A note on Neuberger's double pass algorithm
Ting-Wai Chiu, Tung-Han Hsieh
我们分析了 Neuberger 的矩阵-向量乘法 R(H).Y(其中 R(H) 是 (n-1,n)-th 度的正定运算符 H 的度 rational 多项式算法),并表明浮点运算数独立于度 n,前提是站点的数量远大于共轭梯度中的迭代次数。 这意味着矩阵矢量产品(H)^-1/2 Y ≃ R^(n-1,n)(H) · Y可以近似于非常高的精度,具有足够大的n,而不会产生明显的额外费用。 此外,我们表明存在一个阈值n_T,使得双关比n > n_T的单次传球更快,大多数平台的n_T≃12 - 25。
We analyze Neuberger's double pass algorithm for the matrix-vector multiplication R(H).Y (where R(H) is (n-1,n)-th degree rational polynomial of positive definite operator H), and show that the number of floating point operations is independent of the degree n, provided that the number of sites is much larger than the number of iterations in the conjugate gradient. This implies that the matrix-vector product (H)^-1/2 Y ≃ R^(n-1,n)(H) · Y can be approximated to very high precision with sufficient...