a hot-button topic. In this paper, based on the Spectral Clustering (SC) algorithm, we introduce the idea of Jacobi iteration, and then propose a novel algorithm J-SC for community detection in complex networks. Furthermore, the accuracy and efficiency of this algorithm are tested by some representative real-world networks and several computer-generated networks. The experimental results indicate that the J-SC algorithm can accurately and effectively detect the community structure in these networks. Meanwhile, compared with the state-of-the-art community detecting algorithms SC, SOM, K-means, Walktrap and Fastgreedy, the J-SC algorithm has better performance, reflecting that this new algorithm can acquire higher values of modularity and NMI. Moreover, this new algorithm has faster running time than SOM and Walktrap algorithms.

A novel method is proposed to seek more new exact solutions for coupled Schrodinger–Boussinesq equations. In this present paper, an improved algebraic method based on the generalized Jacobi elliptic function method with symbolic computation is used to construct more new exact solutions for coupled Schrodinger–Boussinesq equations. As a result, several families of new generalized Jacobi double periodic elliptic function wave solutions are obtained by using this method, some of them are degenerated to solitary wave solutions in the limiting cases. The present generalized method is efficient, powerful, straightforward and concise, and it can be used in order to establish more entirely new exact solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

Evidence theory has strong ability to handle epistemic uncertainties whose precise probability distributions cannot be obtained due to limited information. However, the excessive computational cost produced by repetitively extreme value analysis severely influences the practical application of evidence theory. This paper aims to develop an efficient algorithm for epistemic uncertainty analysis of acoustic problem under evidence theory. Based on the orthogonal polynomial approximation theory, a numerical approach named as the evidence-theory-based Jacobi expansion method (ETJEM) is proposed. In ETJEM, the response of acoustic system with evidence variables is approximated by Jacobi expansion, through which the repetitively extreme value analysis needed in evidence theory can be efficiently performed. The parametric Jacobi polynomial of Jacobi expansion holds a large number of polynomials as special cases, such as the Legendre polynomial and Chebyshev polynomial. Thus, the ETJEM permits a much wider choice of polynomial bases to control the error of approximation than the traditional evidence-theory-based orthogonal polynomial approximation method, in which only the Legendre polynomial is used for approximation. Three numerical examples are employed to demonstrate the effectiveness of the proposed methodology, including a mathematic problem with explicit expression and two engineering applications in acoustic field. In these three numerical examples, efficiency and accuracy are fully studied by comparing with Legendre expansion method as well as Monte Carlo simulations.

We present and analyze a modified Newton solver, the so called operator-related damped Jacobian method , with a line search globalization for the solution of the strongly nonlinear momentum equation in a viscous-plastic (VP) sea ice model.Due to large variations in the viscosities, the resulting nonlinear problem is very difficult to solve. The development of fast, robust and converging solvers is subject to present research. There are mainly three approaches for solving the nonlinear momentum equation of the VP model, a fixed-point method denoted as Picard solver, an inexact Newton method and a subcycling procedure based on an elastic-viscous-plastic model approximation. All methods tend to have problems on fine meshes by sharp structures in the solution. Convergence rates deteriorate such that either too many iterations are required to reach sufficient accuracy or convergence is not obtained at all.To improve robustness globalization and acceleration approaches, which increase the area of fast convergence, are needed. We develop an implicit scheme with improved convergence properties by combining an inexact Newton method with a Picard solver. We derive the full Jacobian of the viscous-plastic sea ice momentum equation and show that the Jacobian is a positive definite matrix, guaranteeing global convergence of a properly damped Newton iteration. We compare our modified Newton solver with line search damping to an inexact Newton method with established globalization and acceleration techniques. We present a test case that shows improved robustness of our new approach, in particular on fine meshes.

