an a posteriori parameter choice for tikhonov regularization in hilbert scales leading to optimal convergence rates an a posteriori parameter choice for tikhonov regularization in hilbert scales is proposed that leads to optimal convergence rates the advantage of this method is that knowledge of the smoothness parameters of both the operator and the exact solution is not needed the finite dimensional realization of this method is also discussed numerical examples confirm the theoretical results

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

remarks on marti s method for solving first kind equations a convergence theorem for marti s method for solving the fredholm first kind equation kf g is corrected additional convergence results are given marti s method is shown to be well posed under perturbations in g and under certain types of perturbations in k

high speed double precision computation of reciprocal division square root and inverse square root a new method for the high speed computation of double precision floating point reciprocal division square root and inverse square root operations is presented in this paper this method employs a second degree minimax polynomial approximation to obtain an accurate initial estimate of the reciprocal and the inverse square root values and then performs a modified goldschmidt iteration the high accuracy of the initial approximation allows us to obtain double precision results by computing a single goldschmidt iteration significantly reducing the latency of the algorithm two unfolded architectures are proposed the first one computing only reciprocal and division operations and the second one also including the computation of square root and inverse square root the execution times and area costs for both architectures are estimated and a comparison with other multiplicative based methods is presented the results of this comparison show the achievement of a lower latency than these methods with similar hardware requirements

algorithm and architecture for logarithm exponential and powering computation an architecture for the computation of logarithm exponential and powering operations is presented in this paper based on a high radix composite algorithm for the computation of the powering function x sup y the algorithm consists of a sequence of overlapped operations 1 digit recurrence logarithm 2 left to right carry free lrcf multiplication and 3 online exponential a redundant number system is used and the selection in 1 and 3 is done by rounding except from the first iteration when selection by table look up is necessary to guarantee the convergence of the recurrences a sequential implementation of the algorithm with a control unit which allows the independent computation of logarithm and exponential is proposed and the execution times and hardware requirements are estimated for single and double precision floating point computations these estimates are obtained for radices from r 8 to r 1 024 according to an approximate model for the delay and area of the main logic blocks and help determining the radix values which lead to the most efficient implementations r 32 and r 128

global optimality conditions for discrete and nonconvex optimizationwith applications to lagrangian heuristics and column generation the well known and established global optimality conditions based on the lagrangian formulation of an optimization problem are consistent if and only if the duality gap is zero we develop a set of global optimality conditions that are structurally similar but are consistent for any size of the duality gap this system characterizes a primaldual optimal solution by means of primal and dual feasibility primal lagrangian e optimality and in the presence of inequality constraints a relaxed complementarity condition analogously called complementarity the total size e of those two perturbations equals the size of the duality gap at an optimal solution further the characterization is equivalent to a near saddle point condition which generalizes the classic saddle point characterization of a primaldual optimal solution in convex programming the system developed can be used to explain to a large degree when and why lagrangian heuristics for discrete optimization are successful in reaching near optimal solutions further experiments on a set covering problem illustrate how the new optimality conditions can be utilized as a foundation for the construction of new lagrangian heuristics finally we outline possible uses of the optimality conditions in column generation algorithms and in the construction of core problems

minimum mean running time function generation using read only memory high speed hardware function generation using table look up in rom and high speed multiplication is considered the reduced interval of interest a b is split into several large partitions within each large partition the functionf x is evaluated by piecewise polynomials of the same low degree whose coefficients are stored in rom four basic architectures for such a scheme are considered a nonlinear programming problem is solved for determining the optimal partitioning of the interval a b the objective function is the average number of multiplications which takes into account the probability distribution r x 1 x ln for the mantissas of normalized floating point numbers where is the radix of the number system the constraint is the available number of rom words the particular case of f x 1 x and 2 is considered in detail and results are presented including an estimate of the number of rom units required

double residue modular range reduction for floating point hardware implementations in this paper we present a novel algorithm and the corresponding architecture for performing range reduction which is a preprocessing task required for the evaluation of some elementary functions such as trigonometric and exponential based functions the proposed algorithm introduces a modification to the modular range reduction algorithm which increases the speed of computation and allows us to design an architecture for the floating point case the implementation presented admits as an input argument any representable number of the standard single precision ieee 754 floating point representation and provides the maximum accuracy to the final result this supposes a hardware solution to the problem of having an input argument close to a multiple of the constant a final comparison with other implementations is presented

very high radix division with prescaling and selection by rounding a division algorithm in which the quotient digit selection is performed by rounding the shifted residual in carry save form is presented to allow the use of this simple function the divisor and dividend is prescaled to a range close to one the implementation presented results in a fast iteration because of the use of carry save forms and suitable recodings the execution time is calculated and several convenient values of the radix are selected comparison with other dividers for radices 2 sup 9 to 2 sup 18 is performed using the same assumptions

optimal solution of set covering partitioning problems using dual heuristics we present an algorithm for a mixed set covering partitioning model that includes as special cases the well known set covering problem and set partitioning problem the novel feature of our algorithm is the use of continuous heuristics applied to the dual of the linear programming relaxation to provide lower bounds for a branch and bound algorithm the heuristics are continuous adaptations of the well known greedy and 3 opt methods that have been applied to a variety of combinatorial optimization problems our algorithm has outperformed the current best set covering algorithm of balas and ho 1980 by about a factor of 3 and appears to improve on the best existing set partitioning algorithm by more than an order of magnitude

an efficient mean field approach to the set covering problem a mean field feedback artificial neural network ann algorithm is developed and explored for the set covering problem a convenient encoding of the inequality constraints is achieved by means of a multilinear penalty function an approximate energy minimum is obtained by iterating a set of mean field equations in combination with annealing the approach is numerically tested against a set of publicly available test problems with sizes ranging up to 5 103 rows and 106 columns when comparing the performance with exact results for sizes where these are available the approach yields results within a few percent from the optimal solutions comparisons with other approximate methods also come out well in particular given the very low cpu consumption required typically a few seconds arbitrary problems can be processed using the algorithm via a public domain server 2001 elsevier science b v less

