a low complexity mvdr beamformer for use with short observation times it is demonstrated via simulation examples that the steered minimum variance subarray beamformer is a simple viable and computationally efficient approach to partially adaptive beamforming with the additional advantage that it is available for use with short observation intervals where time is traded for bandwidth a quadratically constrained version is also examined

low complexity bit parallel architectures for polynomial basis multiplication over gf 2m representing the field elements with respect to the polynomial or standard basis we consider bit parallel architectures for multiplication over the finite field gf 2m in this effect first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix q the main advantage of this new formulation is that it can be used with any field defining irreducible polynomial using this formulation we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix q to the best of our knowledge this is the first time that these complexities are given in terms of q unlike most other articles on bit parallel finite field multipliers here we also consider the number of signals to be routed in hardware implementation and we show that compared to the well known mastrovito s multiplier the proposed architecture has fewer routed signals the proposed generalized architecture is further optimized for three special types of polynomials namely equally spaced polynomials trinomials and pentanomials we have obtained explicit formulas and complexities of the multipliers for these three special irreducible polynomials this makes it very easy for a designer to implement the proposed multipliers using hardware description languages like vhdl and verilog with minimum knowledge of finite field arithmetic

direction of arrival estimation using parametric signal models we consider the problem of estimating the directions of arrival doas of narrowband sources with known center frequency the paper evaluates the potential improvement in estimation accuracy by using spatial temporal processing for signals obeying a deterministic parametric model one would expect that prior information about the temporal structure of the signals will yield some gain in performance by deriving the cramer rao bound crb on the doa estimates we quantify this gain and identify the cases for which the gain is significant we show that for the single source case spatial temporal processing does not yield any gain in performance relative to conventional spatial processing for multiple noncoherent signals incorporating temporal processing can achieve the single source performance yielding a significant gain for the case of multiple sources with small spatial separation relative to the beamwidth of the array however spatial temporal processing cannot yield any gain in performance for multiple coherent signals

block recombination approach for subquadratic space complexity binary field multiplication based on toeplitz matrix vector product in this paper we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity the method is based on decomposing the building blocks of the fan hasan subquadratic toeplitz matrix vector multiplier we reduce the space complexity of their architecture by recombining the building blocks in comparison to other similar schemes available in the literature our proposal presents a better space complexity while having the same time complexity we also show that block recombination can be used for efficient implementation of the ghash function of galois counter mode gcm

multiway splitting method for toeplitz matrix vector product computing the product of a toeplitz matrix and a vector arises in various applications including cryptography in this paper we consider toeplitz matrices and vectors with entries in hbox rlap i kern 2 0pt hbox f 2 for improved efficiency in such computations large toeplitz matrices and vectors are recursively split and special formulas with subquadratic arithmetic complexity are applied to this end we first present a formula for the five way splitting and then provide a generalization for the k way splitting where k is an arbitrary integer these formulas can be used to compute a toeplitz matrix vector product tmvp of size n with an arithmetic complexity of o n log k k k 1 2

mastrovito multiplier for general irreducible polynomials we present a new formulation of the mastrovito multiplication matrix for the field gf 2 sup m generated by an arbitrary irreducible polynomial we study in detail several specific types of irreducible polynomials e g trinomials all one polynomials and equally spaced polynomials and obtain the time and space complexity of these designs particular examples illustrating the properties of the proposed architecture are also given the complexity results established in this paper match the best complexity results known to date the most important new result is the space complexity of the mastrovito multiplier for an equally spaced polynomial which is found as m sup 2 spl delta xor gates and m sup 2 and gates where spl delta is the spacing factor

accurate solution of dense linear systems part i algorithms in rounding to nearest we investigate how extra precise accumulation of dot products can be used to solve ill conditioned linear systems accurately for a given p bit working precision extra precise evaluation of a dot product means that the products and summation are executed in 2p bit precision and that the final result is rounded into the p bit working precision denote by u 2 p the relative rounding error unit in a given working precision we treat two types of matrices first up to condition number u 1 and second up to condition number u 2 for both types of matrices we present two types of methods first for calculating an approximate solution and second for calculating rigorous error bounds for the solution together with the proof of non singularity of the matrix of the linear system in the first part of this paper we present algorithms using only rounding to nearest in part ii we use directed rounding to obtain better results all algorithms are given in executable matlab code and are available from my homepage

