numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

adaptive runge kutta methods for nonlinear two point boundary value problems with mild boundary layers abstract a variable mesh deferred correction algorithm based on implicit runge kutta formulae is described for the approximate numerical solution of nonlinear two point boundary value problems a strategy for automatically choosing the variable mesh spacing is described and this seeks to equidistribute an approximation to the global truncation error of the runge kutta formula the facility of being able to use a variable mesh extends considerably the range of applicability of runge kutta methods and in particular allows the possibility of solving rather more difficult problems such as those with mild boundary layers or turning points in an efficient manner an extensive set of numerical results is given to illustrate the various algorithms described and a comparison is made with the deferred correction code of lentini and pereyra appearing in the nag library

on the solution of block tridiagonal systems of linear algebraic equations having a special structure recently the present author has developed some high order finite difference formulae for the approximate numerical integration of general two point boundary value problems for ordinary differential equations the algebraic equations arising from using these formulae in conjunction with a modified newton iteration scheme are block tridiagonal with an additional special structure efficient algorithms for solving these equations are given and these appear to make high order finite difference schemes particularly attractive for the numerical solution of general two point boundary value problems

on the coveting radius of extremal self dual codes it is known that every self dual binary code which is not doubly even is a child of a doubly even parent it will be shown that an n 2 n 2 2 child of an n n 2 d doubly even parent has covering radius geq d 1 every extremal doubly even 32 16 8 code has covering radius 6 and every extremal doubly even 48 24 12 code has covering radius 8 the complete coset weight distribution of the 32 16 8 quadratic residue code is given as well as bounds or exact values for the covering radii of all extremai doubly even codes of length less than or equal to 96

a new approach to the general minimum distance decoding problem the zero neighbors algorithm minimum distance decoding mdd for a general error correcting linear code is a hard computational problem that recently has been shown to be np hard the complexity of known decoding algorithms is determined by min 2 k 2 n k where n is the code length and k is the number of information digits two new algorithms are suggested that reduce substantially the complexity of mdd the algorithms use a new concept of zero neighbors a special set of codewords only these codewords which can be computed in advance should be stored and used in the decoding procedure the number of zero neighbors is shown to be very small compared with min 2 k 2 n k for n gg 1 and a wide range of code rates r k n for example for r approx 0 5 this number grows approximately as a square root of the number of codewords

distributed source coding using abelian group codes a new achievable rate distortion region a distributed source coding problem with a joint distortion criterion that depends on the sources and the reconstruction is considered in this work while the prevalent trend in information theory has been to prove achievability results using shannon s random coding arguments using structured random codes offer rate gains over unstructured random codes for many problems motivated by this a new achievable rate distortion region an inner bound to the performance limit is presented for this problem for discrete memoryless sources based on good structured random nested codes built over abelian groups for certain sources and distortion functions the new rate region is shown to be strictly bigger than the berger tung rate region which has been the best known achievable rate region for this problem till now this is done using numerical plots achievable rates for single user source coding using abelian group codes are also obtained as a corollary of the main coding theorem it is shown that nested linear codes achieve the shannon rate distortion function in the arbitrary discrete memoryless case

a comparison of some global methods for solving two point boundary value problems a numerical comparison is made between three codes which implement global methods for solving two point boundary value problems the codes compared are a collocation method colsys a deferred correction method pasva 3 and a deferred correction method based on mono implicit runge kutta formulae hagron although the difficulty of conducting a fair comparison of complex codes is widely recognised it is nevertheless of great practical importance to try to gauge the relative performance of different codes and the three codes considered in this paper are particularly appropriate for numerical comparison by choosing a test set which includes both smooth and increasingly stiff problems some interesting characteristics of the codes are noted and some recommendations regarding their use are made

a probability ratio approach to approximate binary arithmetic coding we describe an alternative mechanism for approximate binary arithmetic coding the quantity that is approximated is the ratio between the probabilities of the two symbols analysis is given to show that the inefficiency so introduced is less than 0 7 on average and in practice the compression loss is negligible