iterative algorithm and architecture for exponential logarithm powering and root extraction an algorithm and architecture for powering computation and root extraction with fixed point and floating point exponents is presented in this paper the algorithm is based on an optimized iterative sequence of parallel and or overlapped operations 1 reciprocal 2 high radix digit recurrence logarithm 3 left to right carry free multiplication and 4 high radix online exponential a redundant number system is used to allow for the overlapping of the different operations of the algorithm as the logarithm and exponential are part of the sequence of operations some minor changes are made to allow for the independent computation of the logarithm and exponential functions a sequential implementation of the algorithm is proposed and the execution times and hardware requirements are estimated for single and double precision floating point computations these estimates are obtained for several radices according to an approximate model for the delay and area of the main logic blocks and help to determine the radix values which lead to the most efficient implementations

low weight polynomial form integers for efficient modular multiplication in 1999 solinas introduced families of moduli called the generalized mersenne numbers gmns which are expressed in low weight polynomial form p f t where t is limited to a power of 2 gmns are very useful in elliptic curve cryptosystems over prime fields since modular reduction by a gmn requires only integer additions and subtractions however since there are not many gmns and each gmn requires a dedicated implementation gmns are hardly useful for other cryptosystems here we modify gmn by removing restriction on the choice of t and restricting the coefficients of f t to 0 and plusmn1 we call such families of moduli low weight polynomial form integers lwpfis we show an efficient modular multiplication method using lwpfi moduli lwpfis allow general implementation and there exist many lwpfi moduli one may consider lwpfis as a trade off between general integers and gmns

a low error cost efficient design procedure for evaluating logarithms to be used in a logarithmic arithmetic processor based on an error flattened non uniform region linear approximation algorithm this brief proposes a low error and a cost efficient design procedure for realizing an optimized shift and add logarithmic unit lu which uses minimum hardware to meet the desired error constraint for embedded graphics systems mathematically this brief first derives two solutions of the error flattened algorithm subsequently for an error constraint the minimum number of approximation regions n the corresponding i th interpolation coefficients ai bi and the regional endpoints xi 1 xi 1 i n are obtained accordingly using the unique properties of the logarithmic function spacing of xi is non uniform to make errors in each region consistent carefully examining the cost of the applied add sub network a low cost candidate for a shift and add lu is determined next a cost exploration process which gradually increases n is performed a large number of regions results in a more accurate conversion algorithm that might tolerate more implementation errors by using simple hardware whose cost is lower than that of the inferior candidate after exploring the cost based on error tolerance the proposed design procedure finally generates a hardware to meet the desired error constraint slightly modifying the regional endpoint xi increases hardware efficiency at the cost of increased error proposed circuits were synthesized in umc 65rvt cmos technology compared to state of the art shift and add logarithmic converters simulation results reveal that the proposed design saves approximately 12 7 51 1 percent area of polynomial approximation and improves approximately 1 14 db snr gain while achieving a tighter error constraint

an electromagnetism metaheuristic for the unicost set covering problem in this paper we propose a new heuristic algorithm to solve the unicost version of the well known set covering problem the method is based on the electromagnetism metaheuristic approach which after generating a pool of solutions to create the initial population applies a fixed number of local search and movement iterations based on the electromagnetism theory in addition to some random aspects used in the construction and local search phases we also apply mutation in order to further escape from local optima the proposed algorithm has been tested over 80 instances of the literature on the classical benchmark instances where the number of columns is larger than the number of rows the algorithm by using a fixed set of parameters always found the best known solution and for 12 instances it was able to improve the current best solution by using different parameter settings the algorithm improved 4 additional best known solutions moreover we proved the effectiveness of the electromagnetism metaheuristic approach for the unicost set covering problem by embedding the procedures of the proposed algorithm in a genetic algorithm scheme the worse results obtained by the genetic algorithm show the impact of the electromagnetism metaheuristic approach in conducting the search of the solution space by applying the movements based on the electromagnetism theory finally we report the results obtained by modifying the proposed electromagnetism metaheuristic algorithm for solving the non unicost set covering problem

an effective and simple heuristic for the set covering problem abstract this paper investigates the development of an effective heuristic to solve the set covering problem scp by applying the meta heuristic meta raps meta heuristic for randomized priority search in meta raps a feasible solution is generated by introducing random factors into a construction method then the feasible solutions can be improved by an improvement heuristic in addition to applying the basic meta raps the heuristic developed herein integrates the elements of randomizing the selection of priority rules penalizing the worst columns when the searching space is highly condensed and defining the core problem to speedup the algorithm this heuristic has been tested on 80 scp instances from the or library the sizes of the problems are up to 1000 rows 10 000 columns for non unicost scp and 28 160 rows 11 264 columns for the unicost scp this heuristic is only one of two known scp heuristics to find all optimal best known solutions for those non unicost instances in addition this heuristic is the best for unicost problems among the heuristics in terms of solution quality furthermore evolving from a simple greedy heuristic it is simple and easy to code this heuristic enriches the options of practitioners in the optimization area

the multicovering problem abstract the multicovering problem is min cx subject to ax b x 0 where a is a matrix whose elements are all zero or one and b is a vector of positive integers this problem has many applications in scheduling location and business areas a variety of industrial commercial and military contexts give rise to large problems of this kind frequently at or beyond the limits of current computational tractability our research goal is the development of a special purpose algorithm capable of solving large problems of this kind conceptual and computational inputs to the algorithm arise from three sources from the generalization of computational tools developed for more specific problems in particular the set covering problem from the adaptation of general integer programming and combinatorial optimization methods and from the development within this paper of special techniques designed to take advantage of the structure of the problem studied following a description of the algorithm we give some computational results for randomly generated problems

reciprocation square root inverse square root and some elementary functions using small multipliers this paper deals with the computation of reciprocals square roots inverse square roots and some elementary functions using small tables small multipliers and for some functions a final large almost full length multiplication we propose a method based on argument reduction and series expansion that allows fast evaluation of these functions in high precision the strength of this method is that the same scheme allows the computation of all these functions we estimate the delay the size number of tables and the size number of multipliers and compare with other related methods