spatial spectral filtering with linearly constrained minimum variance beamformers in this paper orthogonal low dimension models for broad band sources as observed by an array delay line structure are developed the models are termed broad band source representation spaces and are subspaces of the observation space they are generated from eigenstructures of source sample covariance matrices and represent sources efficiently in the 2nd order statistical sense for a single broad band source propagating in a pure delay environment and observed with an isotropic array an accurate indication of the model dimension is derived which is independent on the array configuration being a function of the observed source time bandwidth product only using these representations procedures are described for controlling the spatial and spectral response of linearly constrained minimum variance broad band array beamformers the procedures employ linear constraints to control complex gain characteristics over selected bands of frequency for multiple points of 3 d regions of source locations and do not require the use of steering delays the constraints are derived from source representation spaces and are termed eigenvector constraints simulations are presented which illustrate the effectiveness of the new linear constraints

a new approach to subquadratic space complexity parallel multipliers for extended binary fields based on toeplitz matrix vector products and coordinate transformation techniques we present a new scheme for subquadratic space complexity parallel multiplication in gf 2n using the shifted polynomial basis both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers for example with n being a power of 2 the space complexity is about 8 percent better while the asymptotic gate delay is about 33 percent better respectively another advantage of the proposed matrix vector product approach is that it can also be used to design subquadratic space complexity polynomial dual weakly dual and triangular basis parallel multipliers to the best of our knowledge this is the first time that subquadratic space complexity parallel multipliers are proposed for dual weakly dual and triangular bases a recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers this design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature

adaptive reduced rank constrained constant modulus algorithms based on joint iterative optimization of filters for beamforming this paper proposes a robust reduced rank scheme for adaptive beamforming based on joint iterative optimization jio of adaptive filters the novel scheme is designed according to the constant modulus cm criterion subject to different constraints the proposed scheme consists of a bank of full rank adaptive filters that forms the transformation matrix and an adaptive reduced rank filter that operates at the output of the bank of filters to estimate the desired signal we describe the proposed scheme for both the direct form processor dfp and the generalized sidelobe canceller gsc structures for each structure we derive stochastic gradient sg and recursive least squares rls algorithms for its adaptive implementation the gram schmidt gs technique is applied to the adaptive algorithms for reformulating the transformation matrix and improving the performance an automatic rank selection technique is developed and employed to determine the most adequate rank for the derived algorithms a detailed complexity study and a convexity analysis are carried out simulation results show that the proposed algorithms outperform the existing full rank and reduced rank methods in convergence and tracking performance

composition constants for raising the orders of unconventional schemes for ordinary differential equations many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general purpose software designed to solve numerically a wide range of differential equations if those properties are to be exploited fully for the sake of better numerical stability accuracy and or speed the differential equations may have to be solved by unconventional methods this short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods called reflexive

computation of the monodromy matrix in floating point arithmetic with the wilkinson model in this study results have been obtained that compute the monodromy matrix in floating point arithmetic using the wilkinson model these results have been applied to the asymptotic stability of systems of linear difference equations with periodic coefficients also the effect of floating point arithmetic has been investigated on numerical examples

pdf s confidence regions and relevant statistics for a class of sample covariance based array processors we add to the many results on sample covariance matrix scm dependent array processors by i weakening the traditional assumption of gaussian data and ii providing for a class of array processors additional performance measures that are of value in practice the data matrix is assumed drawn from a class of multivariate elliptically contoured mec distributions the performance measures include the exact probability density functions pdfs confidence regions and moments of the weight vector matrix beam response and beamformer output of certain scm based scb array processors the array processors considered include the scb i maximum likelihood ml signal vector estimator ii linearly constrained minimum variance beamformer lcmv iii minimum variance distortionless response beamformer mvdr and iv generalized sidelobe canceller gsc implementation of the lcmv beamformer it is shown that the exact joint pdfs for the weight vectors matrices of the aforementioned scb array processors are a linear transformation from a complex multivariate extension of the standardized t distribution the scb beam responses are generalized t distributed and the pdfs of the scb beamformer outputs are given by kummer s function all but the beamformer outputs are shown to be completely invariant statistics over the class of mecs considered

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

the stability of barycentric interpolation at the chebyshev points of the second kind we present a new analysis of the stability of the first and second barycentric formulae for interpolation at the chebyshev points of the second kind our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice we also extend our current understanding regarding the accuracy problems of the first barycentric formula

the accuracy of floating point summation the usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers five summation methods and their variations are analyzed here the accuracy of the methods is compared using rounding error analysis and numerical experiments four of the methods are shown to be special cases of a general class of methods and an error analysis is given for this class no one method is uniformly more accurate than the others but some guidelines are given on the choice of method in particular cases

minimizing roundoff errors of prefix sums via dynamic construction of huffman trees the prefix sum operation which returns all prefix sums on a sequence of numbers plays an important role in many applications we study how to efficiently evaluate prefix sums on positive floating point numbers such that the worst case roundoff error of each sum is minimized a direct approach to this problem builds a huffman tree for each prefix subsequence from scratch requiring exactly quadratic time for every input x we can do better by taking advantage of the current huffman tree to build the next huffman tree using dynamic insertions and deletions on huffman trees consequently subquadratic time suffices for various input patterns we also provide experimental comparisons of all the algorithms studied in this paper on inputs that are randomly and uniformly generated