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

efficient iterative solution of the three dimensional helmholtz equation we examine preconditioners for the discrete indefinite helmholtz equation on a three dimensional box shaped domain with sommerfeld like boundary conditions the preconditioners are of two types the first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions the second is derived by a block toeplitz approximation to the descretized problem the resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete helmholtz operator by an operator of low rank we present experimental results demonstrating that when these methods are combined with krylov subspace iteration convergence rates depend only mildly on both the wave number and discretization mesh size in addition the methods display high efficiencies in an implementation on an ibm sp 2 parallel computer

the rate of convergence of gmres on a tridiagonal toeplitz linear system the generalized minimal residual method gmres is often used to solve a nonsymmetric linear system ax b but its convergence analysis is a rather difficult task in general a commonly used approach is to diagonalize a x x 1 and then separate the study of gmres convergence behavior into optimizing the condition number of x and a polynomial minimization problem over a s spectrum this artificial separation could greatly overestimate gmres residuals and likely yields error bounds that are too far from the actual ones on the other hand considering the effects of both a s spectrum and the conditioning of x at the same time poses a difficult challenge perhaps impossible to deal with in general but only possible for certain particular linear systems this paper will do so for a nonsymmetric tridiagonal toeplitz system sharp error bounds on and sometimes exact expressions for residuals are obtained these expressions and or bounds are in terms of the three parameters that define a and chebyshev polynomials of the first kind

lattices which are good for almost everything we define an ensemble of lattices and show that for asymptotically high dimension most of its members are simultaneously good as sphere packings sphere coverings additive white gaussian noise awgn channel codes and mean squared error mse quantization codes these lattices are generated by applying construction a to a random linear code over a prime field of growing size i e by lifting the code to spl ropf sup n

on the complexity of minimum distance decoding of long linear codes we suggest a decoding algorithm of q ary linear codes which we call supercode decoding it ensures the error probability that approaches the error probability of minimum distance decoding as the length of the code grows for n spl rarr spl infin the algorithm has the maximum likelihood performance the asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known the algorithm develops the ideas of covering set decoding and split syndrome decoding

the covering radius of the 2 15 16 reed muller code is at least 16276 two interesting cosets of the first order reed muller code of block length 2 15 are described they provide counterexamples to a conjecture on the covering radius

a variable order deferred correction algorithm for the numerical solution of nonlinear two point boundary value problems abstract we develop two variable order deferred correction algorithms based on finite difference methods for the approximate numerical solution of nonlinear two point boundary value problems the deferred corrections provide a powerful local error estimator and this estimate can be used to refine the mesh if need be in order to obtain a prescribed tolerance mesh refinement is particularly straightforward since our algorithms are truly one step in nature the approach described is particularly appropriate for problems where only low or engineering precision is required and for very large systems of equations where storage space is an extremely important consideration some numerical results are given and a comparison with the method developed by lentini and pereyra and with richardson extrapolation is made

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

the complexity of information set decoding information set decoding is an algorithm for decoding any linear code expressions for the complexity of the procedure that are logarithmically exact for virtually all codes are presented the expressions cover the cases of complete minimum distance decoding and bounded hard decision decoding as well as the important case of bounded soft decision decoding it is demonstrated that these results are vastly better than those for the trivial algorithms of searching through all codewords or through all syndromes and are significantly better than those for any other general algorithm currently known for codes over large symbol fields the procedure tends towards a complexity that is subexponential in the symbol size

dimension 1 sequences are close to randoms abstract we show that a sequence has effective hausdorff dimension 1 if and only if it is coarsely similar to a martin lof random sequence more generally a sequence has effective dimension s if and only if it is coarsely similar to a weakly s random sequence further for any s t every sequence of effective dimension s can be changed on density at most h 1 t h 1 s of its bits to produce a sequence of effective dimension t and this bound is optimal

the hardness of decoding linear codes with preprocessing the problem of maximum likelihood decoding of linear block codes is known to be hard the fact that the problem remains hard even if the code is known in advance and can be preprocessed for as long as desired in order to device a decoding algorithm is shown the hardness is based on the fact that existence of a polynomial time algorithm implies that the polynomial hierarchy collapses thus some linear block codes probably do not have an efficient decoder the proof is based on results in complexity theory that relate uniform and nonuniform complexity classes

