Algorithms and architectures for high performance recursive filtering

Recently, a number of most significant digit (msd) first bit parallel multipliers for recursive filtering have been reported. However, the design approach which has been used has, in general, been heuristic and consequently, optimality has not always been assured. In this paper, msd first multiply accumulate algorithms are described and important relationships governing the dependencies between latency, number representations, etc are derived. A more systematic approach to designing recursive filters is illustrated by applying the algorithms and associated relationships to the design of cascadable modules for high sample rate IIR filtering and wave digital filtering.

Fast VLSI algorithms for division and square root

Real time digital signal processing demands high performance implementations of division and square root. This can only be achieved by the design of fast and efficient arithmetic algorithms which address practical VLSI architectural design issues. In this paper, new algorithms for division and square root are described. The new schemes are based on pre-scaling the operands and modifying the classical SRT method such that the result digits and the remainders are computed concurrently and the computations in adjacent rows are overlapped. Consequently, their performance exceeds that of the SRT methods. The hardware cost for higher radices is considerably more than that of the SRT methods but for many applications, this is not prohibitive. A system of equations is presented which enables both an analysis of the method for any radix and the parameters of implementations to be easily determined. This is illustrated for the case of radix 2 and radix 4. In addition, a highly regular array architecture combining the division and square root method is described. © 1994 Kluwer Academic Publishers.

Rapid silicon design of discrete orthonormal wavelet transforms based algorithms

A methodology has been developed which allows a non-specialist to rapidly design silicon wavelet transform cores for a variety of specifications. The cores include both forward and inverse orthonormal wavelet transforms. This methodology is based on efficient, modular and scaleable architectures utilising time-interleaved coefficients for the wavelet transform filters. The cores are parameterized in terms of wavelet type and data and coefficient word lengths. The designs have been captured in VHDL and are hence portable across a range of silicon foundries as well as FPGA and PLD implementations.

New algorithms and VLSI architectures for SRT division and square root

In real time digital signal processing, high performance modules for division and square root are essential if many powerful algorithms are to be implemented. In this paper, a new radix 2 algorithms for SRT division and square root are developed. For these new schemes, the result digits and the residuals are computed concurrently and the computations in adjacent rows are overlapped. Consequently, their performance should exceed that of the radix 2 SRT methods. VLSI array architectures to implement the new division and square root schemes are also presented.

On the application of genetic algorithms for optimising composites against impact loading

A genetic algorithm (GA) was adopted to optimise the response of a composite laminate subject to impact. Two different impact scenarios are presented: low-velocity impact of a slender laminated strip and high-velocity impact of a rectangular plate by a spherical impactor. In these cases, the GA's objective was to, respectively, minimise the peak deflection and minimise penetration by varying the ply angles.

The GA was coupled to a commercial finite-element (FE) package LS DYNA to perform the impact analyses. A comparison with a commercial optimisation package, LS OPT, was also made. The results showed that the GA was a robust, capable optimisation tool that produced near optimal designs, and performed well with respect to LS OPT for the more complex high-velocity impact scenario tested.

Kingship by Descent or Kingship by Election? The Contested Title of James VI and I

Throughout the reign of Elizabeth I, a steady stream of tracts appeared in English print to vindicate the succession of the most prominent contenders, Mary and James Stuart of Scotland. This article offers a comprehensive account of the polemical battle between the supporters and opponents of the Stuarts, and further identifies various theories of English kingship, most notably the theory of corporate kingship, developed by the Stuart polemicists to defend the Scottish succession. James's accession to the English throne in March 1603 marked the protracted end of the debate over the succession. The article concludes by suggesting that, while powerfully renouncing the opposition to his succession, over the course of his attempt to unify his two kingdoms, James and his supporters ultimately departed from the polemic of corporate kingship, for a more assertive language of kingship by natural and divine law.

Implementation and testing of Lanczos-based algorithms for Random-Phase Approximation eigenproblems

The treatment of the Random-Phase Approximation Hamiltonians, encountered in different frameworks, like time-dependent density functional theory or Bethe-Salpeter equation, is complicated by their non-Hermicity. Compared to their Hermitian Hamiltonian counterparts, computational methods for the treatment of non-Hermitian Hamiltonians are often less efficient and less stable, sometimes leading to the breakdown of the method. Recently [Gruning et al. Nano Lett. 8 (2009) 28201, we have identified that such Hamiltonians are usually pseudo-Hermitian. Exploiting this property, we have implemented an algorithm of the Lanczos type for Random-Phase Approximation Hamiltonians that benefits from the same stability and computational load as its Hermitian counterpart, and applied it to the study of the optical response of carbon nanotubes. We present here the related theoretical grounds and technical details, and study the performance of the algorithm for the calculation of the optical absorption of a molecule within the Bethe-Salpeter equation framework. (C) 2011 Elsevier B.V. All rights reserved.