convergence of the smi and the diagonally loaded smi algorithms with weak interference adaptive array approximations for the power levels at the output of an adaptive array that uses the diagonally loaded sample matrix inversion smi algorithm are derived diagonal loading is a technique where the diagonal of the covariance matrix is augmented with a positive or negative constant prior to inversion the authors examine how the signal to interference plus noise ratio sinr and signal to interference ratio sir at the array output vary with the number of samples taken when the input signals are continuous wave it is shown that positive loading produces more rapid convergence with a reduction in output sir negative loading provides an improved sir level but it is shown that positive loading produces more rapid convergence with a reduction in output sir negative loading provides an improved sir level but it is shown that the output power levels are erratic and slow to converge simulation results which verify the theoretical procedure are given

efficient maximum likelihood doa estimation for signals with known waveforms in the presence of multipath we present a large sample maximum likelihood ml algorithm for estimating the directions of arrival doa s and signal amplitudes of known possibly coherent signals impinging on an array of sensors the algorithm is an extension of the deml method of li et al that handles coherent multipath that may be present in the signals the algorithm is computationally efficient because the nonlinear minimization step decouples into a set of minimizations of smaller dimension

accurate sum and dot product algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time we show that the computed results are as accurate as if computed in twice or k fold working precision k ge 3 for twice the working precision our algorithms for summation and dot product are some 40 faster than the corresponding xblas routines while sharing similar error estimates our algorithms are widely applicable because they require only addition subtraction and multiplication of floating point numbers in the same working precision as the given data higher precision is unnecessary algorithms are straight loops without branch and no access to mantissa or exponent is necessary

the statistical performance of eigenspace based adaptive array beamformers in this paper we analyze the statistical performance of several eigenspace based adaptive array bearnformers including conventional direct form beamformers and generalized sidelobe canceler gsc it is shown that the convergence rate of eigenspace based adaptive array beamforming depends on the number of array elements the input signal to noise ratio snr and the number of interferers when the desired signal is present during the weight adaptation in contrast the misadjostment of the array output power is a function of the number of interferers and is independent of the number of array elements when the desired signal is absent several simulation examples are also provided to confirm the theoretical results

array signal processing in the known waveform and steering vector case the amplitude estimation of a signal that is known only up to an unknown scaling factor with interference and noise present is of interest in several applications including using the emerging quadrupole resonance qr technology for explosive detection in such applications a sensor array is often deployed for interference suppression this paper considers the complex amplitude estimation of a known waveform signal whose array response is also known a priori two approaches viz the capon and the maximum likelihood ml methods are considered for the signal amplitude estimation in the presence of temporally white but spatially colored interference and noise we derive closed form expressions for the expected values and mean squared errors mses of the two estimators a comparative study shows that the ml estimate is unbiased whereas the capon estimate is biased downwards for finite data sample lengths we show that both methods are asymptotically statistically efficient when the number of data samples is large but not when the signal to noise ratio snr is high furthermore we consider a more general scenario where the interference and noise are both spatially and temporally correlated we model the interference and noise vector as a multichannel autoregressive ar random process an alternating least squares als method for parameter estimation is presented we show that in most cases the als method is superior to the model mismatched ml m sup 3 l method which ignores the temporal correlation of the interference and noise

multi access interference suppression in canonical space time coordinates a decentralized approach we propose a decentralized space time multiuser detection scheme based on the notion of canonical space time coordinates cstcs for representing the received signal the cstc representation provides a natural framework for decentralized multi access interference mai suppression in lower dimensional subspaces that results in complexity reduction relative to existing chip rate filtering schemes the framework is based on a partitioning of the signal space into active and inactive cstcs the active cstcs contain the signal of the desired user facilitate maximal diversity exploitation and minimal complexity interference suppression the inactive cstcs only contain mai and can be included progressively to attain a desired level of mai suppression at the cost of increased complexity we develop cstc based linear coherent multiuser detectors using the linearly constrained minimum variance lcmv criterion we characterize the set of inactive coordinates and analyze the performance of the lcmv receiver as a function of the number of inactive cstcs channel estimation and detector sensitivity to channel estimation errors are discussed we demonstrate that the low complexity adaptive receivers designed via the cstc framework are more robust to channel estimation errors than existing chip domain filtering schemes

