ダブルシフト逆ベキ乗法によるスパース対称行列の中間固有解の一算定法(実用)  [in Japanese] A Method for Determining Intermediate Eigensolutions of Sparse and Symmetric Matrices by the Double Shifted Inverse Power Method(Practice)  [in Japanese]

本論文はスパース対称行列の任意の中間の原点移動点付近の固有解を求める1方法を提案している.固有値の相対残差値がある収束値以下では,反復毎に原点移動量を設定する原点動逆ベキ乗法を使用することにより,反復回数と計算時間を従来の原点移動逆ベキ乗法のほぼ半分以下とした.逐次シフト点の設定値の理論およびアルゴリズムを示すと共に,多重根と極近接根を持つ標準固有値問題に関する数値実験により,その有効性を示す.

This paper proposes a method to find intermediate eigensolutions in the vicinity of a shifted origin in a large sparse symmetric matrix. The shifted origin is held stationary at its new location in the conventional shifted origin inverse power method. In this method, however, when the relative residual error of the just-determined eigenvalue is above some convergence value, the conventional shifted origin inverse power method is employed. Once the relative residual error falls below the threshold, the shift distance is reset at every iteration of the shifted origin inverse power method. This enables the algorithm to reach a solution in approximately half the number of iterations and half the time of the conventional shifted origin inverse power method. Numerical experiments with Helmholtz problems in the standard eigenvalue problem carried out, as these often have multiple roots. This method is shown to be effective.