low power unsigned divider and square root circuit designs using adaptive approximation in this paper an adaptive approximation approach is proposed for the design of a divider and a square root sqr circuit in this design the division sqr is computed by using a reduced width divider sqr circuit and a shifter by adaptively pruning some insignificant input bits specifically for a 2n n 2n n division 2k 2k and k k k n k n consecutive bits are selected starting from the most significant 1 in the dividend and divisor respectively at the same time redundant least significant bits lsbs are truncated or if the number of remaining bits after pruning is smaller than the number of bits to be kept 0 s are appended to the lsbs of the inputs to avoid overflow a 2 k 1 k 1 2 k 1 k 1 divider is used to compute the 2k k 2k k division finally an error correction circuit is proposed to recover the error caused by the shifter using or gates for a 2n 2n bit approximate sqr circuit similar pruning schemes are used to obtain a 2k 2k bit radicand a 2k 2k bit sqr circuit and a shifter are then utilized to compute the sqr this adaptive operation leads to very small maximum error distances of the approximate divider and sqr circuits as shown by a theoretical error analysis the proposed 16 8 approximate divider using an 8 4 exact array divider is 2 5 times 2 5 as fast but only consumes 34 42 percent of the power of the accurate design compared to the accurate 16 bit array sqr circuit the approximate design with a 6 bit radicand is 3 9 times 3 9 as fast and consumes 20 66 percent of the power the approximate sqr circuit using a 6 bit lookup table based sqr circuit consumes 7 15 percent of the power of its corresponding accurate design the proposed designs outperform other approximate designs in image processing applications including change detection for the divider envelope detection for the sqr circuit and image reconstruction for both designs

the trade off between regularity and stability in tikhonov regularization when deriving rates of convergence for the approximations generated by the application of tikhonov regularization to ill posed operator equations assumptions must be made about the nature of the stabilization i e the choice of the seminorm in the tikhonov regularization and the regularity of the least squares solutions which one looks for in fact it is clear from works of hegland engl and neubauer and natterer that in terms of the rate of convergence there is a trade off between stabilization and regularity it is this matter which is examined in this paper by means of the best possible worst error estimates the results of this paper provide better estimates than those of engl and neubauer and also include and extend the best possible rate derived by natterer the paper concludes with an application of these results to first kind integral equations with smooth kernels

a tabu search approach to the constraint satisfaction problem as a general problem solver many combinatorial problems including a variety of combinatorial optimization problems can be naturally formulated as a constraint satisfaction problem csp we develop in this paper a tabu search based algorithm for the csp as a foundation for a general problem solver in addition to the basic components of tabu search we develop a number of elaborations such as an automatic control mechanism for the tabu tenure modification of the penalty function to handle objective functions and enlargement of the neighborhood by allowing swap operations computational results with our algorithm are reported for various problems selected from a wide range of applications i e graph coloring generalized assignment set covering timetabling and nurse scheduling our results appear to be competitive with those of existing algorithms specially developed for the respective problem domains

five six and seven term karatsuba like formulae the karatsuba ofman algorithm starts with a way to multiply two 2 term i e linear polynomials using three scalar multiplications there is also a way to multiply two 3 term i e quadratic polynomials using six scalar multiplications these are used within recursive constructions to multiply two higher degree polynomials in subquadratic time we present division free formulae which multiply two 5 term polynomials with 13 scalar multiplications two 6 term polynomials with 17 scalar multiplications and two 7 term polynomials with 22 scalar multiplications these formulae may be mixed with the 2 term and 3 term formulae within recursive constructions leading to improved bounds for many other degrees using only the 6 term formula leads to better asymptotic performance than standard karatsuba the new formulae work in any characteristic but simplify in characteristic 2 we describe their application to elliptic curve arithmetic over binary fields we include some timing data

a fast segmentation algorithm for piecewise polynomial numeric function generators we give an efficient algorithm for partitioning the domain of a numeric function f into segments the function f is realized as a polynomial in each segment and a lookup table stores the coefficients of the polynomial such an algorithm is an essential part of the design of lookup table methods ercepovac et al 2000 5 lee et al 2003 7 nagayama et al 2007 12 paul et al 2007 6 and sasao et al 2004 8 for realizing numeric functions such as sin px ln x and ln x our algorithm requires many fewer steps than a previous algorithm given in frenzen et al 2010 10 and makes tractable the design of numeric function generators based on table lookup for high accuracy applications we show that an estimate of segment width based on local derivatives greatly reduces the search needed to determine the exact segment width we apply the new algorithm to a suite of 15 numeric functions and show that the estimates are sufficiently accurate to produce a minimum or near minimum number of computational steps

a method for solving ill posed linear operator equations an algorithm is presented to compute least squares solutions of the linear equation tx y where t is a bounded pperator from one hilbert space into another the algorithm is a generalization of marti s method the convergence to a solution minimizing the seminorm lx is proved where the operator l is a closed densely defined linear operator with closed range the method is applied to integral equations and three examples are treated numerically

minimax designs for finite design regions the problem of choosing a design that is representative of a finite candidate set is an important problem in computer experiments the minimax criterion measures the degree of representativeness because it is the maximum distance of a candidate point to the design this article proposes a method for finding minimax designs for finite design regions we establish the relationship between minimax designs and the classical set covering location problem sclp in operations research which is a binary linear program in particular we prove that the set of minimax distances is the set of discontinuities of the function that maps the covering radius to the optimal objective function value we show that solving the sclp at the points of discontinuities which can be determined gives minimax designs these results are employed to design an efficient procedure for finding minimax designs for small sized candidate sets a heuristic procedure is proposed to generate near minimax designs for large candidate sets s

power and area efficient unified computation of vector and elementary functions for handheld 3d graphics systems a unified computation method of vector and elementary functions is proposed for handheld 3d graphics systems it unifies vector operations like vector multiply multiply and add divide divide by square root and dot product and elementary functions like trigonometric inverse trigonometric hyperbolic inverse hyperbolic power xy with two variables and logarithm to an arbitrary base into a single four way arithmetic platform a number system called the fixed point hybrid number system fxp hns which combines the fixed point number system fxp and the logarithmic number system lns is proposed for the power and area efficient unification power and area efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed point and logarithmic numbers in the fxp hns and achieve 0 41 percent and 0 08 percent maximum conversion errors respectively the unified arithmetic unit based on the proposed schemes is presented with less than 6 3 percent operation error its fully pipelined architecture achieves single cycle throughput with maximum four cycle latency for all of the supported operations comparison results show that the proposed arithmetic unit achieves 30 percent power and 10 9 percent area reductions and runs two times faster than the previous approach

a heuristic method for the set covering problem we present a lagrangian based heuristic for the well known set covering problem scp the algorithm was initially designed for solving very large scale scp instances involving up to 5 000 rows an

uniform convergence of regularization methods for linear ill posed problems abstract after a general discussion about convergence and convergence rates for regularization methods in banach spaces we present a general method that can be used to modify regularization methods in l2 in such a way that uniform convergence which is often preferred in concrete applications to just l2 convergence is obtained we prove results about convergence rates in the uniform norm and discuss questions of parameter choice the theoretical results will be supported by numerical examples which indicate that although the results are asymptotic in character they are of some relevance also for actual computations

