order and stepsize control in extrapolation methods the paper presents a new theory for joint order and stepsize control in extrapolation methods this theory defines a locally optimal order that can be determined along any trajectory to be computed in addition shannon s information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non stiff integration by the new device moreover a significant increase in reliability robustness and portability of the extrapolation codes is achieved

some new results concerning index 3 differential algebraic equations on introduit la notion de tractabilite d indice 3 on donne un critere matriciel pour tester cette tractabilite pour l equation differentielle algebrique f x t x t t 0

powering by a table look up and a multiplication with operand modification an efficient method for generating a power of an operand i e x sup p for an operand x and a given p is proposed it is applicable to ps in the form of spl plusmn 2 sup k where k is any integer and of spl plusmn 2 k sub 1 spl plusmn 2 k sub 2 where k sub 1 is any integer and k sub 2 is any nonnegative integer the reciprocal the square root the reciprocal square root the reciprocal square the reciprocal cube and so forth are included the method is a modification of the piecewise linear approximation a power of an operand is generated through a table look up and a multiplication with operand modification the same accuracy is achieved as the piecewise linear approximation the multiplication and an addition required for the piecewise linear approximation are replaced by only one double sized multiplication with a slight modification of the operand and hence one clock cycle may be reduced the required table size is reduced because only one coefficient instead of two has to be stored

brief paper regularizations of linear time varying singular systems singular systems of differential equations arising in reduced order models are often regularized by the introduction of small parameters the types of regularization used for linear time invariant systems need not work on linear time varying systems this paper gives examples and discusses this behavior conditions under which the standard type of regularization works are developed

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

on boundary value problems in differential algebraic equations for linear and nonlinear boundary value problems in dae s solvability is investigated in particular it is shown that only transferable dae s lead to well posed boundary value problems in their n

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

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

a digit by digit algorithm for mth root extraction a general digit recurrence algorithm for the computation of the mth root with an m integer is presented in this paper based on the concept of completing the mth root a detailed analysis of the convergence conditions is performed and iteration independent digit selection rules are obtained for any radix and redundant digit set a radix 2 version for mth rooting is also studied together with closed formulas for both the digit selection rules and the number of bits required to perform correct selections

on the fly rounding computing arithmetic in implementations of operations based on digit recurrence algorithms such as division left to right multiplication and square root the result is obtained in digit serial form from most significant digit to least significant to reduce the complexity of the result digit selection and allow the use of redundant addition the result digit has values from a signed digit set as a consequence the result has to be converted to conventional representation which can be done on the fly as the digits are produced without the use of a carry propagate adder the authors describe three ways to modify this conversion process so that the result is rounded the resulting operation is fast because no carry propagate addition is needed the schemes described apply also to online arithmetic operations

on a regularization of index 3 differential algebraic equations in hanke j math anal appl 151 1990 236 253 boundary value problems for linear differential algebraic equations daes with time varying coefficients a t x t b t x t q t tractable with index 2 are considered in this paper we consider index 3 daes in an analogous way a transferable daes is obtained by the regularization and the convergence rate is derived

fast multiplication without carry propagate addition conventional schemes for fast multiplication accumulate the partial products in redundant form carry save or signed digit and convert the result to conventional representation in the last step this step requires a carry propagate adder which is comparatively slow and occupies a significant area of the chip in a vlsi implementation a report is presented on a multiplication scheme left to right carry free lrcf that does not require this carry propagate step the lrcf scheme performs the multiplication most significant bit first and produces a conventional sign and magnitude product most significant n bits by means of an on the fly conversion the resulting implementation is fast and regular and is very well suited for vlsi the lrcf scheme for general radix r and a radix 4 signed digit implementation are presented

analysis of descriptor systems using numerical algorithms in this paper we analyze numerical methods for the solution of the large scale dynamical system e dot y t ay t g t y t 0 y 0 where e and a are matrices possibly singular systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables special cases of such systems are algebraic differential systems we discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure detect nonsolvable systems resolve initial value consistency problems and are applicable to stiff descriptor systems we also present an algorithm for the control of the local truncation error on only the state variables

one step collocation methods for differential algebraic systems of index 1 abstract in this paper we propose one step collocation methods for linear differential algebraic equation dae systems of index 1 the proposed methods give a continuous approximate solution on the integration interval t0 tn we study the uniform convergence properties of the proposed collocation methods some of the methods show an order reduction phenomenon at the nodes similar to that observed for runge kutta methods petzold 1986 since dae systems may be considered as a kind of very stiff differential systems the collocation methods for dae systems have to verify strong stability properties for example collocation methods at radau points are particularly suitable for solving dae systems however the numerical results will also show that gauss legendre collocation methods may be applied successfully to dae systems also if these are only a stable methods

