Quaternion Generalized Minimal Residual Method With Applications To Image Processing

Many Problems,which are in large-scale engineering and scientific computing area,can be transformed into solving the large sparse linear systems,such as fluid mechanics,the optimization problems and so on.The generalized minimal residual method is one of the most common methods to solve large sparse linear systems,and also has the advantages of converging quickly and stably.The researches,which are on solving large sparse linear systems,have great significance both in theory and experi-ments.However,because of the complex structure of the quaternion matrix,there are no fast and accurate methods to solve the large quaternion sparse linear systems.In this paper,we propose methods based on the quaternion Krylov subspace to solve large-scale quaternion sparse linear systems.First of all,we clearly define the sparse quaternion matrices and storage schemes,generalize the Krylov subspace to quaternion field,and propose the quaternion Arnoldi method and its modified algo-rithms.Second,we apply quaternion Arnoldi method to solve large quaternion sparse linear systems,and present quaternion full orthogonal method and its modified varia-tions.Third,we present the quaternion generalized minimal residual method and modi-fied algorithms,including quaternion Givens-Householder Arnoldi algorithm,restarted quaternion generalized minimal residual algorithm and truncated generalized minimal residual algorithm.Additionally,the numerical results show that this method not only steadily solves quaternion sparse linear systems,also saves the calculation time.Final-ly,the proposed algorithms are applied to deblur image,and the numerical results can show the effectiveness of these algorithms.