In recent years, numerical methods have been introduced to solve two-dimensional Volterra and Fredholm integral equations. In this study, a numerical scheme is constructed to solve classes of linear and nonlinear three-dimensional integral equations (Volterra, Fredholm, and mixed VolterraFredholm). This operational approach is proposed to easily and directly solve these equations at low computational costs. The scheme is based on the Jacobi polynomials on the interval [0, 1] where three-variable Jacobi polynomials are introduced and their operational matrices of integration and product are derived. Compared to other existing methods for multidimensional problems, the Jacobi operational method eliminates the time-consuming computations and solely employs the one-dimensional operational matrix to construct corresponding multidimensional operational matrices. The absolute error of the proposed method is almost constant on the studied interval even at higher dimensions, confirming the stability of the proposed operational Jacobi method. Required theorems on the convergence of the method are proved in Jacobi-weighted Sobolev space. It is established that the error function vanishes as N increases. The method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the BlockPulse functions, differential transform, and degenerate kernel methods.

This paper proposes an adaptive, wavelet-enriched hierarchical finite element model (FEM) to enhance computational efficiency and solution accuracy for heterogeneous domains of anisotropic elastic materials. It is motivated by the need to overcome shortcomings of conventional methods like the standard FEM or fast Fourier transformation (FFT) based method for analyzing deformation in large polycrystalline microstructures. The multi-resolution wavelet functions are advantageous for selection of an optimal set of functions that can adaptively enrich the solution space with a prescribed level of accuracy. The proposed method introduces a discretization space that conforms to the profile of the solution sought. The method engages a second generation family of wavelets with a lifting scheme to generate hierarchical interpolation functions. An iterative algorithm efficiently calculates estimates of the solution from the previous iterate using a modified Jacobi method. The adaptive FE method performs significantly better than uniformly refined FEM in validation tests with focus on the convergence rate and error mitigation. A polycrystalline microstructure with elastically anisotropic grains is simulated by the adaptive wavelet-enriched hierarchical method, showing high convergence rates.

The paper analyzes special cyclic Jacobi methods for symmetric matrices of order $4$. Only those cyclic pivot strategies that enable full parallelization of the method are considered. These strategies, unlike the serial pivot strategies, can force the method to be very slow or very fast within one cycle, depending on the underlying matrix. Hence, for the global convergence proof one has to consider two or three adjacent cycles. It is proved that for any symmetric matrix $A$ of order~$4$ the inequality $S(A^{[2]})\leq(1-10^{-5})S(A)$ holds, where $A^{[2]}$ results from $A$ by applying two cycles of a particular parallel method. Here $S(A)$ stands for the Frobenius norm of the strictly upper-triangular part of $A$. The result holds for two special parallel strategies and implies the global convergence of the method under all possible fully parallel strategies. It is also proved that for every $\epsilon>0$ and $n\geq4$ there exist a symmetric matrix $A(\epsilon)$ of order $n$ and a cyclic strategy, such that upon completion of the first cycle of the appropriate Jacobi method the inequality $S(A^{[1]})> (1-\epsilon)S(A(\epsilon))$ holds., Comment: 27 pages

The MPACT code, being developed collaboratively by the University of Michigan and Oak Ridge National Laboratory, is the primary deterministic neutron transport solver being deployed within the Virtual Environment for Reactor Applications (VERA) as part of the Consortium for Advanced Simulation of Light Water Reactors (CASL). In many applications of the MPACT code, transport-corrected scattering has proven to be an obstacle in terms of stability, and considerable effort has been made to try to resolve the convergence issues that arise from it. Most of the convergence problems seem related to the transport-corrected cross sections, particularly when used in the 2D method of characteristics (MOC) solver, which is the focus of this work. In this paper, the stability and performance of the 2-D MOC solver in MPACT is evaluated for two iteration schemes: Gauss-Seidel and Jacobi. With the Gauss-Seidel approach, as the MOC solver loops over groups, it uses the flux solution from the previous group to construct the inscatter source for the next group. Alternatively, the Jacobi approach uses only the fluxes from the previous outer iteration to determine the inscatter source for each group. Consequently for the Jacobi iteration, the loop over groups can be moved from the outermost loop—as is the case with the Gauss-Seidel sweeper—to the innermost loop, allowing for a substantial increase in efficiency by minimizing the overhead of retrieving segment, region, and surface index information from the ray tracing data. Several test problems are assessed: (1) Babcock & Wilcox 1810 Core I, (2) Dimple S01A-Sq, (3) VERA Progression Problem 5a, and (4) VERA Problem 2a. The Jacobi iteration exhibits better stability than Gauss-Seidel, allowing for converged solutions to be obtained over a much wider range of iteration control parameters. Additionally, the MOC solve time with the Jacobi approach is roughly 2.0–2.5× faster per sweep. While the performance and stability of the Jacobi iteration are substantially improved compared to the Gauss-Seidel iteration, it does yield a roughly 8–10% increase in the overall memory requirement.