schmi gf 2 n shifted polynomial basis multipliers based on subquadratic toeplitz matrix vector product approach for all irreducible pentanomials besides karatsuba s algorithm optimal toeplitz matrix vector product tmvp formulae is another approach to design gf 2 n subquadratic multipliers however when gf 2 n elements are represented using a polynomial basis or its generalization shifted polynomial basis this approach is currently appliable only to fields gf 2 n generated by an irreducible trinomial or a special type of irreducible pentanomials but not to a general irreducible pentanomial the reason is that no transformation matrix which transforms the mastrovito matrix into a toeplitz matrix has been found in this article we propose such a transformation matrix and its inverse matrix for an arbitrary irreducible pentanomial

combinatorial characterization of the null spaces of symmetric h matrices abstract we characterize the structure of null spaces of symmetric diagonally dominant sdd matrices and symmetric h matrices with non negative diagonal entries we show that the structure of the null space of a symmetric sdd matrix or h matrix a depends on the structure of the connected components of its underlying graph each connected component contributes at most one vector to the null space this paper provides a combinatorial characterization of the rank of each connected component and a combinatorial characterization of a null vector if one exists for sdd matrices we also present an efficient combinatorial algorithm for constructing an orthonormal basis for the null space the paper also shows a close connection between gain graphs and h matrices which extends known results regarding the connection between undirected graphs and laplacian matrices and between signed graphs and sdd matrices we show how to exploit these combinatorial algorithms to reliably solve certain singular linear systems in finite precision arithmetic

analysis of adaptive mesh refinement for imex discontinuous galerkin solutions of the compressible euler equations with application to atmospheric simulations the resolutions of interests in atmospheric simulations require prohibitively large computational resources adaptive mesh refinement amr tries to mitigate this problem by putting high resolution in crucial areas of the domain we investigate the performance of a tree based amr algorithm for the high order discontinuous galerkin method on quadrilateral grids with non conforming elements we perform a detailed analysis of the cost of amr by comparing this to uniform reference simulations of two standard atmospheric test cases density current and rising thermal bubble the analysis shows up to 15 times speed up of the amr simulations with the cost of mesh adaptation below 1 of the total runtime we pay particular attention to the implicit explicit imex time integration methods and show that the ark2 method is more robust with respect to dynamically adapting meshes than bdf2 preliminary analysis of preconditioning reveals that it can be an important factor in the amr overhead the compiler optimizations provide significant runtime reduction and positively affect the effectiveness of amr allowing for speed ups greater than it would follow from the simple performance model

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

reducing floating point error in dot product using the superblock family of algorithms this paper discusses both the theoretical and statistical errors obtained by various well known dot products from the canonical to pairwise algorithms and introduces a new and more general framework that we have named superblock which subsumes them and permits a practitioner to make trade offs between computational performance memory usage and error behavior we show that algorithms with lower error bounds tend to behave noticeably better in practice unlike many such error reducing algorithms superblock requires no additional floating point operations and should be implementable with little to no performance loss making it suitable for use as a performance critical building block of a linear algebra kernel

a new architecture for a parallel finite field multiplier with low complexity based on composite fields a bit parallel structure for a multiplier with low complexity in galois fields is introduced the multiplier operates over composite fields gf 2 sup n sup m with k nm the karatsuba ofman algorithm a karatsuba and y ofmanis 1963 is investigated and applied to the multiplication of polynomials over gf 2 sup n it is shown that this operation has a complexity of order o k sup log23 under certain constraints regarding k a complete set of primitive field polynomials for composite fields is provided which perform module reduction with low complexity as a result multipliers for fields gf 2 sup k up to k 32 with low gate counts and low delays are listed the architectures are highly modular and thus well suited for vlsi implementation

a generalized method for constructing subquadratic complexity gf 2 sup k multipliers we introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions the construction is obtained by recursively extending short convolution algorithms and nesting them to obtain the short convolution algorithms the winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication we present a recursive construction technique that extends any d point multiplier into an n d sup k point multiplier with area that is subquadratic and delay that is logarithmic in the bit length n we present a thorough analysis that establishes the exact space and time complexities of these multipliers using the recursive construction method we obtain six new constructions among which one turns out to be identical to the karatsuba multiplier all six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the karatsuba algorithm

efficient bit parallel gf 2 m multiplier for a large class of irreducible pentanomials this work studies efficient bit parallel multiplication in gf 2m for irreducible pentanomials based on the so called shifted polynomial bases spbs we derive a closed expression of the reduced spb product for a class of polynomials xm xk s xk s 1 hellip xk 1 1 with ks k1 les m 1 2 then we apply the above formulation to the case of pentanomials the resulting multiplier outperforms or is as efficient as the best proposals in the technical literature but it is suitable for a much larger class of pentanomials than those studied so far unlike previous works this property enables the choice of pentanomials optimizing different field operations for example inversion yet preserving an optimal implementation of field multiplication as discussed and quantitatively proved in the last part of the paper

