bivariate generalized inverse newton thiele type matrix pade approximation abstract in this paper a practical bivariate newton thiele type matrix pade approximation is introduced by using the generalized inverse of a matrix the approximants are expressed in the form of newton expansion and branched continued fractions which can be computed by an efficient recursive algorithm some algebraic properties are also discussed and the application in state space realization of the 2 d filters is given in the end

design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

switch type hybrid hard decision decoding algorithms for regular low density parity check codes this correspondence presents the switch strategies for switch type hybrid hard decision decoding algorithms for regular low density parity check ldpc codes after the piecewise analysis of extrinsic information transfer functions for gallager decoding algorithm b gb the normalized switch scheme nss for which the majority based algorithms and gb are two examples is proposed then several other examples of nsss are presented and their convergence properties are analyzed based on the extrinsic information transfer exit functions in simulations the proposed nsss show meaningful performance improvements and less sensitivity to channel parameter underestimations compared with gb for small and moderate block length codes

on the convergence of rows of vector pade approximants we state and prove a de montessus like theorem for vector valued meromorphic functions using vector pade approximants which are defined in the context of clifford algebras the proof is based on that used by saff for the scalar case the theorem may be regarded as an answer to a general problem posed by graves morris and saff

clifford algebras and vector valued rational forms i we demonstrate how the use of clifford algebras in the theory of vector valued rational forms leads to practical recurrence relations which do not involve representations of the algebras

a note on matrix valued rational interpolants in this paper by defining a kind of transformation from matrix to vector we succeed in transferring some results on vector valued rational interpolants to those corresponding to the matrix valued rational interpolants moreover it is pointed out through a numerical example that the statement of the so called uniqueness theorem in 4 is incorrect and what is more the proof is also wrong a new uniqueness theorem is given

matrix pade approximation definitions and properties abstract some proposals are made to give a general definition of matrix pade approximants depending on the normalization of the denominator we define type i constant term is the unit matrix and type ii by conditions on the leading coefficient approximants existence and uniqueness are considered determinant expressions are given and relations among types i ii and left right approximants are considered

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

an extension of a row convergence theorem for vector pade approximants abstract a row convergence theorem of de montessus type is established for vector pade approximants to a vector valued meromorphic function f while in a previous paper the authors established such a theorem for the case when f has simple poles the essential feature of the present paper is to treat the situation when f has multiple poles

bivariate composite vector valued rational interpolation in this paper we point out that bivariate vector valued rational interpolants bvri have much to do with the vector grid to be interpolated when a vector grid is well defined one can directly design an algorithm to compute the bvri however the algorithm no longer works if a vector grid is ill defined taking the policy of divide and conquer we define a kind of bivariate composite vector valued rational interpolant and establish the corresponding algorithm a numerical example shows our algorithm still works even if a vector grid is ill defined

the qd algorithm and multivariate pade approximants the quotient difference qd algorithm can be used to construct univariate pade approximants 1 in this paper we see that it can also be used to construct the multivariate pade approximants introduced in 3 just by reformulating the quotient difference algorithm as in sect 1 the multivariate pade approximants and the multivariate qd scheme are treated in the sects 2 and 3 respectively thus for this type of multivariate pade approximants a link with the theory of multivariate continued fractions is established

data driven parameterization of the generalized langevin equation we present a data driven approach to determine the memory kernel and random noise in generalized langevin equations to facilitate practical implementations we parameterize the kernel function in the laplace domain by a rational function with coefficients directly linked to the equilibrium statistics of the coarse grain variables we show that such an approximation can be constructed to arbitrarily high order and the resulting generalized langevin dynamics can be embedded in an extended stochastic model without explicit memory we demonstrate how to introduce the stochastic noise so that the second fluctuation dissipation theorem is exactly satisfied results from several numerical tests are presented to demonstrate the effectiveness of the proposed method

an implementation friendly binary ldpc decoding algorithm we introduce an implementation friendly binary message passing decoding method for low density parity check ldpc codes that does not require the degree information of variable nodes or degree dependent parameters for hard decision decoding given its low complexity the implementation cost for variable node degree information is an important consideration we develop an estimation method for the extrinsic error probability eep as well as its analysis the proposed method offers similar performance as the existing methods for time invariant decoding in most cases while it facilitates efficient circuit implementations of the ldpc decoder

row convergence theorems for generalised inverse vector valued pade approximants abstract the approximants mentioned in the title are related to vector valued continued fractions and the vector algorithm devised by wynn in 1963 here we establish a unitary invariance property of these approximants and describe how the classical 1 dimensional pade approximants can be obtained as a special case the main results of the paper consist of de montessus de ballore type convergence theorems for row sequences having fixed denominator degree of vector valued approximants to meromorphic vector functions

a backward three term recurrence relation for vector valued continued fractions and its applications this paper presents a backward three term recurrence relation which is a solution of an open problem in graves morris numer math 42 1983 331 by means of this relation we succeed in establishing some conclusions about the two backward successive convergents of vector valued continued fractions moreover an expression for the vector valued continued fraction in series form is obtained and some theorems of the convergence of the vector valued continued fraction are proved by virtue of the recurrence relation

minimal vector pade approximation abstract we describe the minimal vector pade approximation problem which consists in finding pade approximants with a common denominator for a number of series these approximants are minimal in the sense that for a given order of approximation and a given discrepancy in numerator and denominator degrees the degree of the rational approximant is minimal properties and solution methods are derived from an associated partial realization problem

a sleszyňski pringsheim theorem for vector valued continued fractions and its optimal error bounds the first purpose of this work is to prove a convergence theorem for vector valued continued fractions which is an exact extension of a well known theorem of sleszynski pringsheim and our proof is based on the truncation error analysis for such continued fractions the second purpose is to give refined error bounds for such continued fractions these bounds are also improvements of the scalar results and the numerical example given explains the effect of our results