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

a comparison of lagrangean and surrogate relaxations for the maximal covering location problem abstract we compare heuristics based on lagrangean and surrogate relaxations of the maximal covering location problem mclp the lagrangean relaxation of mclp used in this paper has the integrality property and the surrogate relaxed problem we solve is the lp relaxation of the original 0 1 knapsack problem the heuristics were compared using 331 test problems available in the literature corresponding to networks ranging from 55 to 900 vertices the gaps obtained with both heuristics were very low and did not differ substantially among themselves for the several problem sets used in accordance with theoretical results reviewed in the paper when the initial set of multipliers was determined using a valid bound for mclp the computing times did not differ significantly between the lagrangean and surrogate heuristics

hardware designs for exactly rounded elementary functions this paper presents hardware designs that produce exactly rounded results for the functions of reciprocal square root 2 sup x and log sub 2 x these designs use polynomial approximation in which the terms in the approximation are generated in parallel and then summed by using a multi operand adder to reduce the number of terms in the approximation the input interval is partitioned into subintervals of equal size and different coefficients are used for each subinterval the coefficients used in the approximation are initially determined based on the chebyshev series approximation they are then adjusted to obtain exactly rounded results for all inputs hardware designs are presented and delay and area comparisons are made based on the degree of the approximating polynomial and the accuracy of the final result for single precision floating point numbers a design that produces exactly rounded results for all four functions has an estimated delay of 80 ns and a total chip area of 98 mm sup 2 in a 1 0 micron cmos technology allowing the results to have a maximum error of one unit in the last place reduces the computational delay by 5 to 30 and the area requirements by 33 to 77

complex variable and regularization methods of inversion of the laplace transform in this paper three methods are derived for approximating f given its laplace transform g on 0 oo i e fo f t exp st dt g s assuming that g e l 0 oo the first method is based on a sinc like rational approximation of 9 the second on a sinc solution of the integral equation fo0 f t exp st dt g s via standard regularization and the third method is based on first converting fo0 f t exp st dt g s to a convolution integral over r and then finding a sinc approximation to f via the application of a special regularization procedure to solve the fourier transform problem we also obtain bounds on the error of approximation which depend on both the method of approximation and the regularization parameter

fast hardware based algorithms for elementary function computations using rectangular multipliers as the name suggests elementary functions play a vital role in scientific computations yet due to their inherent nature they are a considerable computing task by themselves not surprisingly since the dawn of computing the goal of speeding up elementary function computation has been pursued this paper describes new hardware based algorithms for the computation of the common elementary functions namely division logarithm reciprocal square root arc tangent sine and cosine these algorithms exploit microscopic parallelism using specialized hardware with heavy use of truncation based on detailed accuracy analysis the contribution of this work lies in the fact that these algorithms are very fast and yet are accurate if we let the time to perform an ieee standard 754 double precision floating point multiplication be spl tau sub spl times our algorithms to achieve roughly 3 68 spl tau sub spl times 4 56 spl tau sub spl times 5 25 spl tau sub spl times 3 69 spl tau sub spl times 7 06 spl tau sub spl times and 6 5 spl tau sub spl times for division logarithm square root exponential are tangent and complex exponential sine and cosine respectively the trade off is the need for tables and some specialized hardware the total amount of tables required however is less than 128 kbytes we discuss the hardware algorithmic and accuracy aspects of these algorithms

a new range reduction algorithm range reduction is a key point for getting accurate elementary function routines we introduce a new algorithm that is fast for input arguments belonging to the most common domains yet accurate over the full double precision range

an efficient local search heuristic with row weighting for the unicost set covering problem abstract the set covering problem scp is np hard we propose a new row weighting local search rwls algorithm for solving the unicost variant of the scp i e uscps where the costs of all sets are identical rwls is a heuristic algorithm that has three major components united in its local search framework 1 a weighting scheme which updates the weights of uncovered elements to prevent convergence to local optima 2 tabu strategies to avoid possible cycles during the search and 3 a timestamp method to break ties when prioritizing sets rwls has been evaluated on a large number of problem instances from the or library and compared with other approaches it is able to find all the best known solutions bks and improve 14 of them although requiring a higher computational effort on several instances rwls is especially effective on the combinatorial or library instances and can improve the best known solution to the hardest instance cyc11 considerably rwls is conceptually simple and has no instance dependent parameters which makes it a practical and easy to use uscp solver

on newton raphson iteration for multiplicative inverses modulo prime powers we study algorithms for the fast computation of modular inverses newton raphson iteration over p adic numbers gives a recurrence relation computing modular inverse modulo p m that is logarithmic in m we solve the recurrence to obtain an explicit formula for the inverse then we study different implementation variants of this iteration and show that our explicit formula is interesting for small exponent values but slower or large exponent say of more than 700 bits overall we thus propose a hybrid combination of our explicit formula and the best asymptotic variants this hybrid combination yields then a constant factor improvement also for large exponents

an algorithm for multiple precision floating point multiplication we present an algorithm for multiple precision floating point multiplication the conventional algorithms based on the fast fourier transform fft multiply two n bit numbers to obtain a 2n bit result in multiple precision floating point multiplication we need only the returned result whose precision is equal to the multiple precision floating point number we show that the overall arithmetic operations for fft based multiple precision floating point multiplication are reduced by decomposition of the full length multiplication into shorter length multiplication

improved handling of uncertainty and robustness in set covering problems abstract set covering problems are one of the classical combinatorial optimization problems and part of many different practical applications e g the location of emergency service facilities in many real world problems the required statistical parameters are not precisely known and the obtained solutions may reveal a non adequate performance we introduce a robust formulation of the uncertain probabilistic set covering problem which combines the concepts of robust and probabilistic optimization by introducing robust covering constraints this robust uncertain set covering problem can be stated as a compact mixed integer linear programming model additionally two non compact integer linear model formulations are developed as the strength of these formulations is not known a priori we analyze the performance of these formulations in an extensive computational study a case study for the location of emergency service facilities highlights the benefits of our approach in comparison to a formulation neglecting these uncertainties

