一种调和Ritz向量的精化算法及应用

利用调和Arnoldi算法的一种等价形式,用较少的运算量将大规模矩阵特征值问题转化成一个小型的标准特征值问题来求解调和Ritz对。针对调和Arnoldi算法中调和Ritz值收敛而相应的调和Ritz向量往往不收敛的情况,保持调和Ritz值不变,结合精化Arnoldi算法的思想给出了一种在位移Krylov子空间上对调和Ritz向量进行精化求解的精化变形算法,以寻求使残量范数达到极小的近似特征向量。理论分析和数值实验表明这种精化变形算法的可行性、有效性以及更快的收敛速度,利用此算法可以更快求解满足精度要求的大规模矩阵的特征值和特征向量。同时,将这种算法应用于图像K-L变换的协方差矩阵的特征值和特征向量的求解,克服了K-L变换中由于图像矩阵过大而求解过程困难的问题,选取前若干个较大的特征值所对应的特征向量构成变换矩阵进行K-L变换来压缩图像,能直接应用于实时的图像压缩,较对图像分块在每个小块上进行K-L变换的方法更有效。

Refined algorithm of harmonic Ritz vectors and its application

Using an equivalent form of the harmonic Amoldi algorithm and less calculation quantity,this paper transforms the large-scale matrix eigenvalue problem into a small-scale eigenvalue problem to solve the harmonic Ritz pair.According to the case that the harmonic Ritz value is convergent and the corresponding harmonic Ritz vector is often not convergent in the harmonic Arnoldi algorithm,the harmonic Ritz value is maintained,and the theory to refine the Arnoldi algorithm,this paper presents a variant refined method.This method makes harmonic Ritz-vectors to be refined in the displacement Krylov subspace to find the approximate eigenvector of matrix which makes the residual norm of the Krylov subspace minimal.The Theoretic analysis and experiment results show that the proposed method is feasible and effective and has a faster speed of convergence.The proposed algorithm can satisfy the precision of calculating the eigenvalue and eigenvector of a large-scale matrix more fast.At the same time,the proposed algorithm is applied to the solution of the eigenvalue and eigenvector of covariance matrix of image K-L transform to compress the images.This algorithm can overcome the difficulty to solve a large image matrix in K-L transform and select the eigenvectors to that the first several larger eigenvalues corresponds to pose transformation matrix for K-L transform to compress the image.This can be applied to the image compression of real-timing directly.Compared with to divide image into blocks and apply K-L transform on each block,the proposed method in this paper is more effective.