Estimation of the bilinear form y^{ⁿ/sup>f(A)x for Hermitian matrices}

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• 作者：Paraskevi Fika ^{pfika@math.uoa.gr" class="auth_mail" title="E-mail the corresponding author} ; Marilena Mitrouli ; ^{mmitroul@math.uoa.gr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65F15 ; 65F30 ; 65B05 ; 65C05 ; 65J10 ; 15A18 ; 15A45
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：502
• 期：Complete
• 页码：140-158
• 全文大小：601 K

文摘

For a Hermitian matrix A∈C^p×p$A\in {\mathbb{C}}^{p\times p}$, given vectors x  , y∈C^p$y\in {\mathbb{C}}^p$ and for suitable functions f  , the bilinear form y^⁎f(A)x$y^{⁎}f(A)x$ is estimated by extending the extrapolation method proposed by C. Brezinski in 1999. Families of one term and two term estimates e_f,ν$e_{f,\nu }$, ν∈C$\nu \in \mathbb{C}$ and ${\hat{e}}_{f,n,k}$, n,k∈Z$n,k\in \mathbb{Z}$, respectively, are derived by extrapolation of the moments of the matrix A. For the positive definite case, bounds for the optimal value of ν  , which leads to an efficient one term estimate in only one matrix vector product, are derived. For f(A)=A⁻¹$f(A)=A^{-1}$, a formula approximating this optimal value of ν is specified. Numerical results for several matrix functions and comparisons are provided to demonstrate the effectiveness of the extrapolation method.