on computational efficiency for multi precision zero finding methods an adapted definition related to multi precision arithmetic zero finding methods is considered and several improvements to iterative methods to compute nonlinear equation solutions which increase the local order of convergence are revisited furthermore a higher efficiency when we use a definition related to adaptive multi precision arithmetic is also given numerical results computed with a floating point system representing 200 and 1000 decimal digits support this theory

fast evaluation of the elementary functions in single precision in this paper we introduce a new method for the fast evaluation of the elementary functions in single precision based on the evaluation of truncated taylor series using a difference method we assume the availability of large and fast at least for read purposes memory we call this method the ata add table lookup add method as the name implies the hardware required for the method are adders both two and multi operand adders and fast tables for ieee single precision numbers our initial estimates indicate that we can calculate the basic elementary functions namely reciprocal square root logarithm exponential trigonometric and inverse trigonometric functions within the latency of two to four floating point multiplies

graph theoretic relaxations of set covering and set partitioning problems in this paper we review the graph theoretic relaxations of the set covering problem scp and the set partitioning problem ssp these are a network relaxation which can be solved by the greedy method a matching relaxation and a graph covering relaxation other relaxations such as assignment shortest route and minimum spanning tree are also presented

a low latency pipelined 2d and 3d cordic processors the unfolded and pipelined cordic is a high performance hardware element that produces a wide variety of one and two argument functions with high throughput the reduction in delay power and area cost are of significant interest regarding this module due to its high demand for resources the linear approximation to rotation has been proposed to achieve such reductions however the schemes for rotation multiplication and vectoring division complicate the implementation in a single unit in this work we improve the linear approximation scheme leading to a unified implementation for rotation and vectoring where fully parallel tree multipliers are used instead of the second half of cordic iterations we also combine the linear approximation to rotation with the scale factor compensation so that the compensation is concurrently performed with the rotation process we then extend the method to 3d cordic such an extension is not straightforward due to the lack of existing analytical expressions for the convergence of the algorithm a comparison using a rough area time model and synthesis results shows that our proposals may achieve significant reductions in delay with no increase in area in actual implementations

a genetic algorithm for the set covering problem abstract in this paper we present a genetic algorithm based heuristic for non unicost set covering problems we propose several modifications to the basic genetic procedures including a new fitness based crossover operator fusion a variable mutation rate and a heuristic feasibility operator tailored specifically for the set covering problem the performance of our algorithm was evaluated on a large set of randomly generated problems computational results showed that the genetic algorithm based heuristic is capable of producing high quality solutions

a unified architecture for the accurate and high throughput implementation of six key elementary functions this paper presents a unified architecture for the compact implementation of several key elementary functions including reciprocal square root and logarithm in single precision floating point arithmetic the proposed high throughput design is based on uniform domain segmentation and curve fitting techniques numerically accurate least squares regression is utilized to calculate the polynomial coefficients the architecture is optimized by analyzing the trade off between the size of the required memory and the precision of intermediate variables to achieve the minimum 23 bit accuracy required for single precision floating point representation the efficiency of the proposed unified data path is demonstrated on a common field programmable gate array

toward correctly rounded transcendentals the table maker s dilemma is the problem of always getting correctly rounded results when computing the elementary functions after a brief presentation of this problem we present new developments that have helped us to solve this problem for the double precision exponential function in a small domain these new results show that this problem can be solved at least for the double precision format for the most usual functions

computational experience with approximation algorithms for the set covering problem the set covering problem scp is a well known combinatorial optimization problem which is np hard we conducted a comparative study of eight different approximation algorithms for the scp including several greedy variants fractional relaxations randomized algorithms and a neural network algorithm the algorithms were tested on a set of random generated problems with up to 500 rows and 5000 columns and on two sets of problems originating in combinatorial questions with up to 28160 rows and 11264 columns on the random problems and on one set of combinatorial problems the best algorithm among those we tested was the neural network algorithm with greedy variants very close in second and third place on the other set of combinatorial problems the best algorithm was a greedy variant and the neural network performed quite poorly the other algorithms we tested were always inferior to the ones mentioned above

radix 4 reciprocal square root and its combination with division and square root in this work we present a reciprocal square root algorithm by digit recurrence and selection by a staircase function and the radix 4 implementation as in similar algorithms for division and square root the results are obtained correctly rounded in a straightforward manner in contrast to existing methods to compute the reciprocal square root although apparently a single selection function can only be used for j spl ges 2 the selection constants are different for j 0 j 1 and j spl ges 2 we show that it is possible to use a single selection function for all iterations we perform a rough comparison with existing methods and we conclude that our implementation is a low hardware complexity solution with moderate latency especially for exactly rounded results we also extend the unit to support division and square root with the same selection function and with slight modifications in the initialization of the reciprocal square root unit

approximation algorithms for the set covering and vertex cover problems we propose a heuristic that delivers in o n 3 steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value

elementary functions hardware implementation using constrained piecewise polynomial approximations a novel technique for designing piecewise polynomial interpolators for hardware implementation of elementary functions is investigated in this paper in the proposed approach the interval where the function is approximated is subdivided in equal length segments and two adjacent segments are grouped in a segment pair suitable constraints are then imposed between the coefficients of the two interpolating polynomials in each segment pair this allows reducing the total number of stored coefficients it is found that the increase in the approximation error due to constraints between polynomial coefficients can easily be overcome by increasing the fractional bits of the coefficients overall compared with standard unconstrained piecewise polynomial approximation having the same accuracy the proposed method results in a considerable advantage in terms of the size of the lookup table needed to store polynomial coefficients the calculus of the coefficients of constrained polynomials and the optimization of coefficients bit width is also investigated in this paper results for several elementary functions and target precision ranging from 12 to 42 bits are presented the paper also presents vlsi implementation results targeting a 90 nm cmos technology and using both direct and horner architectures for constrained degree 1 degree 2 and degree 3 approximations

