Numerical Algorithms最新文献

In this paper, the algorithms and applications of the dual quaternion QR decomposition are studied. The direct algorithm and dual structure-preserving algorithm of dual quaternion QR decomposition utilizing Householder transformation of dual quaternion vector are proposed. Numerical experiments show that two algorithms are feasible, and the dual structure-preserving algorithm is superior to the direct algorithm in terms of computational efficiency. Therefore, the dual structure-preserving algorithm of dual quaternion QR decomposition is used to color image watermarking. Experiments illustrate that our method is feasible and better than the compared methods in anti-aggression.

This paper concentrates on the quaternion indefinite least squares (QILS) problem. Firstly, we define the quaternion J-unitary matrix and the quaternion hyperbolic Givens rotation, and study their properties. Then, based on these, we investigate the quaternion hyperbolic QR factorization, and purpose its real structure-preserving (SP) algorithm by the real representation (Q-RR) matrix of the quaternion matrix. Immediately after, we explore the solution of the QILS problem, and give a real SP algorithm of solving the QILS problem. Eventually, to illustrate the effectiveness of proposed algorithms, we offer numerical examples.

The well-known INdividual Differences SCALing (INDSCAL) model is intended for the simultaneous metric multidimensional scaling (MDS) of several doubly centered matrices of squared dissimilarities. An alternative approach, called for short DINDSCAL (direct INDSCAL), is proposed for analyzing directly the input matrices of squared dissimilarities. In the present work, the problem of fitting the DINDSCAL model to the data is formulated as a Riemannian optimization problem on a product matrix manifold comprised of the Stiefel sub-manifold of zero-sum matrices and non-negative diagonal matrices. A practical algorithm, based on the generic Riemannian trust-region method by Absil et al., is presented to address the underlying problem, which is characterized by global convergence and local superlinear convergence rate. Numerical experiments are conducted to illustrate the efficiency of the proposed method. Furthermore, comparisons with the existing projected gradient approach and some classical methods in the MATLAB toolbox Manopt are also provided to demonstrate the merits of the proposed approach.

The noncentral beta distribution function is a generalization of the central beta distribution (the regularized incomplete beta function) that includes a noncentrality parameter. This paper describes an algorithm and provides a Matlab implementation for calculating the noncentral beta distribution function. Through a series of numerical tests, we demonstrate that the algorithm is accurate and efficient across a wide range of parameters.

This paper explores numerical methods for recovering a density term in a fractional Sturm-Liouville problem using a set of spectra. By applying Lidskii’s theorem, a sequence of trace formulae are derived to elucidate the connections between the unknown coefficients and the complex eigenvalues of a fractional spectra problem. Two efficient algorithms are proposed based on these trace formulae, and their effectiveness is demonstrated through numerical experiments.

The main purpose of this paper is to develop and analyze a partially truncated Euler-Maruyama method for numerically solving SDEs with super-linear piecewise continuous drift coefficients and (varvec{(1/2+alpha )})-Hölder diffusion coefficients (PTEMH), for (varvec{alpha in [0,1/2]}). We first present an analytical form for the unique solution of such problems. Then we establish the strong convergence theory of the PTEMH scheme. We show that the convergence rate of the proposed method in the case (varvec{alpha in (0,1/2]}) reaches (varvec{alpha }), which is optimal compared to the explicit Euler-Maruyama method. Finally, numerical results are given to confirm the theoretical convergence rate.

This paper studies a type of multiterm fractional stochastic delay integro-differential equations (FSDIDEs). First, the Euler-Maruyama (EM) method is developed for solving the equations, and the strong convergence order of this method is obtained, which is (varvec{min left{ alpha _{l}-frac{1}{2}, alpha _{l}-alpha _{l-1}right} }). Then, a fast EM method is also presented based on the exponential-sum-approximation with trapezoid rule to cut down the computational cost of the EM method. In the end, some concrete numerical experiments are used to substantiate these theoretical results and show the effectiveness of the fast method.

