Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic
Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic
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DOI：
10.1002/nla.1680020303
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发表时间：
1995-05
期刊：
影响因子：
--
通讯作者：
V. Druskin;L. Knizhnerman
中科院分区：
文献类型：
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作者：
V. Druskin;L. Knizhnerman
许多研究人员现在正在使用Krylov子空间中的近似来计算矩阵函数和向量的乘积。本文回顾了对对称矩阵的谱Lanczos分解方法（SLDM）的分析结果，并证明了一个一般的收敛估计，将SLDM的误差界与用一部分Chebyshev级数逼近矩阵函数的误差界联系起来。从而得到了解抛物型、双曲型和椭圆型偏微分方程时矩阵函数的有效估计。我们集中在抛物的情况下，在那里我们得到的估计，表明超收敛的SLDM。对于这种情况，我们还考虑了SLDM和splittingmethod的组合，并给出了一些数值结果，我们实现了我们的一般估计，得到了Lanczos逼近的收敛界的特征值的内部部分的频谱。与Kaniel-Saad估计不同，我们的估计与所需特征值和最近谱界之间的特征值集无关。我们考虑将我们的一般估计扩展到夜间计算机算术中简单Lanczos方法（无需重新正交化）的情况，这表明对于中等维度的Krylov子空间，对于精确算法证明的结果在大约是稳定的。
Many researchers are now working on computing the product of a matrix function and a vector,using approximations in a Krylov subspace. We review our results on the analysis of one implemen-tation of that approach for symmetric matrices, which we call the Spectral Lanczos DecompositionMethod (SLDM).We have proved a general convergence estimate, relating SLDM error bounds to those obtainedthrough approximation of the matrix function by a part of its Chebyshev series. Thus, we arrivedat e ective estimates for matrix functions arising when solving parabolic, hyperbolic and ellipticpartial di erential equations. We concentrate on the parabolic case, where we obtain estimatesthat indicate superconvergence of SLDM. For this case a combination of SLDM and splittingmethods is also considered and some numerical results are presented.We implement our general estimates to obtain convergence bounds of Lanczos approximationsto eigenvalues in the internal part of the spectrum. Unlike Kaniel-Saad estimates, our estimatesare independent of the set of eigenvalues between the required one and the nearest spectrumbound.We consider an extension of our general estimate to the case of the simple Lanczos method(without reorthogonalization) in nite computer arithmetic which shows that for a moderatedimension of the Krylov subspace the results, proved for the exact arithmetic, are stable up toroundo .