Time-spectral methods show a huge potential for decreasing computation time of time-periodic flows. While time-spectral methods are often used for compressible flows, applications to incompressible flows are rare. This paper presents an extension of the time-spectral method (TSM) to incompressible, viscous fluid flows using a pressure-correction algorithm in a finite volume flow solver. Several algorithmic treatments of the time-spectral operator in a pressure-correction algorithm have been investigated. Initially the single time instances were solved using the Jacobi method as preconditioner. While the existing fluid code is easily adapted, the solver shows a fast degradation in stability. Thus the solution matrix was reordered with respect to time and a block Gauss–Seidel preconditioner was applied. The single time blocks were directly solved using the Cholesky algorithm. The solver is more robust, but the current implementation is inefficient. To alleviate this problem an approach, coupling all time instances and control volumes, was developed. For the complete time and spatial system two different treatments in the preconditioner were researched. To outline the advantages and disadvantages of the proposed solution strategies the laminar flow around the pitching NACA0012 airfoil was investigated. Moreover, unsteady simulations using first and second order time-stepping techniques were used and the time-spectral results were compared to regular time-stepping approaches. It is shown that the time-spectral implementations solving the whole temporal-spatial system are faster than the regular time-stepping schemes. The efficiency of the time-spectral solver decreases with increasing number of harmonics. Furthermore, with a small number of harmonics the lift coefficient over time is not accurately predicted.

In the present paper, a matrix-free, implicit finite volume lattice Boltzmann method for steady flow on unstructured mesh is proposed. The approximate linear system arising from the implicit finite volume discretization of lattice Boltzmann equation (LBE) is solved by a novel algorithm, which combines the lower-upper symmetric Gauss–Seidel (LU-SGS) and the Jacobi iteration schemes. A remarkable feature of the present implicit method is that the storage of the Jacobian matrixes of the convection and collision terms can be completely eliminated by approximating the Jacobian matrix-solution incremental vector product with appropriate numerical flux incremental vector and the numerical increment of collision term, resulting a matrix-free implicit scheme. The present method is validated by several two-dimensional testing cases.

We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and show that they satisfy strong non-subordinacy condition., 34 pages

The matrix completion problem can be found in many applications such as classification, image inpainting and collaborative filtering. In recent years, the emerging field of graph signal processing (GSP) has shed new light on this problem, deriving the graph signal matrix completion problem which incorporates the correlation of data elements. The nuclear-norm based methods possess satisfactory recovery performance, while they suffer from high computational cost and usually have slow convergence rate. In this paper, we propose two new iterative algorithms for solving the nuclear-norm regularization based graph signal matrix completion (NRGSMC) problem. By adopting approximate diagonalization approaches to estimate singular value decomposition (SVD), we obtain two generalized Newton algorithms, the generalized Newton with truncated Jacobi method (GNTJM) and the generalized Newton with parallel truncated Jacobi method (GNPTJM). The proposed methods are with low complexity and fast convergence by using the second-order information associated with the problem. Numerical results on three real-world data sets demonstrate that our schemes have evidently faster convergence rate than the gradient method with exact SVD, while maintain the similar completion performance.