optimal soft decision block decoders based on fast hadamard transform an approach for efficient utilization of fast hadamard transform in decoding binary linear block codes is presented computational gain is obtained by employing various types of concurring codewords and memory reduction is also achieved by appropriately selecting rows for the generator matrix the availability of these codewords in general and particularly in some of the most frequently encountered codes is discussed

constrained designs for maximum likelihood soft decoding of rm 2 m and the extended golay codes we present algorithms for maximum likelihood soft decoding of the second order reed muller codes rm 2 m and the extended 24 12 8 golay code the decoding procedures are based on a concise representation of appropriately selected cosets of a subcode of the code considered such representation enables application of certain elimination rules a structure called constrained design is established to support the elimination procedures remarkably efficient algorithms are obtainable by this approach the method of elimination is applicable also in combination with existing coset decoding schemes

a stability criterion for explicit boundary value runge kutta methods abstract a function characterizing the stability of explicit boundary value runge kutta methods for the solution of ordinary differential equations is introduced it can be used to design methods with better stability and also in the selection of mesh points it is based on a generalization of the algebraic stability criterion numerical examples demonstrating the use of the proposed function are given

distribution of the discretization and algebraic error in numerical solution of partial differential equations abstract in the adaptive numerical solution of partial differential equations local mesh refinement is used together with a posteriori error analysis in order to equilibrate the discretization error distribution over the domain since the discretized algebraic problems are not solved exactly a natural question is whether the spatial distribution of the algebraic error is analogous to the spatial distribution of the discretization error the main goal of this paper is to illustrate using standard boundary value model problems that this may not hold on the contrary the algebraic error can have large local components which can significantly dominate the total error in some parts of the domain the illustrated phenomenon is of general significance and it is not restricted to some particular problems or dimensions to our knowledge the discrepancy between the spatial distribution of the discretization and algebraic errors has not been studied in detail elsewhere

on the complexity of some coding problems corresp it is shown that the problem of finding a codeword with least weight and whose weight is not a multiple of k in a binary linear code belongs to the class of nondeterministic polynomial np hard problems for any k geq 2 some other related problems are shown to belong to the same class these results were motivated by a conjecture due to berlekamp mceliece and van tilborg that the problem of finding the hamming distance of a code is np hard

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

do most binary linear codes achieve the goblick bound on the covering radius corresp the following two problems are dealt with p1 finding the smallest rate r of a binary code of length n admitting a prescribed covering radius rho n p2 discovering whether a majority of codes with any rate larger than r admits the given covering radius for the class of unrestricted nonlinear codes a solution to both problems is obtained by an elementary averaging argument the solution to p1 is r 1 h rho o n 1 log n and the answer to p2 is positive as for the more interesting class of linear codes goblick s extension method shows that the solution to p1 is the same as in the unrestricted case in contrast p2 seems to remain an open question a simple derivation of goblick s result is presented and a discussion is made of the positive conjecture concerning p2 for linear codes

the complexity of computing the covering radius of a code the problem of finding the covering radius of a binary linear code is shown to be nondeterministic polynomial np hard in fact a problem that is complete for the class prod 2 p in the polynomial hierarchy is shown to be reducible to the covering radius problem so that finding the covering radius is strictly harder than any np complete problem unless the polynomial hierarchy collapses with np prod 2 p

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

a note on conjugate gradient convergence part ii summary continuing our previous analysis we derive the exact number of conjugate gradient iterations needed to achieve a given tolerance for the one dimensional discrete poisson equation on a uniform grid and a particularly smooth solution vector

minimal vectors in linear codes minimal vectors in linear codes arise in numerous applications particularly in constructing decoding algorithms and studying linear secret sharing schemes however properties and structure of minimal vectors have been largely unknown we prove basic properties of minimal vectors in general linear codes then we characterize minimal vectors of a given weight and compute their number in several classes of codes including the hamming codes and second order reed muller codes further we extend the concept of minimal vectors to codes over rings and compute them for several examples turning to applications we introduce a general gradient like decoding algorithm of which minimal vectors decoding is an example the complexity of minimal vectors decoding for long codes is determined by the size of the set of minimal vectors therefore we compute this size for long randomly chosen codes another example of algorithms in this class is given by zero neighbors decoding we discuss relations between the two decoding methods in particular we show that for even codes the set of zero neighbors is strictly optimal in this class of algorithms this also implies that general asymptotic improvements of the zero neighbors algorithm in the frame of gradient like approach are impossible we also discuss a link to secret sharing schemes