on girth conditioning for low density parity check codes low density parity check ldpc codes are gaining interest for high data rate applications in both terrestrial and spatial communications they can be designed and studied through a bipartite graph whose characteristics affect the performance this paper proposes a low complexity method to improve the performance of ldpc codes by selectively removing some cycles from the associated bipartite graph the method is based on a modified version of the breadth first search bfs algorithm that we call modified bfs mbfs which is applied to find cycles and a greedy procedure to eliminate them throughout the paper we will give a detailed description of the algorithm proposed and analytically study its complexity simulation results show that this girth conditioning method applied to some classes of codes whose structure allows further optimization can lead to a significative complexity reduction and a performance improvements with respect to other methods

an algorithm for set covering problem in this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent subgradient optimisation and linear programming computational results are presented for problems involving up to 400 rows and 4000 columns

a dynamic subgradient based branch and bound procedure for set covering we discuss a branch and bound algorithm for set covering whose centerpiece is a new integrated upper bounding lower bounding procedure called dynamic subgradient optimization dynsgrad this new procedure applied to a lagrangean dual at every node of the search tree combines the standard subgradient method with primal and dual heuristics that interact to change the lagrange multipliers and tighten the upper and lower bounds fix variables and periodically restate the lagrangean itself extensive computational testing is reported as a stand alone heuristic dynsgrad performs significantly better than other procedures in terms of the quality of solutions obtainable with a certain computational effort when incorporated into a branch and bound algorithm dynsgrad considerably advances the state of the art in solving set covering problems

a low complexity lut based squaring algorithm the computation of large modular multi exponentiation is a time consuming arithmetic operation used in cryptography the standard squaring algorithm is well known and the guajardo paar algorithm fixes the improper carry handling bug produced by the standard squaring algorithm but produces error indexing bug in this paper a novel squaring algorithm is proposed which stores base products in the look up table before the squaring computation and base size comparison method the proposed algorithm can not only evaluate squaring efficiently but also avoid bugs produced in other proposed algorithms the guajardo paar algorithm and the yang heih laih algorithm the performance of the proposed algorithm is 1 615 times faster than the standard squaring algorithm and much faster than other algorithms

estimation of two phase petroleum reservoir properties by regularization abstract an algorithm is developed based on the theory of regularization and on spline approximation to estimate the absolute permeability in two phase petroleum reservoirs from noisy well pressure data the regularization feature of the algorithm converts the ill posed estimation problem to a well posed one the algorithm which employs the partial conjugate gradient method of nazareth as its core minimization technique automatically chooses the regularization parameter based on the non regularized estimation it is shown that regularized estimation is more stable and insensitive to the choice of initial guess as compared to non regularized conventional estimation

an adaptation of sh heuristic to the location set covering problem abstract in a recent paper a new surrogate heuristic sh has been proposed for the set covering problem here we present an adaptation of it in order to solve more efficiently the location set covering problem we will show that our new version not only outperforms algorithm sh but that it is more accurate than the pair cma fmc its power is experimentally tested over a set of 65 randomly generated problems

an optimal order regularization method which does not use additional smoothness assumptions this paper defines an optimal method to reconstruct the solutions of operator equations of the first kind only the case of compact operators is considered the method is in principle a discrepancy method it does not require any additional knowledge about the solution and is optimal for all standard smoothness assumptions in order to analyze the properties of the new regularization method variable hilbert scales are introduced and several well known results for hilbert scales are generalized convergence theorems for classes of optimal and suboptimal methods are derived from a generalization of the interpolation inequality

an interpolating memory unit for function evaluation analysis and design a technique for the evaluation of a general continuous function f x is presented and the design of an interpolating memory as an implementation of the technique is described the technique partitions the domain of f x into segments and defines an interpolating or approximating function for each the implementation is a memory subsystem that holds the parameters of the approximating functions and yields an interpolated function value on each read reference polynomial interpolating functions are considered in particular hardware requirements memory and computational logic are analyzed in terms of the required precision it is shown that as long as f x has d 1 derivatives where d is the degree of the interpolating polynomial d 1 additional bits of precision of the computed f x are obtained for each additional address bit used in the interpolating memory this establishes a tradeoff between memory and computational logic which can be exploited in the design of a unit for a specific function for any precision requirement furthermore a single unit can be designed for any class of functions that have the required derivatives two examples of implementations for particular functions are presented

identification of parabolic distributed parameter systems by regularization with differential operators abstract the theory of identification of variable coefficients in parabolic distributed parameter systems by regularization is extended to the case in which the stabilizing functional is the norm of a differential operator stability and convergence of the method are proved a detailed computational evaluation of the approach is presented via the estimation of a spatially varying conductivity in the two dimensional heat equation

surrogate constraint normalization for the set covering problem the set covering problem scp is central in a wide variety of practical applications for which finding good feasible solutions quickly often in real time is crucial surrogate constraint normalization is a classical technique used to derive appropriate weights for surrogate constraint relaxations in mathematical programming this framework remains the core of the most effective constructive heuristics for the solution of the scp chiefly represented by the widely used chvatal method this paper introduces a number of normalization rules and demonstrates their superiority to the classical chvatal rule especially when solving large scale and real world instances directions for new advances on the creation of more elaborate normalization rules for surrogate heuristics are also provided

an a posteriori parameter choice for ordinary and iterated tikhonov regularization of ill posed problems leading to optimal convergence rates on propose un choix a posteriori du parametre pour la regularisation de tikhonov ordinaire et iteree des problemes mal poses conduisant des vitesses de convergence optimales

area efficient multipliers based on multiple radix representations in this paper we shall introduce several new algorithms for integer multiplication that are based on specific multiple radix representation of one of the multiplicands we provide extensive theoretical analysis and experimental results for multipliers based on the new representations on 0 18 m cmos technology they provide a clear picture about the advantages of the new method in 64 bit hardware implementations compared to array based classical multiplier and radix 8 based multiplier the proposed multipliers have better area and power consumption compared to reference multipliers

covering problems and core percolations on hypergraphs we introduce two generalizations of core percolation in graphs to hypergraphs related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs respectively we offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments we find that for several real world hypergraphs their two cores tend to be much smaller than those of their null models suggesting that covering problems in those real world hypergraphs can actually be solved in polynomial time

