Structure preserving quaternion full orthogonalization method with applications

Abstract

This article proposes a structure‐preserving quaternion full orthogonalization method (QFOM) for solving quaternion linear systems arising from color image restoration. The method is based on the quaternion Arnoldi procedure preserving the quaternion Hessenberg form. Combining with the preconditioning techniques, we further derive a variant of the QFOM for solving the linear systems, which can greatly improve the rate of convergence of QFOM. Numerical experiments on randomly generated data and color image restoration problems illustrate the effectiveness of the proposed algorithms in comparison with some existing methods.

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

“…The value of the derivative was recalculated using the finite difference formulas and the new error was obtained. With the values of the two errors with different steps, we could obtain the error order of the approximation by finite differences, making the logarithm of the ratio of errors on the basis of the ratio of the steps and rounding the result of the logarithm to the nearest integer [21]. Thus, we obtained:…”

“…Many other applications and tests of finite difference derivative approximations are provided in the literature [21,[32][33][34][35][36][37][38][39][40][41].…”

A Global Krylov Subspace Method for the Sylvester Quaternion Matrix Equation

“…Due to these wide ranges of applications for Sylvester equations, as well as quaternions, the problem of obtaining solutions to (1.1), specifically over the skew-field of quaternions, has attracted considerable attention [12][13][14][15]. Matrix equations other than Sylvester equations over quaternions have also been studied in the literature [16][17][18][19][20][21].…”

“…However, this conversion is usually not desirable since it results in matrices in the converted matrix equation that are twice or four times as large as the matrices in the original problem. On the other hand, structure-preserving Krylov subspace methods have become popular recently to overcome the increase in the size of the matrices when such a conversion is applied [19][20][21]. Some other approaches for solving linear quaternion matrix equations work on a right or a left Hilbert space over quaternions equipped with a proper inner product [17,18].…”