This work is focused on developing a hybrid numerical method that combines a higher-order finite difference method and multi-dimensional Hermite wavelets to address two-dimensional multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels having bounded and unbounded time derivatives at the initial time (t=0). Specifically, the multi-term fractional operators are discretized using a higher-order approximation designed by employing different interpolation schemes based on linear, quadratic, and cubic interpolation leading to (mathcal {O}(N^{-(4-alpha _1)})) accuracy on a suitably chosen nonuniform mesh and (mathcal {O}(N^{-alpha _1})) accuracy on a uniformly distributed mesh. The weakly singular integral operators are approximated by a modified numerical quadrature, which is a combination of the composite trapezoidal approximation and the midpoint rule. The effects of the exponents of the weakly singular kernels over fractional orders are analyzed in terms of accuracy over uniform and nonuniform meshes for the solution having both bounded and unbounded time derivatives. The stability of the proposed semi-discrete scheme is derived based on (L^infty )-norm for uniformly distributed temporal mesh. Further, we employ the uniformly distributed collocation points in spatial directions to estimate the tensor-based wavelet coefficients. Moreover, the convergence analysis of the fully discrete scheme is carried out based on (L^2)-norm leading to (mathcal {O}(N^{-alpha _1})) accuracy on a uniform mesh. It also highlights the higher-order accuracy over nonuniform mesh. Additionally, we discuss the convergence analysis of the proposed scheme in the context of the multi-term time-fractional diffusion equations involving time singularity demonstrating a (mathcal {O}(N^{-(4-alpha _1)})) accuracy on a nonuniform mesh with suitably chosen grading parameter. Note that the scheme reduces to (mathcal {O}(N^{-alpha _1})) accuracy on a uniform mesh. Several tests are performed on numerous examples in (L^infty )- and (L^2)-norm to show the efficiency of the proposed method. Further, the solutions’ nature and accuracy in terms of absolute point-wise error are illustrated through several isosurface plots for different regularities of the exact solution. These experiments confirm the theoretical accuracy and guarantee the convergence of approximations to the functions having time singularity, and the higher-order accuracy for a suitably chosen nonuniform mesh.

We present a Virtual Element MATLAB solver for elliptic and parabolic, linear and semilinear Partial Differential Equations (PDEs) in two and three space dimensions, which is coined VEMcomp. Such PDEs are widely applicable to describing problems in material sciences, engineering, cellular and developmental biology, among many other applications. The library covers linear and nonlinear models posed on different simple and complex geometries, involving time-dependent bulk, surface, and bulk-surface PDEs. The solver employs the Virtual Element Method (VEM) of lowest polynomial order ({k=1}) on general polygonal and polyhedral meshes, including the Finite Element Method (FEM) of order ({k=1}) as a special case when the considered mesh is simplicial. VEMcomp has three main purposes. First, VEMcomp generates polygonal and polyhedral meshes optimized for fast matrix assembly. Triangular and tetrahedral meshes are encompassed as special cases. For surface PDEs, VEMcomp is compatible with the well-known Matlab package DistMesh for mesh generation. Second, given a mesh for the considered geometry, possibly generated with an external package, VEMcomp computes all the matrices (mass and stiffness) required by the VEM or FEM method. Third, for multiple classes of stationary and time-dependent bulk, surface and bulk-surface PDEs, VEMcomp solves the considered PDE problem with the VEM or FEM in space and IMEX Euler in time, through a user-friendly interface. As an optional post-processing, VEMcomp comes with its own functions for plotting the numerical solutions and evaluating the error when possible. An extensive set of examples illustrates the usage of the library.

In this present work, in order to solve the numerical solution of quaternion matrix equation (EY=F), the quaternion modified gradient-based algorithm (QMGI) is proposed by applying the real presentation of quaternion matrix. The proposed method can be applied to solve the quaternion solution, pure imaginary solution, and real solution of the studied equation (EY=F). If the studied equation is consistent, it is proved that the proposed algorithm converges to the exact solution for given any initial quaternion matrix under appropriate conditions. If the studied equation is not consistent, it is found that the QMGI algorithm converges to the least squares solution. And some numerical examples are examined to confirm the feasibility and efficiency of the proposed algorithms, which all indicate that the proposed QMGI algorithm is much more effective than QGI algorithm and QRGI algorithm in computational time and accuracy. Moreover, QMGl algorithm is applied to color image encryption and evaluated the encryption effectiveness from four aspects. All metrics are close to the ideal values. lt is demonstrated that the effectiveness of the encryption scheme and the accuracy of the obtained theory results in this paper.