求解大型对称线性方程组的循环收缩Lanczos算法

Restarted and Deflated Lanczos Algorithm for Solving Large Symmetric Systems

向 Krylov子空间中加入一些模接近于零的特征值对应的特征向量能够加快收敛速度 ,事实上 ,对于这些模接近于零的特征值对应的特征向量 ,可以用 Krylov子空间方法得到 ,并且在新的 Krylov子空间形成的过程中 ,近似特征向量的近似度会不断提高 ,特别在标准 Krylov子空间方法中 ,如果因为这些特征向量而减缓了收敛速度 ,则随着这些特征向量的近似度的提高 ,用增广 Krylov子空间方法解线性方程组的收敛速度会明显加快。Lanczos算法是求解大型对称不定线性方程组的有效方法之一。但在计算过程中由于 Lanczos向量失去正交性减慢了收敛速度。本文根据增广 Krylov子空间方法提出循环收缩 Lanczos算法 ,新算法充分利用 Lanczos过程所得到的谱信息 ,确定预处理 ,从而加速

Many papers discussed the benefits using eigenvalues deflation in recent years when solving linear systems with Krylov subspace methods. Significant improvements in convergence rates can be achieved from Krylov subspace methods by adding to these subspaces to a few approximate eigenvectors associated with the eigenvalues closed to zero. In practice, approximations to the eigenvectors closed to zero are obtained from the new Krylov subspace. Experimental results indicate that the improvement in convergence over standard Krylov subspace of the same dimension can sometimes be substantial. The Lanczos method is well known for solving large systems of liner equations. However, loss of orthogonality of the Lanczos vectors can reduce the convergence of the method. A restarted and deflated Lanczos algorithm is developed. The algorithm improves the convergence rate of Lanczos algorithm by using the spectral information obtained from the previous Lanczos process.