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

a self stabilized algorithm for enforcing constraints in multibody systems a new algorithm is developed for the enforcement of constraints within the framework of nonlinear flexible multibody system modeled with the finite element approach the proposed algorithm exactly satisfies the constraints at the displacement and velocity levels and furthermore it achieves nonlinear unconditional stability by imposing the vanishing of the work done by the constraint forces when combined with specific discretizations of the inertial and elastic forces identical convergence rates are observed for the displacements velocities and lagrange multipliers the proposed algorithm is closely related to the stabilized index 2 or ggl method although no additional multipliers are introduced in the present approach these desirable characteristics are obtained without resorting to numerically dissipative algorithms if high frequencies are present in the system i e the system is physically stiff dissipative schemes become necessary the proposed algorithm is extended to deal with this situation

operational tau method for singular system of volterra integro differential equations the legendre spectral tau matrix formulation is proposed to approximate solution of singular system of volterra integro differential equations the existence and uniqueness solution of this system are investigated by means of the smoothing property of a volterra integral operator and some projectors the l 2 convergence of the numerical method is analyzed it is proved theoretically and demonstrated numerically that the proposed method converges exponentially finally two numerical examples illustrate the theoretical results

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

on boundary value problems in differential algebraic equations for linear and nonlinear boundary value problems in dae s solvability is investigated in particular it is shown that only transferable dae s lead to well posed boundary value problems in their natural setting for those problems appropriate modifications of well known ode methods are proposed further some regularizations of non transferable dae s which are closely related to singular perturbations are discussed

automatic design of frequency sampling filters by hybrid genetic algorithm techniques a new method of designing recursive and nonrecursive frequency sampling filters is presented we investigate the use of a hybrid real coded genetic algorithm ga for optimising transition sample values to give the maximum stopband attenuation a modification allows the coefficient wordlength to be optimized concurrently thereby reducing the overall number of design steps and simplifying the design process the technique is able to consistently optimize filters with up to six transition samples designing digital filters is a complex process involving optimization at several discrete design steps the techniques presented could form the basis for integrating several of the optimizations investigations into increasing this integration by using a binary coded ga to optimize nonlinear phase quantized coefficient fir filters are introduced with an analysis of the difficulty of the problem from a ga perspective

one step and extrapolation methods for differential algebraic systems the paper analyzes one step methods for differential algebraic equations dae in terms of convergence order in view of extrapolation methods certain perturbed asymptotic expansions are shown to hold for the special dae extrapolation solver based on the semi implicit euler discretization the perturbed order pattern of the extrapolation tableau is derived in detail the theoretical results lead to modifications of the known code the efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion

the simulation problem for high order linear differential systems in this paper we consider linear constant coefficient ordinary differential equations of arbitrary order of the form r ddt w m ddt f with r and m given but otherwise arbitrary polynomial matrices to these equations we associate a behavior b defined as the set of all vector valued distributions w f that solve the equations themselves we think of f as a given distribution while w is to be found in such a way that the equations defining the behavior are satisfied alongside r ddt w m ddt f we also consider initial conditions obtained by imposing the value at time t 0 of a linear combination of the variables w and their derivatives yielding s ddt w 0 ta with s a polynomial matrix t a fixed real matrix and a a real vector the three main issues we address are i solvability of the behavior meaning the problem of finding conditions that assure that corresponding to a or any given vector distribution f a trajectory w exists such that w f belong to b this question is formalized and answered in section 4 ii the index of the behavior which means investigating how the smoothness of the given distribution f is related to smoothness of the solution w section 5 is dedicated to this issue iii compatibility of initial conditions in which case the initial conditions s ddt w 0 ta are considered alongside the behavior b we first check that s ddt w 0 is well defined if it is we provide conditions under which there exists a unique distribution w such that w f belong to b and such that s ddt w 0 ta are also satisfied for a or any given a this problem is addressed in section 6 we show how our results generalize to behaviors defined by equations of arbitrary order classical properties of behaviors defined by first order systems such as eddtw aw f

a frequency sampling filter design method which accounts for finite word length effects system functions of frequency sampling filters require pole zero cancellations on the unit circle for practical implementations finite word length effects usually prevent pole zero cancellations which can result in filter instability the article develops an optimization method for designing a modified frequency sampling filter which is guaranteed to be stable