on the numerical integration of nonlinear two point boundary value problems using iterated deferred corrections part 2 the development and analysis of highly stable deferred correction formulae iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two point boundary value problems in ordinary differential equations however there

the rate of convergence of gmres on a tridiagonal toeplitz linear system ii abstract this paper continues the recent work of the authors r c li w zhang the rate of convergence of gmres on a tridiagonal toeplitz linear system numer math 112 2009 267 293 electronically published on 19 december 2008 on the rate of convergence of gmres for a tridiagonal toeplitz linear system ax b much simpler formulas than the earlier ones for gmres residuals when b is the first or the last column of the identity matrix are established and these formulas allow us to confirm the rate of convergence that was conjectured but only partially proven earlier simpler and sharper bounds than earlier ones when all b s entries except its first and last ones are zeros are also obtained

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

explicit codes with low covering radius a series of explicit low rate binary linear codes that have relatively low covering radius and can be rapidly decoded is exhibited these codes can be derived from higher dimensional analogs of the gale berlekamp switching game conjectures of independent interest involving hadamard matrices are given that could yield semiexplicit covering codes of very low density

suboptimal decoding of linear codes partition technique general symmetric channels are introduced and near maximum likelihood decoding in these channels is studied first we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis it is proved that the decoding error probability of suboptimal decoding is bounded above for any q ary code of length n and code rate r by twice the error probability of its maximum likelihood decoding and tends to the latter as n grows second we design a suboptimal trellis like algorithm which reduces the known decoding complexity of the order of q sup n min r 1 r operations to that of q sup nr i r operations for all cyclic codes and virtually all long linear codes we also consider the corresponding bounds for concatenated codes an important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability

another upper bound on covering radius a new upper bound on the covering radius of binary linear codes improves on previously known bounds for some codes the bound is w 2 rho 2 where w is one of the weights occurring among the codewords and rho 2 is the covering radius of an associated code some comparisons with other bounds are made

defect correction for two point boundary value problems on nonequidistant meshes new finite difference formulae of arbitrary order are derived for the speciai class of second order two point boundary value problems y f x y x a x b variable mesh spacing is possible and the required accuracy is achieved under a very mild mesh condition a naturai defect correction framework is set up to compute the higher order approximations

asymptotic bounds on the covering radius of binary codes asymptotic covering properties of families of binary codes are studied a gaussian approximation to the weight distribution of translates for codes of high strength is used from this an upper bound on the covering radius of these codes is deduced applications include reed muller codes quadratic residue codes and bch codes sufficient conditions for a family of codes to have best possible covering radius asymptotically perfect codes are derived

nonexistence of a uniformly packed 70 58 5 code corresp the nonexistence of a uniformly packed 70 58 5 code c perp is proved by examining geometries associated with the 3 weight code c

uniform fourth order difference scheme for a singular perturbation problem the numerical solution of a nonlinear singularly perturbed two point boundary value problem is studied the developed method is based on hermitian approximation of the second derivative on special discretization mesh numerical examples which demonstrate the effectiveness of the method are presented

the zero guards algorithm for general minimum distance decoding problems we present some properties of an improved version of the zero neighbors algorithm the zero guards algorithm these properties can be used to find a zero guards a new decoding procedure using a zero guards is also given

a note on conjugate gradient convergence the one dimensional discrete poisson equation on a uniform grid with n points produces a linear system of equations with a symmetric positive definite coefficient matrix hence the conjugate gradient method can be used and standard analysis gives an upper bound of o n on the number of iterations required for convergence this paper introduces a systematically defined set of solutions dependent on a parameter beta and for several values of beta presents exact analytic expressions for the number of steps k beta tau n needed to achieve accuracy tau the asymptotic behavior of these expressions has the form o n alpha as n rightarrow infty and o tau gamma as tau rightarrow 0 in particular two choices of beta corresponding to nonsmooth solutions give alpha 0 i e iteration counts independent of n this is in contrast to the standard bounds the standard asymptotic convergence behavior alpha 1 is seen for a relatively smooth solution numerical examples illustrate and supplement the analysis