a distillation algorithm for floating point summation the addition of two or more floating point numbers is fundamental to numerical computations this paper describes an efficient distillation style algorithm which produces a precise sum by exploiting the natural accuracy of compensated cancellation the algorithm is applicable to all sets of data but is particularly appropriate for ill conditioned data where standard methods fail due to the accumulation of rounding error and its subsequent exposure by cancellation the method uses only standard floating point arithmetic and does not rely on the radix used by the arithmetic model the architecture of specific machines or the use of accumulators

an application of the discrete time toda lattice to the progressive algorithm by lanczos and related problems the finite or the semi infinite discrete time toda lattice has many applications to various areas in applied mathematics the purpose of this paper is to review how the toda lattice appears in the lanczos algorithm through the quotient difference algorithm and its progressive form pqd then a multistep progressive algorithm mpa for solving linear systems is presented the extended lanczos parameters can be given not by computing inner products of the extended lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost the asymptotic behavior of the pqd algorithm brings us some applications of mpa related to eigenvectors

on floating point summation in this paper we focus on some general error analysis results in floating point summation we emphasize analysis that is useful from both a scientific and a teaching point of view

computation and storage of the bianisotropic scalar green s function and its derivatives a set of algorithms is proposed for the accurate and efficient computation and storage of the bianisotropic scalar green s function the computation is based on an expansion of the green s function into chebyshev polynomials the analytical properties of these polynomials are exploited to allow the accurate computation of the derivatives of the green s function as well as the green s function itself for lossy materials the proposed computation strategy is provably robust in addition a multilevel storage scheme with a favorable complexity based on the chebyshev polynomial expansion is proposed for the storage of the expansion coefficients numerical results showcase the accuracy and computational complexity of the proposed algorithms

parallel algorithms to evaluate orthogonal polynomial series in this paper four parallel algorithms for the evaluation of finite series of orthogonal polynomials are introduced the algorithms are based on the forsythe and clenshaw sequential algorithms several tests carried out on a cray t3d are presented

a new distillation algorithm for floating point summation the summation of n floating point numbers is ubiquitous in numerical computations we present a new distillation algorithm for floating point summation which is stable efficient and accurate the algorithm iteratively distills the summands without discarding any significant digit until the partial sums cannot change the whole sum it uses standard floating point arithmetic and does not rely on the choice of radix or any other specific assumption furthermore the error bound of our algorithm is independent of n and less than 1 ulp

recursive short data record estimation of av and mmse mvdr linear filters for ds cdma antenna array systems the presence of the desired signal during estimation of the minimum mean square error mmse minimum variance distortionless response mvdr and auxiliary vector av filters under limited data support leads to significant signal to interference plus noise ratio sinr performance degradation we quantify this observation in the context of direct sequence code division multiple access ds cdma communications by deriving close approximations for the mean square filter estimation error the probability density function of the output sinr and the probability density function of the symbol error rate ser of the sample matrix inversion smi receiver evaluated using both a desired signal present and desired signal absent input covariance matrix to avoid such performance degradation we propose a ds cdma receiver that utilizes a simple pilot assisted algorithm that estimates and then subtracts the desired signal component from the received signal prior to filter estimation then to accommodate decision directed operation we develop two recursive algorithms for the on line estimation of the av and mmse mvdr filter and we study their convergence properties finally simulation studies illustrate the ser performance of the overall receiver structures

comments on a new architecture for a parallel finite field multiplier with low complexity based on composite field in this comment we show that the gate complexity of karatsuba polynomial multipliers presented in the above paper can be reduced by exploring the common subexpressions

architecture for a low complexity rate adaptive reed solomon encoder multiple error correcting reed solomon rs codes have many practical applications the complexity of an rs encoder depends on multiplications in the finite field over which the code is defined we consider a triangular basis for representing the field elements and present an architecture for a rate adaptive rs encoder using a triangular basis multiplication algorithm the architecture supports pipeline and bit serial operations and has a low circuit complexity

a modular structure for implementation of linearly constrained minimum variance beamformers a description is given of a cascade decomposition of the generalized sidelobe canceller gsc implementation for linearly constrained minimum variance beamformers the gsc is initially separated into an adaptive interference cancellation module followed by a nonadaptive beamformer the authors prove that the adaptive interference cancellation module can be decomposed into a cascade of first or higher order adaptive interference cancellation modules where the order corresponds to the number of adaptive degrees of freedom represented in the module this distributes the computational burden associated with determining the adaptive weights over several lower order problems and facilitates simultaneous implementation of beamformers with differing numbers of adaptive degrees of freedom

