Efficient Scheme for Computing the k-th Eigenvalue

• Main Authors: You-Huai Xia, 夏佑槐
• Other Authors: Weichung Wang
• Format: Others
• Language: zh-TW
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/99874458773901629469

碩士 === 國立臺灣大學 === 數學研究所 === 102 === In some applications of physics problems, we may want to know certain eigenvalues we''re interested in. Here our target is an arbitrarily chosen k-th eigenvalue of a generalized eigenvalue problems. According to the Sylvester''s law of inertia in Linear algebra, we may get the eigenvalue counts before an arbitrarily given number via $LDL^T$ decomposition. If we view the eigenvalue counts before a given number as a step function f, then our problem of finding the k-th eigenvalue can be transformed into a root-finding problem ( solving $f(x)=k$ for x ). As far as our problem concerned, our goals is to find a real value $s$ such that $f(s)=k$ or $f(s)=k-1$ so that we can find the exact k-th eigenvalue by an shift-invert eigenvalue solver. In this paper, we do not emphasize the choice of a specific eigenvalue solver, but focus on developing the root-finding algorithm and introduce an approximating method which can reduce time consuming relative to $LDL^T$ decomposition. In the literature, the Bisection method is an existing stable root-finding algorithm which can be applied to step functions. Here we will first extend the Bisection method to the Multiple-section method which can be naturally parallelized. We will also developed a sequential method based on piecewise Monotone cubic interpolation and then combine it with Multiple-section method to get a parallel algorithm. With the same number of MPI processes, it can reduce the total number of function evaluations for one time root-solving in most cases relative to the pure Multiple-section method. On the other hand, besides computing the exact function value via $LDL^T$ decomposition, we also took a stochastic scheme for estimating the eigenvalue counts as an alternative to achieve the aim for reducing the time consuming for one-time function evaluation. The result of numerical experiment shows that in the case of large sparse matrix, we can save function evaluation time by sacrificing some exactness of function value.