the DOS of the Holstein t-J model [3], to the dielectric constants of Si quantum dots [4], to linear scaling algo-Chebyshev polynomial approximations are an efficient and numerically stable way to calculate properties of the very large Hamil-rithms for tight-binding molecular dynamics [5], to projectonians important in computational condensed matter physics. The tion methods for the electronic structure problem [6], and present paper derives an optimal kernel polynomial which enforces to provide alternatives to path integral methods for statistipositivity of density of states and spectral estimates, achieves the cal mechanics [7]. Compared with other methods, the KPM best energy resolution, and preserves normalization. This kernel is easy to implement, interpret, and manipulate. Unlike polynomial method (KPM) is demonstrated for electronic structure and dynamic magnetic susceptibility calculations. For tight binding competing Lanczos recursion methods (LRM), the KPM Hamiltonians of Si, we show how to achieve high precision and avoids accumulation of numerical roundoff errors even for rapid convergence of the cohesive energy and vacancy formation large numbers of MVMs.
energy by careful attention to the order of approximation. For disor-In view of these recent applications to condensed matter dered XXZ-magnets, we show that the KPM provides a simpler physics and the prospect for expanded use in the future, it and more reliable procedure for calculating spectral functions than Lanczos recursion methods. Polynomial approximations to Fermi is important to optimize the KPM. Chebyshev polynomial projection operators are also proposed.