analytical and numerical solutions to higher index linear variable coefficient dae systems abstract analytical and numerical solutions to the singular linear system of differential equations a t x t b t x t f t are investigated closed form analytical solutions of systems transformable to standard canonical form scf see campbell 1983 are derived equivalent ways to compute the analytic solution are presented which are of interest in a symbolic numeric computing environment many numerical methods for solving dae systems are based on gear s backward difference formula bdf method it is proved that the bdf method and the modified bdf method mbdf see clark 1986 are stable and o hk accurate after a maximum of s 1 k 1 steps for linear variable coefficient systems in scf where s is the dimension of the algebraic part these new results are valid for arbitrarily higher index systems and generalize well known results for linear constant coefficient singular systems to the variable coefficient case numerical experiments confirm the order and stability results

on the regularization of a class of nontransferable differential algebraic equations abstract this paper deals with linear differential algebraic equations which arise by coupling ordinary differential equations with solution invariants boundary value problems for these equations are essentially ill posed a parametrization into transferable equations is shown to present a regularization results about convergence rates for exact and noisy data are presented

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

sigmoid generators for neural computing using piecewise approximations a piecewise second order approximation scheme is proposed for computing the sigmoid function the scheme provides high performance with low implementation cost thus it is suitable for hardwired cost effective neural emulators it is shown that an implementation of the sigmoid generator outperforms in both precision and speed existing schemes using a bit serial pipelined implementation the proposed generator requires one multiplication no look up table and no addition it has been estimated that the sigmoid output is generated with a maximum computation delay of 21 bit serial machine cycles representing a speedup of 1 57 to 2 23 over other proposals

the application of rosenbrock wanner type methods with stepsize control in differential algebraic equations two rosenbrock wanner type methods for the numerical treatment of differential algebraic equations are presented both methods possess a stepsize control and an index 1 monitor the first method dae34 is of order 3 4 and uses a full semi implicit rosenbrock wanner scheme the second method rkf4da is derived from the runge kutta fehlberg 4 5 pair where a semi implicit rosenbrock wanner method is embedded in order to solve the nonlinear equations the performance of both methods is discussed in artificial test problems and in technical applications

rosenbrock type methods adapted to differential algebraic systems we consider the numerical solution of differential algebraic systems of index one given in kronecker canonical form the methods described here are derived from the rosenbrock approach hence they do not require the solution of nonlinear systems of equations but one evaluation of the jacobian and one lu decomposition per step by construction the s stage method coincides with a solver for nonlinear equations of order s 1 if the stepsize is set to zero in this sense the adaptation to differential algebraic equations is performed the special structure of the method leads to simplified order conditions and to an easy implementation some particular methods up to order 4 are given especially an embedded 4 stage method of order 4 3 is derived

on the regularization of index 2 differential algebraic equations abstract boundary value problems for linear differential algebraic equations daes with time varying coefficients a t x t b t x t q t tractable with index 2 are considered these daes contain differentiation problems and lead therefore to essentially ill posed problems we show that a parametrization proposed by marz is a regularization in the sense of tikhonov convergence rates for noisy data are derived moreover for the so called pencil regularization analogous results are derived in the case of a time independent nullspace n a t

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

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

differential algebraic equations with after effect in this paper we are concerned with the solution of delay differential algebraic equations these are differential algebraic equations with after effect or constrained delay differential equations the general semi explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments even simply stated problems of this type can give rise to difficult analytical and numerical problems the more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems and to indicate problems that require further research

frequency sampling filters an improved model structure for step response identification this paper presents a new methodology for step response identification using the frequency sampling filter fsf model the advantages of the fsf model structure over the popular finite impulse response fir model are i the number of parameters to be estimated is independent of the choice of sampling interval and is generally far fewer than the number required by the fir model to obtain an accurate estimate of the step response and ii the general conditioning of the correlation matrix is better with the fsf model because of the reduction in the number of parameters that need to be estimated and to the narrow bandpass nature of the frequency sampling filters

linear differential algebraic equations in spaces of integrable functions on developpe une l 2 theorie pour des equations differentielles algebriques lineaires t x t a t x t b t x t f t t a b x t r m

backward differentiation approximations of nonlinear differential algebraic systems finite difference approximations of dynamical systems modeled by nonlinear semiexplicit differential algebraic equations are analyzed convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints the appropriate asymptotic convergence rates and the leading error terms are determined

integral algebraic equations theory of collocation methods i our analysis of collocation solutions for general systems of linear integral algebraic equations iaes is based on the notions of the tractability index and the nu smoothing property of a volterra integral operator these are used to decouple the given iae system into the inherent system of regular second kind volterra integral equations vies and a system of first kind vies this decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions numerical examples illustrate the theoretical results

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems