求实对称矩阵部分特征值的并行算法

Parallel algorithm for solving few eigenvalues of real symmetric matrix

提出了并行求解实对称稠密矩阵部分特征值的反幂法的预处理方法。该方法基于带状矩阵特征问题反幂法的信息传递复杂度低的特点,采用Householder变换并行算法约化大型实对称稠密矩阵为一定带宽的带状矩阵,针对带状矩阵用反幂法求解矩阵的在某一点的近似特征值;其中针对反幂法迭代中遇到的线性方程组,采用文献中的并行预处理共轭梯度算法求解。最后在Lenovo深腾1800集群上进行数值实验,并与预处理前反幂法的计算结果进行了比较,实验结果表明,经过预处理后的并行性远高于直接采用反幂法的并行性。

The preconditioned method for parallel solution is presented to some eigenvalues of real symmetric matrices by the anti-power method. It is based on the low complexity information transmission of the banded matrix satisfying some bandwidth condition. We reduce the large-scale real symmetric dense matrix to a banded matrix by parallel Householder transformation, then the approximation eigenvalue is solved to a certain point of the banded matrix by the anti-power method, of which we use the preconditioning conjugate gradient parallel algorithm of the Ref to solve linear equations. Finally, some numerical experiments on Lenovo ShenTeng 1800 cluster are shown for comparing our method with the calculation results by the anti-power method without pretreatment. It is shown that our method has the highest parallel efficiency.