on the mean square error performance of adaptive minimum variance beamformers based on the sample covariance matrix the authors examine the mean square error mse performance of two common implementations of adaptive linearly constrained minimum variance lcmv beamformers that employ the sample covariance matrix the type i beamformer is representative of block processing methods where the same input data is used both to compute the adaptive weights and to form the beamformer output the type ii beamformer as in many recursive schemes applies adaptive weights computed from previous data to the current input due to correlation between the adaptive weights and the input data the type i lcmv beamformer exhibits signal cancellation which is shown here to cause signal estimate bias to explicitly account for signal cancellation the mean square error mse and output signal to noise ratio snr measures of the bias corrected type i beamformer are analyzed thus extending previous results further new analytical results for these performance measures are given for the type ii lcmv beamformer comparison of bias corrected type i and type ii implementations indicate that both methods yield exactly the same mse and output snr performance

ultimately fast accurate summation we present two new algorithms fastaccsum and fastprecsum one to compute a faithful rounding of the sum of floating point numbers and the other for a result as if computed in k fold precision faithful rounding means the computed result either is one of the immediate floating point neighbors of the exact result or is equal to the exact sum if this is a floating point number the algorithms are based on our previous algorithms accsum and precsum and improve them by up to 25 the first algorithm adapts to the condition number of the sum i e the computing time is proportional to the difficulty of the problem the second algorithm does not need extra memory and the computing time depends only on the number of summands and k both algorithms are the fastest known in terms of flops they allow good instruction level parallelism so that they are also fast in terms of measured computing time the algorithms require only standard floating point addition subtraction and multiplication in one working precision for example double precision

exact computation of the sign of a finite sum many geometric algorithms are dependent on the sign of a simple expression like a finite sum examples of such algorithms are left turn test orientation questions boolean algorithms point in circle versus not in circle etc implementing such algorithms in fixed length floating point arithmetic can lead to inaccurate or wrong geometric configurations due to falsification of the computation by rounding errors in this paper we present an algorithm for determining the sign of a sum of a finite set of real numbers the result is exact i e the computation is free of rounding errors when the real numbers are machine numbers the algorithm is extremely simple only computation with fixed word length is required for example single precision floating point arithmetic no splitting or other mantissa manipulations are necessary one only needs to know the exponential parts of the represented summands and it is almost never necessary to compute the value of the whole sum

parallel multipliers based on special irreducible pentanomials the state of the art galois field gf 2 sup m multipliers offer advantageous space and time complexities when the field is generated by so special irreducible polynomial to date the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equally spaced polynomial esp unfortunately there exist only a few irreducible esps in the range of interest for most of the applications e g error correcting codes computer algebra and elliptic curve cryptography furthermore it is not always possible to find an irreducible trinomial of degree m in this range for those cases where neither an irreducible trinomial nor an irreducible esp exists the use of irreducible pentanomials has been suggested irreducible pentanomials are abundant and there are several eligible candidates for a given m we promote the use of two special types of irreducible pentanomials we propose new mastrovito and dual basis multiplier architectures based on these special irreducible pentanomials and give rigorous analyses of their space and time complexity

fast parallel gf 2 m polynomial multiplication for all degrees numerous works have addressed efficient parallel gf 2m multiplication based on polynomial basis or some of its variants for those field degrees where neither irreducible trinomials nor equally spaced polynomials epss exist the best area time performance has been achieved for special type irreducible pentanomials which however do not exist for all degrees in other words no multiplier architecture has been proposed so far achieving the best performance and at the same time being general enough to support any field degrees in this paper we propose a new representation based on what we called generalized polynomial bases gpbs covering polynomial bases and the so called shifted polynomial bases spbs as special cases in order to study the new representation we introduce a novel formulation for polynomial basis and its variants which is able to express concisely all implementation aspects of interest i e gate count subexpression sharing and time delay the methodology enabled by the new formulation is completely general and repetitive in its application allowing the development of an ad hoc software tool to derive proofs for area complexity and time delays automatically as the central contribution of this paper we introduce some new types of irreducible pentanomials and an associated gpb based on the above formulation we prove that carefully chosen gpbs yield multiplier architectures matching or even outperforming the best special type pentanomials from both the area and time point of view most importantly the proposed gpb architectures require pentanomials existing for all degrees of practical interest a list of suitable irreducible pentanomials for all degrees less than 1 000 is given in the appendix fig 5 and tables 4 11 are provided in a separate file containing the body of appendix which can be found on the computer society digital library at http doi ieeecomputersociety org 10 1109 tc 2012 63

improved three way split formulas for binary polynomial and toeplitz matrix vector products in this paper we consider three way split formulas for binary polynomial multiplication and toeplitz matrix vector product tmvp we first recall the best known three way split formulas for polynomial multiplication the formulas with six recursive multiplications given by sunar in a 2006 ieee transactions on computers paper and the formula with five recursive multiplications proposed by bernstein at crypto 2009 second we propose a new set of three way split formulas for polynomial multiplication that are an optimization of sunar s formulas then we present formulas with five recursive multiplications based on field extension in addition we extend the latter formulas to tmvp we evaluate the space and delay complexities when computations are performed in parallel and provide a comparison with best known methods

