求解大规模Hamilton矩阵特征问题的辛Lanczos算法的误差分析

Error Analysis of Symplectic Lanczos Method for Hamiltonian Eigenvalue Problem

对求解大规模稀疏Hamilton矩阵特征问题的辛Lanczos算法给出了舍入误差分析.分析表明辛Lanczos算法在无中断时,保Hamilton结构的限制没有破坏非对称Lanczos算法的本质特性.本文还讨论了辛Lanczos算法计算出的辛Lanczos向量的J一正交性的损失与Ritz值收敛的关系.结论正如所料,当某些Ritz值开始收敛时.计算出的辛Lanczos向量的J-正交性损失是必然的.以上结果对辛Lanczos算法的改进具有理论指导意义.

A rounding error analysis of the symplectic Lanczos method is given for the Hamil-tonian eigenvalue problem. It is applicable when no break down occurs and shows that the restriction of preserving the Hamiltonian structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. All analysis follows the lines of Bai's analysis of the nonsymmetric Lanczos algorithm and the lines of H. FaBbender's analysis of the symplectic Lanczos algorithm for the symplectic eigenvalue problem. As is expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthog-onality when some Ritz values begin to converge.