Lower-order Matrix Research Articles

Fast 3D simulation of magnetotelluric data in anisotropic media using a rational Krylov method

This study introduces a computational method aimed at accelerating multifrequency 3D magnetotelluric (MT) forward modeling in 3D conductivity structures with general anisotropy. The approach combines an edge-based finite-element (FE) method with a rational Krylov subspace method. In this method, the MT source term is expressed as a planar current source, and the frequency-dependent electric field response is the product of a transfer function and a constant vector of the current source. Consequently, the rapid approximate electric field for multifrequency calculations can be obtained by constructing the orthogonal basis of the rational Krylov subspace. By applying this technique, the large sparse matrix resulting from FE discretization can be projected onto a much lower-order matrix, significantly reducing its size to tens or hundreds. Incorporating the direct solver PARDISO to construct the orthogonal basis enables a considerably faster solution compared with the previous FE method. Moreover, the algorithm is implemented using Julia, a high-level programming language known for its readability, maintainability, and extensibility. The validity of the algorithm is demonstrated through a 1D anisotropic model, and two additional 3D models are designed to further evaluate the applicability and effectiveness of our algorithm. Numerical experiments illustrate a notable increase in modeling speed compared with the traditional edge-based FE method.

A matrix Yajima–Oikawa long-wave-short-wave resonance equation, Darboux transformations and rogue wave solutions

Based on an introduced (2m+n)×(2m+n) matrix spectral problem, a matrix Yajima–Oikawa long-wave-short-wave resonance equation is proposed, which can be reduced to an (n+1)-component Yajima–Oikawa long-wave-short-wave resonance equation. Multi-fold generalized Darboux transformations for these two equations are constructed by using the gauge transformation between Lax pairs and their Riccati equations. Every solution to the Riccati equations can be transformed into a new solution of the matrix Yajima–Oikawa long-wave-short-wave resonance equation through the Darboux transformations. As an application of the obtained Darboux transformation, first, we derive high-order rogue wave solutions of the Yajima–Oikawa long-wave-short-wave resonance equation. Second, we obtain explicit solutions for the lower-order matrix Yajima–Oikawa long-wave-short-wave resonance equation, including soliton solutions, rogue wave solutions, wave solutions that are constant along the x-direction, and wave solutions traveling at varying speeds.

Polynomial eigenvalue bounds from companion matrix polynomials

We show how -ifications, namely, lower order matrix polynomials with the same eigenvalues as a given complex square matrix polynomial, can be used in combination with other recent results, namely, a generalization to matrix polynomials of the Cauchy radius and its enhancement, to produce eigenvalue bounds. In addition, we present a simplified derivation of -ifications for square matrices. We found that reasonably good bounds can be obtained by working with any companion matrix polynomial, and that these bounds were less computationally demanding for higher degree companion matrix polynomials than for such polynomials of degree one (linearizations).

An exact solution method for perturbation analysis of standard eigenvalue problems

In the presence of perturbation, the accurate method of calculating the perturbed values of eigenvalues is an important issue to ensure the safety in engineering structures. Based on a power series expansion for the eigenvalues and eigenvectors, all the present perturbation methods neglect the higher-order terms in the calculating process, which may result in the deterioration of accuracy. However, the present methods for improving the accuracy usually result in amazing amounts of computation. The contradiction between accuracy and efficiency has existed for a long time, which made the research stagnant in recent years. In this paper, an exact solution method for perturbation analysis of standard eigenvalue problem is proposed to calculate the exact perturbed values of eigenvalues, overcoming the drawbacks of lack of accuracy due to neglecting the higher-order terms in matrix perturbation series expansion method. In the presented method, the perturbation solution equation of the standard eigenvalue problem is derived without any approximations in the derivation process. By substituting the normal eigenvalues obtained by solving the standard eigenvalue problem equation of the normal matrix into the derived perturbation solution equation, the perturbed values of eigenvalues can be calculated efficiently, which can satisfy the requirement of high accuracy and high efficiency for standard eigenvalue analysis with perturbed parameters. In three numerical examples with perturbed parameters of the undamped spring-mass system, the Bernoulli-Euler cantilever beam and the plane frame, the accuracy and efficiency of the proposed method are verified by comparing numerical results with those obtained by matrix perturbation series expansion method. The results show: (1) The relative errors of the perturbed values of eigenvalues yielded by the proposed method and the lower-order matrix perturbation series expansion method are much bigger and can be reduced reduce remarkably with the increase of the order. The results obtained by higher-order matrix perturbation series expansion method are quite close to those obtained by the proposed method. It is known that as the order increases, the results obtained by the lower-order matrix perturbation series expansion method become much closer to the exact values. It is indicated that the accuracy of the proposed method is much higher. (2) Although the results obtained by higher-order matrix perturbation series expansion method are quite close to those obtained by the proposed method, the execution time of the proposed method is much shorter than that of higher-order matrix perturbation series expansion method. It is indicated that the proposed method is much more efficient, especially for large and complex structures. To sum up, the accuracy and efficiency of the proposed method are both remarkably higher than those of the matrix perturbation series expansion method for calculating the exact perturbed values of real eigenvalues. The reason for the proposed method of high accuracy is that the perturbed values of eigenvalues are exact without approximation while the higher-order terms in the matrix perturbation series expansion method are neglected. Predictably, this method will become a powerful tool for standard eigenvalue analysis with perturbed parameters in the future. Moreover, it can be highly applicable to the fields of engineering practice and will make great contributions.

Research on Lower-Order Body Matrix Method and the Configuration Transformation of Planar Metamorphic Mechanism

The lower-order body matrix method is proposed as to topological configuration issue of planar metamorphic mechanism. Through combining lower-order operator and type code of kinematic pair, the lower-order body matrix is built to describe the information of adjacent body and kinematic pair. In the research on configuration transformation of planar metamorphic mechanism, step-description method is put forward which adapts to the analysis of various planar metamorphic ways. Step mathematical equation is established based on the generalized operational rule of lower-order body matrix. Application example shows that the lower-order matrix built with lower-order body matrix method is simpler and more informative, which has many advantages over the adjacency matrix and the correlation matrix that cannot be combined with the kinematics and dynamics equations, and solve the issue that the low-order sequences can only describe the open chain metamorphic mechanism. Step-description method can describe various planar metamorphic configuration transformation process comprehensively, which the EU matrix method cannot complete, providing a new method on the research of planar metamorphic mechanism.

Improved finite-element method for the analysis of multilayer coupled cylindrical striplines and microstrip lines

A novel method, the finite-element method (FEM) using Fourier series as the interpolating function, is presented in this paper for calculating the quasi-static characteristics of coupled cylindrical striplines and microstrip lines filled with multilayered dielectrics. This method is very similar to the normal FEM, but has a much lower-order matrix equation, smaller computational contents, and higher accuracy. It is simpler and more flexible for application than the conformal transformation method. Compared with the integral-equation and spectral-domain methods, this new method is more obvious with regard to its physics and, also, it is not involved in the difficult task of selecting the base functions and relative convergence. The numerical results shown here for several examples are in good agreement with those in the references. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 47: 265–270, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21143