improved decimal floating point logarithmic converter based on selection by rounding this paper presents the algorithm and architecture of the decimal floating point dfp logarithmic converter based on the digit recurrence algorithm with selection by rounding the proposed approach can compute faithful dfp logarithm results for any one of the three dfp formats specified in the ieee 754 2008 standard in order to optimize the latency for the proposed design we mainly integrate the following novel features 1 using the redundant carry save representation of the data path 2 reducing the number of iterations by determining the number of initial iteration and 3 retiming and balancing the delay of the proposed architecture the proposed architecture is synthesized with stm 90 nm standard cell library and the results show that the critical path delay and the number of clock cycles of the proposed decimal64 logarithmic converter are 1 55 ns 34 4 fo4 and 19 respectively and the total hardware complexity is 43 572 nand2 gates the delay estimation results of the proposed architecture show that its latency is close to that of the binary radix 16 logarithmic converter and that it has a significant decrease on latency compared with a recently published high performance cordic implementation

hardware implementation trade offs of polynomial approximations and interpolations this paper examines the hardware implementation trade offs when evaluating functions via piecewise polynomial approximations and interpolations for precisions of up to 24 bits in polynomial approximations polynomials are evaluated using stored coefficients polynomial interpolations however require the coefficients to be computed on the fly by using stored function values although it is known that interpolations require less memory than approximations but at the expense of additional computations the trade offs in memory area delay and power consumption between the two approaches have not been examined in detail this work quantitatively analyzes these trade offs for optimized approximations and interpolations across different functions and target precisions hardware architectures for degree 1 and degree 2 approximations and interpolations are described the results show that the extent of memory savings realized by using interpolation is significantly lower than what is commonly believed furthermore experimental results on a field programmable gate array fpga show that for high output precision degree 1 interpolations offer considerable area and power savings over degree 1 approximations but similar savings are not realized when degree 2 interpolations and approximations are compared the availability of both interpolation based and approximation based designs offers a richer set of design trade offs than what is available using either interpolation or approximation alone

high speed function approximation using a minimax quadratic interpolator a table based method for high speed function approximation in single precision floating point format is presented in this paper our focus is the approximation of reciprocal square root square root reciprocal exponentials logarithms trigonometric functions powering with a fixed exponent p or special functions the algorithm presented here combines table look up an enhanced minimax quadratic approximation and an efficient evaluation of the second degree polynomial using a specialized squaring unit redundant arithmetic and multioperand addition the execution times and area costs of an architecture implementing our method are estimated showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second degree interpolation algorithms moreover the use of an enhanced minimax approximation which through an iterative process takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look up tables to be used making our method very suitable for the implementation of an elementary function generator in state of the art dsps or graphics processing units gpus

optimized custom precision function evaluation for embedded processors fixed point processors are utilized in an enormous variety of applications often for tasks that require the evaluation of mathematical functions we present an automated method for mapping functions to such processors via polynomials that explicitly targets the native word length of the processor thereby significantly reducing the execution time relative to commonly used floating point emulation approaches based on traditional mathematical libraries the methods presented here also contrast with hand tuned processor specific code which has the potential to deliver efficient implementations but at the cost of significant design time we describe an automated design flow utilizing multi word arithmetic to provide overflow protection and precision accurate to one unit in the last place ulp analytical approaches are used to minimize the number of fixed width operands required for each operation and to ensure that precision requirements are met this allows automated generation of processor optimized code and characterization of a design space representing a rich range of tradeoffs among precision latency and memory cost

identifying redundant constraints and implicit equalities in systems of linear constraints redundant constraints are constraints that can be omitted from a system of linear constraints without changing the feasible region implicit equalities are inequality constraints that can be replaced by equalities without changing the feasible region we prove some theorems concerning the identification of both kinds of constraints based on these theorems two methods are developed that allow identification of all redundant constraints and all implicit equalities computational experience on small problems indicates a behavior at least competitive with other methods in this field

finite dimensional approximation of constrained tikhonov regularized solutions of ill posed linear operator equations in this paper we derive conditions under which the finite dimensional constrained tikhonov regularized solutions x c of an ill posed linear operator equation tx y i e xac is the minimizing element of the functional iitx y112 alix112 a 0 in the closed convex set c which is a finite dimensional approximation of a closed convex set c converge to the best approximate solution of the equation in c moreover we develop an estimate for the approximation error which is optimal for certain sets c and c we present numerical results that verify the theoretical results

algorithm architecture and floating point unit codesign of a matrix factorization accelerator this paper examines the mapping of algorithms encountered when solving dense linear systems and linear least squares problems to a custom linear algebra processor specifically the focus is on cholesky lu with partial pivoting and qr factorizations and their blocked algorithms as part of the study we expose the benefits of redesigning floating point units and their surrounding data paths to support these complicated operations we show how adding moderate complexity to the architecture greatly alleviates complexities in the algorithm we study design tradeoffs and the effectiveness of architectural modifications to demonstrate that we can improve power and performance efficiency to a level that can otherwise only be expected of full custom asic designs a feasibility study of inner kernels is extended to blocked level and shows that at block level the linear algebra core lac can achieve high efficiencies with up to 45 gflops w for both cholesky and lu factorization and over 35 gflops w for qr factorization while maintaining such efficiencies our extensions to the mac units can achieve up to 10 12 and 20 percent speedup for the blocked algorithms of cholesky lu and qr factorization respectively

an improved version of marti s method for solving ill posed linear integral equations we propose an algorithm for solving linear integral equations of the first kind that can be viewed as a variant of marti s method as opposed to that method our algorithm leads to optimal convergence rates also with noisy data 1 introduction throughout this paper let x and y be real hilbert spaces t x y a bounded linear operator with nonclosed range then the problem of determining the best approximate solution of 1 1 tx y is ill posed the best approximate solution exists only for y e d tt r t r t which we assume from now on and depends discontinuously on the right hand side here tt is the moore penrose inverse of t see 12 the best approximate solution is defined as the element of minimal norm that minimizes the residual tx yll and can be written as tty a prominent example for the ill posed