fast bit parallel gf 2 sup n multiplier for all trinomials based on a new representation of gf 2 sup n we present two multipliers for all irreducible trinomials space complexities of the multipliers match the best results the time complexity of one multiplier is t sub a 1 log sub 2 n t sub x for all irreducible trinomials where t sub a and t sub x are the delay of one 2 input and and xor gates respectively

mass conservation of the unified continuous and discontinuous element based galerkin methods on dynamically adaptive grids with application to atmospheric simulations we perform a comparison of mass conservation properties of the continuous cg and discontinuous dg galerkin methods on non conforming dynamically adaptive meshes for two atmospheric test cases the two methods are implemented in a unified way which allows for a direct comparison of the non conforming edge treatment we outline the implementation details of the non conforming direct stiffness summation algorithm for the cg method and show that the mass conservation error is similar to the dg method both methods conserve to machine precision regardless of the presence of the non conforming edges for lower order polynomials the cg method requires additional stabilization to run for very long simulation times we addressed this issue by using filters and or additional artificial viscosity the mathematical proof of mass conservation for cg with non conforming meshes is presented in appendix b

estimating the number of stable configurations for the generalized thomson problem given a natural number n one may ask what configuration of n points on the two sphere minimizes the discrete generalized coulomb energy if one applies a gradient based numerical optimization to this problem one encounters many configurations that are stable but not globally minimal this led the authors of this manuscript to the question how many stable configurations are there in this manuscript we report methods for identifying and counting observed stable configurations and estimating the actual number of stable configurations these estimates indicate that for n approaching two hundred there are at least tens of thousands of stable configurations

accurate and efficient floating point summation we present and analyze several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator let f and f be the number of significant bits in the summands and the accumulator respectively then assuming gradual underflow no overflow and round to nearest arithmetic up to approximately 2f f numbers can be added accurately by simply summing the terms in decreasing order of exponents yielding a sum correct to within about 1 5 units in the last place ulps we apply this result to the floating point formats in the ieee floating point standard for example a dot product of single precision vectors of length at most 33 computed using double precision and sorting is guaranteed correct to nearly 1 5 ulps if double extended precision is used the vector length can be as large as 65 537 we also investigate how the cost of sorting can be reduced or eliminated while retaining accuracy

accurate floating point summation a new approach the aim of this paper is to find an accurate and efficient algorithm for evaluating the summation of large sets of floating point numbers we present a new representation of the floating point number system in which a number is represented as a linear combination of integers and the coefficients are powers of the base of the floating point system the approach allows to build up an accurate floating point summation algorithm based on the fact that no rounding error occurs whenever two integer numbers are summed or a floating point number is multiplied by powers of the base of the floating point system the proposed algorithm seems to be competitive in terms of computational effort and under some assumptions the computed sum is greatly accurate with such assumptions less conservative in the practical applications we prove that the relative error of the computed sum is bounded by the unit roundoff

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

design and analysis of supervised and decision directed estimators of the mmse lcmv filter in data limited environments we consider sample matrix inversion smi type estimates of the minimum mean square error mmse and the linearly constrained minimum variance lcmv linear filters obtained from data records of limited size we quantify theoretically the detrimental effect of the desired signal energy level on the mean square ms filter estimation error and the normalized output signal to interference plus noise ratio sinr by deriving a new exact analytical expression and a lower bound respectively for cases where accumulation of pure disturbance observations is not possible we show theoretically how certain intuitive pilot assisted and decision directed adaptive filter implementations that utilize desired signal present data observations perform close to their desired signal absent counterparts simulation studies illustrate our theoretical developments in the context of spread spectrum communications over multipath fading channels under perfect and nonperfect synchronization

an lll algorithm with quadratic complexity the lenstra lenstra lovasz lattice basis reduction algorithm called lll or rm l 3 is a fundamental tool in computational number theory and theoretical computer science which can be viewed as an efficient algorithmic version of hermite s inequality on hermite s constant given an integer d dimensional lattice basis with vectors of euclidean norm less than b in an n dimensional space the rm l 3 algorithm outputs a reduced basis in o d 3n rm log b cdot mathcal m d rm log b bit operations where mathcal m k denotes the time required to multiply k bit integers this worst case complexity is problematic for applications where d or and rm log b are often large as a result the original rm l 3 algorithm is almost never used in practice except in tiny dimension instead one applies floating point variants where the long integer arithmetic required by gram schmidt orthogonalization is replaced by floating point arithmetic unfortunately this is known to be unstable in the worst case the usual floating point rm l 3 algorithm is not even guaranteed to terminate and the output basis may not be rm l 3 reduced at all in this article we introduce the rm l 2 algorithm a new and natural floating point variant of the rm l 3 algorithm which provably outputs rm l 3 reduced bases in polynomial time o d 2n d rm log b rm log b cdot mathcal m d this is the first rm l 3 algorithm whose running time without fast integer arithmetic provably grows only quadratically with respect to rm log b like euclid s gcd algorithm and lagrange s two dimensional algorithm