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations ($Au=b$) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of $P$ relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of $M$ consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method. Our method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix $A$ is replaced by the (much easier) calculation of the maximum and minimum frequencies derived from a von Neumann analysis. This set of weights is also optimal for the general problem, resulting in the fastest convergence of all possible SRJ schemes for a given grid structure. We also show that with the set of weights computed for the optimal SRJ scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter $\omega$ in the Successive Overtaxation (SOR) method in some cases. Finally, we demonstrate with practical examples that our method also works very well for Poisson-like problems in which a high-order discretization of the Laplacian operator is employed. This is of interest since the former discretizations do not yield consistently ordered $A$ matrices. Furthermore, the optimal SRJ schemes here deduced, are advantageous over existing SOR implementations for high-order discretizations of the Laplacian operator in as much as they do not need to resort to multi-coloring schemes for their parallel implementation. (abridged), Comment: 28 pages, 5 figures, submitted to JCP

In this paper, the Jacobi and Gauss-Seidel-type iteration methods are proposed for solving the matrix equation A X B = C , which are based on the splitting schemes of the matrices A and B. The convergence and computational cost of these iteration methods are discussed. Furthermore, we give the preconditioned Jacobi and Gauss-Seidel-type iteration methods. Numerical examples are given to demonstrate the efficiency of these methods proposed in this paper.

We present a simple and efficient finite difference method for the phase-field crystal (PFC) equation on curved surfaces embedded in R 3 . We employ a narrow band neighborhood of a curved surface that is defined as a zero level set of a signed distance function. The PFC equation on the surface is extended to the three-dimensional narrow band domain. By using the closest point method and applying a pseudo-Neumann boundary condition, we can use the standard seven-point discrete Laplacian operator instead of the discrete Laplace–Beltrami operator on the surface. The PFC equation on the narrow band domain is discretized using an unconditionally stable scheme and the resulting implicit discrete system of equations is solved by using the Jacobi iterative method. Computational results are presented to demonstrate the efficiency and usefulness of the proposed method.

Lacking labeled examples of working numerical strategies, adapting an iterative solver to accommodate a numerical issue, e.g., density discontinuities in the pressure Poisson equation, is non-trivial and usually involves a lot of trial and error. Here, we resort to evolutionary neural network. A evolutionary neural network observes the outcome of an action and adapts its strategy accordingly. The process requires no labeled data but only a measure of a network’s performance at a task. Applying neuro-evolution and adapting the Jacobi iterative method for the pressure Poisson equation with density discontinuities, we show that the adapted Jacobi method is able to accommodate density discontinuities.

Natural frequencies of laminates during its progressive failure are investigated using finite element method. The finite element formulation is based on seven degrees of freedom displacement field, deduced for an orthotropic material. Free vibration analysis is carried out during ply-by-ply failure for a laminate subjected to uniformly distributed load. The variation of fundamental frequency with material and geometric properties is also presented. This study illustrates the effect of ply failure on the natural frequencies of a laminate. The prediction of failed ply and failure mode is accomplished via. Hashin failure criterion. Ply-discount material degradation model is adopted to modify the elastic constants of the failed ply. The effect of fibre orientation, failure mode and failed-ply on natural frequencies is highlighted in this paper. Jacobi iterative method is used to evaluate the natural frequencies of laminate. For the analysis, a computer code is written using FORTRAN based on the finite element formulation.

We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Anderson-Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speed-ups that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. Overall, the proposed method represents a simple yet efficient technique that is particularly attractive for large-scale parallel solutions of linear systems of equations.

We show that a weighted least squares approximation of Bezier coefficients with factored Hahn weights provides the best constrained polynomial degree reduction with respect to the Jacobi L 2 -norm. This result affords generalizations to many previous findings in the field of polynomial degree reduction. A solution method to the constrained multi-degree reduction with respect to the Jacobi L 2 -norm is presented. We study the problem of constrained degree reduction with respect to Jacobi norms.Our results generalize several previous findings on polynomial degree reduction.We explore the space of Jacobi parameters on the reduced polynomial approximation.

We present an improved version of the famous orthogonal joint diagonalization (JD) of a set of matrices, on the basis of successive Jacobi-like transformations. In particular, the Jacobi transformation matrix in each rotation step is only dependent on a single parameter that has analytical solution in real-value case. If this algorithm is improved, it can indirectly deal with the complex target matrices that can be converted into real symmetric ones. Moreover, this algorithm can be modified to handle the complex target matrices by a bi-Givens rotation procedure. The overall algorithm performance is evaluated through numerical simulations, and compared favorably with some existing state-of-the-art methods in terms of speed of convergence, complexity and accuracy. The Jacobi rotation matrix depends on a single parameter.The complex target matrices are transformed into real symmetric ones.Joint diagonalization can be obtained by one-dimensional optimization.It can handle the complex matrices by modifying algorithm implementation.The proposed algorithms have the good performance like the SOBI algorithm.

