计算大规模矩阵最大最小奇异值和奇异向量的两个精化Lanczos算法

1.引言 在科学工程计算中经常需要计算大规模矩阵的少数最大或最小的奇异值及其所对应的奇异子空间。例如图像处理中要计算矩阵端部奇异值之比作为图像的分辨率,诸如此类的问题还存在于最小二乘问题、控制理论、量子化学中等等。然而大多实际问题中的矩阵是大型稀疏矩阵,且需要的是矩阵的部分奇异对。如果计算A的完全奇异值分解(SVD),则运算量和存储量极大,甚至不可能。因此必须寻求其它有效可靠的算法。 假设A的SVD为

TWO REFINED LANCZOS ALGORITHMS FOR COMPUTING THE LARGEST/SMALLEST SINGULAR VALUES AND ASSOCITED SINGULAR VECTORS OF A LARGE MATRIX

This paper concerns the computation of a few large (or small) singular values and the associated singular vectors of an l×n matrix A. They are the square roots of the large (or the small) eigenvalues and the eigenvectors of the cross-product matrix AT A. So instead of solving the full SVD problem we solve the eigenproblem of the cross-product matrix using projection methods, and then revert it to the original one. For the cross-product matrix ATA, an explicitly restarted refined Lanczos algorithm and an implicitly restarted refined Lanczos algorithm are proposed. A convergence analysis is presented for the Ritz value, Ritz vector and refined Ritz vector. Numerical experiments show that two refined algorithms are far superior to their conventional counterparts.