Improved description of chemical barriers with generalized gradient approximations (GGAs) and meta-GGAs

The performance of exchange and correlation (xc) functionals of the generalized gradient approximation (GGA) type and of the meta-GGA type in the calculation of chemical reactions is related to topological features of the electron density which, in turn, are connected to the orbital structure of chemical bonds within the Kohn-Sham (KS) theory. Seventeen GGA and meta-GGA xc functionals are assessed for 15 hydrogen abstraction reactions and 3 symmetrical S(N)2 reactions. Systems that are problematic for standard GGAs characteristically have enhanced values of the dimensionless gradient argument s(sigma)(2) with local maxima in the bonding region. The origin of this topological feature is the occupation of valence KS orbitals with an antibonding or essentially nonbonding character. The local enhancement of s(sigma)(2) yields too negative exchange-correlation energies with standard GGAs for the transition state of the S(N)2 reaction, which leads to the reduced calculated reaction barriers. The unwarranted localization of the effective xc hole of the standard GGAs, i.e., the nondynamical correlation that is built into them but is spurious in this case, wields its effect by their s(sigma)(2) dependence. Barriers are improved for xc functionals with the exchange functional OPTX as x component, which has a modified dependence on s(sigma)(2). Standard GGAs also underestimate the barriers for the hydrogen abstraction reactions. In this case the barriers are improved by correlation functionals, such as the Laplacian-dependent (LAP3) functional, which has a modified dependence on the Coulomb correlation of the opposite- and like-spin electrons. The best overall performance is established for the combination OLAP3 of OPTX and LAP3.

On the required shape corrections to the local density and generalized gradient approximations to the Kohn-Sham potentials for molecular response calculations of (hyper)polarizabilities and excitation energies

It is well known that shape corrections have to be applied to the local-density (LDA) and generalized gradient (GGA) approximations to the Kohn-Sham exchange-correlation potential in order to obtain reliable response properties in time dependent density functional theory calculations. Here we demonstrate that it is an oversimplified view that these shape corrections concern primarily the asymptotic part of the potential, and that they affect only Rydberg type transitions. The performance is assessed of two shape-corrected Kohn-Sham potentials, the gradient-regulated asymptotic connection procedure applied to the Becke-Perdew potential (BP-GRAC) and the statistical averaging of (model) orbital potentials (SAOP), versus LDA and GGA potentials, in molecular response calculations of the static average polarizability alpha, the Cauchy coefficient S-4, and the static average hyperpolarizability beta. The nature of the distortions of the LDA/GGA potentials is highlighted and it is shown that they introduce many spurious excited states at too low energy which may mix with valence excited states, resulting in wrong excited state compositions. They also lead to wrong oscillator strengths and thus to a wrong spectral structure of properties like the polarizability. LDA, Becke-Lee-Yang-Parr (BLYP), and Becke-Perdew (BP) characteristically underestimate contributions to alpha and S-4 from bound Rydberg-type states and overestimate those from the continuum. Cancellation of the errors in these contributions occasionally produces fortuitously good results. The distortions of the LDA, BLYP, and BP spectra are related to the deficiencies of the LDA/GGA potentials in both the bulk and outer molecular regions. In contrast, both SAOP and BP-GRAC potentials produce high quality polarizabilities for 21 molecules and also reliable Cauchy moments and hyperpolarizabilities for the selected molecules. The analysis for the N-2 molecule shows, that both SAOP and BP-GRAC yield reliable energies omega(i) and oscillator strengths f(i) of individual excitations, so that they reproduce well the spectral structure of alpha and S-4.(C) 2002 American Institute of Physics.

The failure of generalized gradient approximations (GGAs) and meta-GGAs for the two-center three-electron bonds in He-2(+), (H2O)(2)(+), and (NH3)(2)(+)

The radical cations He-2(+) (H2O)(2)(+), and (NH3)(2)(+) with two-center three-electron A-A bonds are investigated at the configuration interaction (CI), accurate Kohn-Sham (KS), generalized gradient approximation (GGA), and meta-GGA levels. Assessment of seven different GGA and six meta-GGA methods shows that the A(2)(+) systems remain a difficult case for density functional theory (DFT). All methods tested consistently overestimate the stability of A(2)(+): the corresponding D-e errors decrease for more diffuse valence densities in the series He-2(+) > (H2O)(2)(+) > (NH3)(2)(+). Upon comparison to the energy terms of the accurate Kohn-Sham solutions, the approximate exchange functionals are found to be responsible for the errors of GGA-type methods, which characteristically overestimate the exchange in A(2)(+). These so-called exchange functionals implicitly use localized holes. Such localized holes do occur if there is left-right correlation, i.e., the exchange functionals then also describe nondynamical correlation. However, in the hemibonded A(2)(+) systems the typical molecular (left-right, nondynamical) correlation of the two-electron pair bond is absent. The nondynamical correlation built into the exchange functionals is then spurious and yields too low energies.

Shape corrections to exchange-correlation potentials by gradient-regulated seamless connection of model potentials for inner and outer region

Shape corrections to the standard approximate Kohn-Sham exchange-correlation (xc) potentials are considered with the aim to improve the excitation energies (especially for higher excitations) calculated with time-dependent density functional perturbation theory. A scheme of gradient-regulated connection (GRAC) of inner to outer parts of a model potential is developed. Asymptotic corrections based either on the potential of Fermi and Amaldi or van Leeuwen and Baerends (LB) are seamlessly connected to the (shifted) xc potential of Becke and Perdew (BP) with the GRAC procedure, and are employed to calculate the vertical excitation energies of the prototype molecules N-2, CO, CH2O, C2H4, C5NH5, C6H6, Li-2, Na-2, K-2. The results are compared with those of the alternative interpolation scheme of Tozer and Handy as well as with the results of the potential obtained with the statistical averaging of (model) orbital potentials. Various asymptotically corrected potentials produce high quality excitation energies, which in quite a few cases approach the benchmark accuracy of 0.1 eV for the electronic spectra. Based on these results, the potential BP-GRAC-LB is proposed for molecular response calculations, which is a smooth potential and a genuine "local" density functional with an analytical representation. (C) 2001 American Institute of Physics.