Approximating spectral densities of large matrices

In physics, it is sometimes desirable to compute the so-called \emph{Density Of States} (DOS), also known as the \emph{spectral density}, of a real symmetric matrix $A$. The spectral density can be viewed as a probability density distribution that measures the likelihood of finding eigenvalues near some point on the real line. The most straightforward way to obtain this density is to compute all eigenvalues of $A$. But this approach is generally costly and wasteful, especially for matrices of large dimension. There exists alternative methods that allow us to estimate the spectral density function at much lower cost. The major computational cost of these methods is in multiplying $A$ with a number of vectors, which makes them appealing for large-scale problems where products of the matrix $A$ with arbitrary vectors are relatively inexpensive. This paper defines the problem of estimating the spectral density carefully, and discusses how to measure the accuracy of an approximate spectral density. It then surveys a few known methods for estimating the spectral density, and proposes some new variations of existing methods. All methods are discussed from a numerical linear algebra point of view

New high fill-factor triangular micro-lens array fabrication method using UV proximity printing

A simple and effective method to fabricate a high fill-factor triangular microlens array using the proximity printing in lithography process is reported. The technology utilizes the UV proximity printing by controlling a printing gap between the mask and substrate. The designed approximate triangle microlens array pattern can be fabricated the high fill-factor triangular microlens array in photoresist. It is due to the UV light diffraction to deflect away from the aperture edges and produce a certain exposure in photoresist material outside the aperture edges. This method can precisely control the geometric profile of high fill factor triangular microlens array. The experimental results showed that the triangular micro-lens array in photoresist could be formed automatically when the printing gap ranged from 240 micrometers to 840 micrometers. The gapless triangular microlens array will be used to increases of luminance for backlight module of liquid crystal displays.Comment: Submitted on behalf of EDA Publishing Association (http://irevues.inist.fr/handle/2042/16838