Communication costs are an important factor in the performance of massively parallel algorithms. We present a new asynchronous parallel algorithm for solving sparse linear systems that reduces communication compared to other algorithms on distributed memory machines. Implemented using passive one-sided remote memory access (RMA) MPI functions, the new method is a variation of the Southwell method, where rows are relaxed greedily, instead of sequentially, by choosing the row with the maximum residual norm. A process relaxes its rows if it holds the maximum residual norm among its neighbors at any given moment. Experimental results show that this method reduces communication costs compared to several other asynchronous iterative methods and the classic synchronous Jacobi method.

In this study, we propose shifted fractional-order Jacobi orthogonal functions (SFJFs) based on the definition of the classical Jacobi polynomials. We derive a new formula that explicitly expresses any Caputo fractional-order derivatives of SFJFs in terms of the SFJFs themselves. We also propose a shifted fractional-order Jacobi tau technique based on the derived fractional-order derivative formula of SFJFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). A shifted fractional-order Jacobi pseudo-spectral approximation is investigated for solving the nonlinear initial value problem of fractional order ν. An extension of the fractional-order Jacobi pseudo-spectral method is given to solve systems of FDEs. We describe the advantages of using the spectral schemes based on SFJFs and we compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and efficiency of the proposed techniques.

The paper presents a mathematical model of the thermal load on the aquatic environment under various operational capacities of thermal power plant. It is solved by the Navier–Stokes and temperature equations for an incompressible fluid in a stratified medium based on the splitting method by physical parameters which approximated by the finite volume method. The numerical solution of the equation system is divided into four stages. At the first step it is assumed that the momentum transfer carried out only by convection and diffusion. Intermediate velocity field is solved by five-step Runge–Kutta method. At the second stage, the pressure field is solved by the found intermediate velocity field. Poisson equation for the pressure field is solved by Jacobi method. The third step assumes that the transfer is carried out only by pressure gradient. The fourth step of the temperature equation is also solved as motion equations, with five-step Runge–Kutta method. The algorithm is parallelized on high-performance computer. The obtained numerical results of three-dimensional stratified turbulent flow were compared with experimental data. What revealed qualitatively and quantitatively approximately the basic laws of hydrothermal processes occurring in the reservoir-cooler.

A class of iterative schemes based on the regular splitting is proposed for PageRank computation. By some specific splitting, these methods will reduce to some well-known iterative schemes, like Jacobi method and successive overrelaxation method. To suit computational requirements of the modern high-speed multiprocessor environments, we further study synchronous parallel counterparts for the new splitting iteration methods. By using the M-splitting of the coefficient matrix, we establish the convergence theory of these synchronous multisplitting iteration methods. Finally, some numerical experiments are presented to illustrate the feasibility, stability and effectiveness of the proposed algorithms.

We present an emulation-based accelerator for EM simulations in metamaterials.We propose two hardware designs to solve systems of linear equations from the FEM.Both presented designs are run on a physical Veloce 1 emulator from Mentor Graphics.Timing results exceed the performance of several software-based solvers. Display Omitted Finding new techniques to accelerate electromagnetic (EM) simulations has become a necessity nowadays due to its frequent usage in industry. As they are mainly based on domain discretization, EM simulations require solving enormous systems of linear equations simultaneously. Available software-based solutions do not scale well with the increasing number of equations to be solved. As a result, hardware accelerators have been utilized to speed up the process. We introduce using hardware emulation as an efficient solution for EM simulation core solvers. Two different scalable architectures are implemented to accelerate the solver part of an EM simulator based on the Gaussian Elimination and the Jacobi iterative methods. Results show that the performance gap between presented solutions and software-based ones increases as the number of equations increases. For example, solving 2,002,000 equations using our Clustered Jacobi design in single floating-point precision achieved a speed-up of 100.88x and 35.24x over pure software implementations represented by MATLAB and the ALGLIB C++ package, respectively.

