Home – Home – Jornals – Numerical Analysis and Applications 2026 number 3

Suppose one needs to check whether given square matrices A and B are congruent. If A and B are nonsingular, one can use such useful instruments as the cosquares of both matrices. How can one reduce the case of singular A and B to a nonsingular one? Sergeichuk and Horn in 2006 proposed a method that brings a given singular matrix A by a finite sequence of congruences to a direct sum of a nonsingular submatrix and one or several nilpotent Jordan blocks. The blocks constitute the singular part of the canonical form of A with respect to congruences. Removing them from A yields the desired method that replaces the input singular matrix by a nonsingular submatrix of smaller order. The method of Horn-Sergeichuk was not implemented by its authors as a working procedure. In this paper, we describe our implementation of this method in the form of an algorithm for rational matrices or complex matrices whose entries are rational Gaussian numbers. The algorithm is based on the error-free arithmetic of rational numbers, which is provided to the users of the computational environment Matlab. The performance of the algorithm is illustrated by two numerical examples.

In this paper, we study a numerical method to approximate a solution with weak singularity at the initial time of non-linear fractional integro-differential equations. For this we use fractional order Legendre polynomials as a basis and reduce the given problem to a finite dimensional problem using a spectral collocation method. We theoretically prove the exponential convergence rate of a function approximation based on the fractional order Legendre polynomials and, on this basis, present an error analysis of the proposed method. We provide some numerical examples to show the convergence rate and efficiency of the proposed method.

In this article, we introduce a class of nonlinear fractional Lane-Emden equations incorporating Jumarie's fractional derivative with Neumann boundary conditions. To compute approximate numerical solutions, we propose an efficient modification of a variational iteration method (VIM) termed the fractional variational iteration method (FrVIM). Notably, for the limiting case our problem reduces to the classical Lane-Emden equation. We illustrate the effectiveness of our approach through graphical representations of solutions obtained using both Jumarie and Caputo derivatives. Furthermore, we observe that when the order of derivative approaches 1, the fractional solution plots gradually converge to the classical solution, demonstrating the accuracy and reliability of our method.

In this paper, a new fast compressive Fourier collocation algorithm for high-dimensional diffusion equations is constructed. A new algorithm is based on the high-dimensional sparse and fast Fourier transform and the compressive sampling matching pursuit algorithm. The computational complexity of the proposed algorithm is proportional to d², where d is the dimension of PDE domain. The error of the approximate solution is estimated and some numerical experiments are given.

The hyperbolicity preserving numerical solving of equations is a fundamental problem of computational mathematics. In the presence of relativistic effects and a magnetic field, the problem becomes even more complicated. The construction of numerical methods for solving continuum mechanics problems that allow one to control the hyperbolicity of a system of equations in solving the equations is an important problem formulated by Sergey Konstantinovich Godunov. This paper proposes an approach to constructing a Godunov-type scheme using a method of formulating the equations in the form of symmetric hyperbolic systems. An important property of this formalism, in addition to hyperbolicity control, is the use of all waves in solving the Riemann problem, which allows one to reproduce the tangential discontinuities with the quality of the Roe and HLLC schemes. The use of an operator splitting into the pressure forces and the advective transfer in the Riemann problem makes it possible to eliminate the ``sonic point'' problem, which may occur in Roe-type schemes. The scheme is verified by using some classical tests on the breakdown of a discontinuity in one-dimensional and two-dimensional formulations.

Based on the idea of Ogita and Aishima [1] and Ulm's method [2], we designed an inexact Ulm-type method based on matrix multiplication for inverse singular value problems, which avoids the shortcomings of solving a system of linear equations in each iteration of the algorithm in [3]. Thus, it looks more stable and requires less computations. Under the condition that the given singular values are distinct, the algorithm has the quadratic convergence in the sense of root convergence. Moreover, numerical experiments in the last section show that the new method is better than some known algorithms.

This paper presents the results of a dissipation and dispersion analysis of a linearized version of a novel finite difference scheme for the fully nonlinear weakly dispersive shallow water equations. It is a modification of the Rusanov scheme, which is a third-order approximation of the gas dynamics equations. We derive stability conditions, phase change and phase error formulas, and investigate the behavior of the harmonic attenuation coefficient. The study uncovers some previously unknown properties of the Rusanov scheme. Notably, both the original and the modified schemes exhibit high order smallness in the phase error for long waves. Furthermore, we identify the optimal parameter values that ensure monotonicity of the harmonic attenuation coefficient, strong suppression of short-wave harmonics, and small change in their phase.

With the swift advancement of artificial intelligence technology, automatically solving plane geometry problems has emerged as a popular area of research. To improve the automatic solution effect of mathematical plane geometry problems, an efficient deterministic prediction model is constructed by combining the powerful feature extraction ability of transformer, the time series processing ability of long short-term memory network, and the generation ability based on the bidirectional encoder representation of transformer. The outcomes of the ablation experiment show that the model proposed by the research performs the best in regard to accuracy and score, with 92.5% and 0.91%, respectively. The inference time is 0.45 seconds, significantly better than other models. In addition, the accuracy of this model in text relationship extraction, geometric element detection, and graphic annоtation is 97.88%, 93.68%, and 95.36%, which is significantly better than other comparative models. The application effect analysis results show that the proposed model maintains a low average number of steps and the highest accuracy of 96.21% as the number of theorems increases. The model proposed by the research exhibits higher robustness and stability when dealing with problems with different step numbers, providing a new approach for automatically solving plane geometry problems.