simulating low precision floating point arithmetic the half precision fp16 floating point format defined in the 2008 revision of the ieee standard for floating point arithmetic and a more recently proposed half precision format bfloat16 are in

adaptive convergence of linearly constrained beamformers based on the sample covariance matrix a statistical analysis of the adaptive convergence behavior of linearly constrained beamformers is given assuming the sample covariance estimator is used to estimate the covariance matrix the sensor data are assumed to be gaussian distributed and independent from data vector to data vector the output power and mean squared error in the absence of the desired signal are shown to be multiples of chi squared random variables the presence of the desired signal results in an excess mean squared error that is beta distributed and depends only on the signal power number of data vectors and number of adaptive degrees of freedom the expected value of the excess mean squared error resulting from the signal presence is directly proportional to the signal power and number of adaptive degrees of freedom but is inversely proportional to the number of data vectors

large system transient analysis of adaptive least squares filtering the performance of adaptive least squares ls filtering is analyzed for the suppression of multiple access interference both full rank ls filters and reduced rank ls filters which reside in a lower dimensional krylov space are considered with training and without training but with known signature for the desired user we compute the large system limit of output signal to interference plus noise ratio sinr as a function of normalized observations load and noise level specifically the number of users k the degrees of freedom n and the number of training symbols or observations i all tend to infinity with fixed ratios k n and i n our results account for an arbitrary power distribution over the users data windowing e g recursive ls rls with exponential windowing and initial diagonal loading of the covariance matrix to prevent ill conditioning numerical results show that the large system analysis accurately predicts the simulated convergence performance of the algorithms considered with moderate degrees of freedom typically n 32 given a fixed short training length the relative performance of full and reduced rank filters depends on the selected rank and diagonal loading with an optimized diagonal loading factor the performance of full and reduced rank filters are similar however full rank performance is generally much more sensitive to the choice of diagonal loading factor than reduced rank performance

numerical implementation of complex orthogonalization parallel transport on stiefel bundles and analyticity abstract numerical integration of complex linear systems of odes depending analytically on an eigenvalue parameter are considered complex orthogonalization which is required to stabilize the numerical integration results in non analytic systems it is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non analytic characteristic function the orthonormal systems are constructed using the geometry of stiefel bundles different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one form on the stiefel bundle for the numerical integration gauss legendre runge kutta algorithms are the principal choice for preserving orthogonality and performance results are shown for a range of glrk methods the theory and methods are tested by application to example boundary value problems including the orr sommerfeld equation in hydrodynamic stability

linear time approximation algorithms for computing numerical summation with provably small errors given a multiset x x1 xn of real numbers the floating point set summation problem asks for sn x1 xn let e n denote the minimum worst case error over all possible orderings of evaluating sn we prove that if x has both positive and negative numbers it is np hard to compute sn with the worst case error equal to e n we then give the first known polynomial time approximation algorithm that has a provably small error for arbitrary x our algorithm incurs a worst case error at most 2 lceil log n 1 rceil 1 e n all logarithms log in this paper are base 2 after x is sorted it runs in o n time for the case where x is either all positive or all negative we give another approximation algorithm with a worst case error at most lceil log log n rceil e n even for unsorted x this algorithm runs in o n time previously the best linear time approximation algorithm had a worst case error at most lceil log n rceil e n while e n was known to be attainable in o n log n time using huffman coding

efficient bit parallel multiplier for irreducible pentanomials using a shifted polynomial basis in this paper we present a bit parallel multiplier for gf 2m defined by an irreducible pentanomial xm xk 3 xk 2 xk 1 1 where 1lesk 1 lesk2 lesk3 lesm 2 in order to design an efficient bit parallel multiplier we introduce a shifted polynomial basis and modify a reduction matrix presented by reyhani masoleh and hasan as a result the time complexity of the proposed multiplier is ta 3 log2 m 1 tx where ta and tx are the delay of one and and one xor gate respectively this result matches or outperforms the previously known results on the other hand the proposed multiplier has the same space complexity as the previously known multipliers except for special types of irreducible pentanomials note that its hardware architecture is similar to that presented by reyhani masoleh and hasan

reducing round off errors in rigid body dynamics in several recent publications numerical integrators based on jacobi elliptic functions are proposed for solving the equations of motion of the rigid body although this approach yields theoretically the exact solution a standard implementation shows an unexpected linear propagation of round off errors we explain how deterministic error contribution can be avoided so that round off behaves like a random walk