finite element approach to clustering of multidimensional time series we present a new approach to clustering of time series based on a minimization of the averaged clustering functional the proposed functional describes the mean distance between observation data and its representation in terms of mathbf k abstract models of a certain predefined class not necessarily given by some probability distribution for a fixed time series x t this functional depends on mathbf k sets of model parameters theta theta 1 dots theta mathbf k and mathbf k functions of cluster affiliations gamma gamma 1 t dots gamma mathbf k t characterizing the affiliation of any element x t of the analyzed time series to one of the mathbf k clusters defined by the considered model parameters we demonstrate that for a fixed set of model parameters theta the appropriate tykhonov type regularization of this functional with some regularization factor epsilon 2 results in a minimization problem similar to a variational problem usually associated with one dimensional nonhomogeneous partial differential equations this analogy allows us to apply the finite element framework to the problem of time series analysis and to propose a numerical scheme for time series clustering we investigate the conditions under which the proposed scheme allows a monotone improvement of the initial parameter guess with respect to the minimization of the discretized version of the regularized functional we also discuss the interpretation of the regularization factor in the markovian case and show its connection to metastability and exit times the computational performance of the resulting method is investigated numerically on multidimensional test data and is applied to the analysis of multidimensional historical stock market data

more on squaring and multiplying large integers methods of squaring and multiplying large integers are discussed the obvious o n sup 2 methods turn out to be best for small numbers existing o n sup log sup 3 log sup 2 spl ap o n sup 1 585 methods become better as the numbers get bigger new methods that are o sup log5 log sup 3 spl ap 0 n sup 1 465 o n sup log sup 7 log sup 4 spl ap o n sup 1 404 and o n sup log sup 9 log sup 5 spl ap o n sup 1 365 presented in actual experiments all of these methods turn out to be faster than fft multipliers for numbers that can be quite large 37 000 000 bits squaring seems to be fundamentally faster than multiplying but it is shown that t sub multiplyspl les 2t sub square o n

cmos vlsi implementation of a low power logarithmic converter we present a unique 32 bit binary to binary logarithm converter including its cmos vlsi implementation the converter is implemented using combinational logic only and it calculates a logarithm approximation in a single clock cycle unlike other complex logarithm correcting algorithms three unique algorithms are developed and implemented with low power and fast circuits that reduce the maximum percent errors that result from binary to binary logarithm conversion to 0 9299 percent 0 4314 percent and 0 1538 percent fast 4 16 and 32 bit leading one detector circuits are designed to obtain the leading one position of an input binary word a 32 word spl times 5 bit mos rom is used to provide 5 bit integers based on the corresponding leading one position both converter area and speed have been considered in the design approach resulting in the use of a very efficient 32 bit logarithmic shifter in the 32 bit logarithmic converter the converter is implemented using 0 6 spl mu m cmos technology and it requires 1 600 spl lambda spl times 2 800 spl lambda of chip area simulations of the cmos design for the 32 bit logarithmic converter operating at v sub dd equal to 5 volts run at 55 mhz and the converter consumes 20 milliwatts

a 3 flip neighborhood local search for the set covering problem the set covering problem scp calls for a minimum cost family of subsets from n given subsets which together covers the entire ground set in this paper we propose a local search algorithm for scp which has the following three characteristics 1 the use of 3 flip neighborhood which is the set of solutions obtainable from the current solution by exchanging at most three subsets as the size of 3 flip neighborhood is o n3 the neighborhood search becomes expensive if implemented naively to overcome this we propose an efficient implementation that reduces the number of candidates in the neighborhood without sacrificing the solution quality 2 we allow the search to visit the infeasible region and incorporate the strategic oscillation technique realized by adaptive control of penalty weights 3 the size reduction of the problem by using the information from the lagrangian relaxation is incorporated which is indispensable for solving very large instances according to computational comparisons on benchmark instances with other existing heuristic algorithms for scp our algorithm performs quite effectively for various types of problems especially for very large scale instances

very high radix square root with prescaling and rounding and a combined division square root unit an algorithm for square root with prescaling and selection by rounding is developed and combined with a similar scheme for division since division is usually more frequent than square root the main concern of the combined implementation is to maintain the low execution time of division while accepting a somewhat larger execution time for square root the algorithm is presented in detail including the mathematical development of bounds for the first square root digit and for the scaling factor the proposed implementation is described evaluated and compared with other combined div sqrt units the comparisons show that the proposed scheme potentially produces a significant speed up for division whereas for square root the speed up is small

a surrogate heuristic for set covering problems abstract the purpose of this paper is to present a new heuristic for set covering problems based upon continuous surrogate relaxations and subgradient optimization the algorithm combines problem reduction tests an adequate step size control and avoid preliminary sorting in solving the continuous surrogate relaxations computational tests for large scale set covering problems up to 1 000 rows and 12 000 columns indicate better quality results than algorithms based on lagrangian relaxations in terms of final solutions and mainly in computer times although the solving of a single surrogate optimization problem is slower than a corresponding lagrangian optimization the overall performance is almost twice as fast this is due to the smaller number of iterations which is a result of faster convergence and less oscillation

a continued fraction analysis of trigonometric argument reduction the calculation of a trigonometric function of a large argument x is effectively carried out by finding the integer n and 0

the optimal diversity management problem in some industries a certain part can be needed in a very large number of different configurations this is the case e g for the electrical wirings in european car factories a given configuration can be replaced by a more complete therefore more expensive one the diversity management problem consists of choosing an optimal set of some given number k of configurations that will be produced any nonproduced configuration being replaced by the cheapest produced one that is compatible with it we model the problem as an integer linear program our aim is to solve those problems to optimality the large scale instances we are interested in lead to difficult lp relaxations which seem to be intractable by the best direct methods currently available most of this paper deals with the use of lagrangean relaxation to reduce the size of the problem in order to be able subsequently to solve it to optimality via classical integer optimization

simple lagrangian heuristic for the set covering problem abstract in this paper a simple lagrangian heuristic is proposed for the set covering problem it is based on simple and classical ideas lagrangian duality greedy heuristic for the set covering problem subgradient optimization and redundant covers the main interesting point of the paper lies in the fact that the proposed heuristic turns out to be efficient for low density set covering problems with a large number of variables

approximating elementary functions with symmetric bipartite tables this paper presents a high speed method for function approximation that employs symmetric bipartite tables this method performs two parallel table lookups to obtain a carry save borrow save function approximation which is either converted to a two s complement number or is booth encoded compared to previous methods for bipartite table approximations this method uses less memory by taking advantage of symmetry and leading zeros in one of the two tables it also has a closed form solution for the table entries provides tight bounds on the maximum absolute error and can be applied to a wide range of functions a variation of this method provides accurate initial approximations that are useful in multiplicative divide and square root algorithms