In this paper, we propose a partially smoothing function for solving the nonlinear complementarity problems (NCP). Some properties of such a smoothing approach are analyzed and are employed to develop a well defined and efficient Jacobian Newton algorithm to find the solution of NCP. Under the condition that the level set of a merit function is bounded, global convergence and super-linear convergence are established for the developed algorithm. Compared with the similar theoretical results available in the literature, the assumption of nonsingularity is removed in virtue of P 0 property and the smoothing approach. Numerical experiments show that the proposed smoothing method outperforms the existent ones, particularly in comparison with the state-of-art methods derived from the classical Fischer-Burmeister smoothing function and the aggregation function.

The Jacobi spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ω α , β 2 -norm and the L ∞ -norm) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate theoretical results.

In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. An efficient numerical scheme for time fractional diffusion-wave equations is proposed.A time-space Jacobi tau approximation is developed for such equations.Several tau-spectral methods can be achieved as special cases.The validity and applicability of the proposed method are demonstrated.Accurate numerical results are obtained by selecting limited collocation nodes.

This study develops the energy balance based on Jacobi collocation method for accurate prediction of conservative nonlinear oscillator models with a single collocation point. The node points are taken as the roots of Jacobi orthogonal polynomials. Several examples are included to demonstrate the applicability and accuracy of the proposed algorithm, and some comparisons are made with the existing results. The method is suitable and the approximate frequencies are valid for small as well as large amplitudes of oscillation. Excellent agreement with exact ones is presented for the first order approximation.

The relation between two Morse functions defined on a smooth, compact, and orientable 2-manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces, have shown to be useful in various applications. In practice, unfortunately, functions often contain noise and discretization artifacts, causing their Jacobi set to become unmanageably large and complex. Although there exist techniques to simplify Jacobi sets, they are unsuitable for most applications as they lack fine-grained control over the process, and heavily restrict the type of simplifications possible. This paper introduces the theoretical foundations of a new simplification framework for Jacobi sets. We present a new interpretation of Jacobi set simplification based on the perspective of domain segmentation. Generalizing the cancellation of critical points from scalar functions to Jacobi sets, we focus on simplifications that can be realized by smooth approximations of the corresponding functions, and show how these cancellations imply simultaneous simplification of contiguous subsets of the Jacobi set. Using these extended cancellations as atomic operations, we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications to some user-defined metric. We show that for simply connected domains, our algorithm reduces a given Jacobi set to its minimal configuration, that is, one with no birth–death points (a birth–death point is a specific type of singularity within the Jacobi set where the level sets of the two functions and the Jacobi set have a common normal direction).

In this paper, a class of linear and nonlinear functional integro-differential equations are considered that can be found in the various fields of sciences such as: stress-strain states of materials, motion of rigid bodies and models of polymer crystallization. The operational collocation method with shifted Jacobi polynomial bases is applied to approximate the solution of these equations. In addition, some theoretical results are given to simplify and reduce the computational costs. Finally, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

In this paper, we propose two new iterative methods for solving the nonsingular saddle point problem based on partitioning the coefficient matrix. One is combining the block Gauss-Seidel iterative method with the Uzawa iterative method, and the other one is combining the block Jacobi iterative method with the Uzawa iterative method. Then we study the convergence of the two novel methods under suitable restrictions on the iteration parameters, respectively. Numerical experiments are also presented to illustrate the behavior of the considered algorithms.

In this paper, approximation techniques based on the shifted Jacobi together with spectral tau technique are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Jacobi polynomials. Using the operational matrix of the fractional derivative, the problem can be reduced to a set of linear algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results to show that the proposed method is easy to implement and produce accurate results.

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from its eigenvalues, its leading principal submatrix and part of the eigenvalues of its submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time-space fractional differential equation with Dirichlet boundary conditions. The shifted Jacobi operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives are presented. By using these operational matrices, we propose a shifted Jacobi tau method for both temporal and spatial discretizations, which allows us to present an efficient spectral method for solving such problem. Furthermore, the error is estimated and the proposed method has reasonable convergence rates in spatial and temporal discretizations. In addition, some known spectral tau approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters ? and ?. Finally, in order to demonstrate its accuracy, we compare our method with those reported in the literature.