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

adaptive osculatory rational interpolation for image processing image interpolation is a common problem in image applications although many interpolation algorithms have been proposed in the literature these methods suffer from the effects of imperfect reconstruction to some degree most often these effects manifest themselves as jagged contours or blurred edges in the image this paper presents a method for preserving the contours or edges based on adaptive osculatory rational interpolation kernel function which is built up by approximating the ideal interpolating kernel function by continued fractions it is a more accurate approximation for the ideal interpolation in space domain or frequency domain than by other linear polynomial interpolation kernel functions simulation results are also presented to demonstrate the superior performance of image magnification

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

l2 overline partial cohomology groups of some singular complex spaces let x be a pure n dimensional where n 2 complex analytic subset in n with an isolated singularity at 0 in this paper we express the l 2 0 q overline partial cohomology groups for all q with 1 q n of a sufficiently small deleted neighborhood of the singular point in terms of resolution data we also obtain identifications of the l 2 0 q overline partial cohomology groups of the smooth points of x in terms of resolution data when x is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

analysis and design of binary message passing decoders binary message passing decoders for low density parity check codes are studied by using extrinsic information transfer charts the channel delivers hard or soft decisions and the variable node decoder performs all computations in the log likelihood ratio l value domain a hard decision results in the gallager b algorithm and examples show that increasing the channel output alphabet to two bits gains more than 1 0 db in signal to noise ratio when using optimized codes finally it is shown that errors on cycles consisting only of degree two and three variable nodes cannot be corrected and a necessary and sufficient condition for the existence of a cycle free subgraph is derived

a neville like method via continued fractions as we know the classical neville s algorithm is an effective method used to solve the interpolation problem by polynomials in this paper we adopt the idea of the neville s algorithm to construct a kind of blending rational interpolants via continued fractions for a given set of support points there are many ways to build up the interpolation schemes by which we mean that there are many choices to make to determine the initial interpolants on subsets of support points and then update them step by step to form a solution to the full interpolation problem numerical examples are given to show the advantage of our method and a multivariate analogy is also discussed

properties of optimum binary message passing decoders we consider a class of message passing decoders for low density parity check ldpc codes whose messages are binary valued we prove that if the channel is symmetric and all codewords are equally likely to be transmitted an optimum decoding rule in the sense of minimizing message error rate should satisfy certain symmetry and isotropy conditions using this result we prove that gallager s algorithm b achieves the optimum decoding threshold among all binary message passing decoding algorithms for regular codes for irregular codes we argue that when the nodes of the message passing decoder do not exploit knowledge of their decoding neighborhood optimality of gallager s algorithm b is preserved we also consider the problem of designing irregular ldpc codes and find a bound on the achievable rates with gallager s algorithm b using this bound we study the case of low error rate channels and analytically find good degree distributions for them

problems and progress in vector pade approximation abstract some open problems in vector pade approximation are stated and some recent de montessus type theorems governing convergence of rows of the vector pade table are contrasted weshow how these results indicate when it is more appropriate to use generalised inverse vector pade approximants or their hybridised form we show how a straightforward analogue of the berlekamp massey algorithm may be used to calculate generalised inverse vector pade approximants this algorithm is applied to the derivation of a new low order hybrid vector approximant related results include the case of a row convergence theorem for a complex valued power series using a clifford inverse instead of the moore penrose inverse

on the theory and computation of nonperfect pade hermite approximants abstract for a vector of k 1 power series we introduce two new types of rational approximations the weak pade hermite form and the weak pade hermite fraction a recurrence relation is then presented which computes pade hermite forms along with their weak counterparts along a sequence of perfect points in the pade hermite table the recurrence relation results in a fast algorithm for calculating a pade hermite approximant of any given type when the vector of power series is perfect the algorithm is shown to calculate a pade hermite form of type n0 nk in o kn 2 operations where n n0 nk this complexity is the same as that of other fast algorithms the new algorithm also succeeds in the nonperfect case usually with only a moderate increase in cost

extrapolation methods for vector sequences an analogue of aitken s 2 method suitable for vector sequences is proposed aspects of the numerical performance of the vector algorithm based on using the moore penrose inverse are investigated the fact that the denominator polynomial associated with a vector pade approximant is the square of its equivalent in the scalar case is shown to be a source of approximation error in cases where the convergence of the vector sequence is dominated by real eigenvalues a hybrid form of the vector pade approximant having a denominator polynomial of minimal degree is proposed and its effectiveness is demonstrated on several standard examples

geometric methods in rational interpolation theory abstract this paper studies the minimal rational interpolation problem using algebrogeometric methods specifically we deal in a unified framework with the following problems scalar and matrix interpolation and left and right tangential interpolation while the results we prove are known our methods of proof are quite different in addition our proof shows that these problems can be solved by an application of the grobner basis algorithm a popular computational tool in computer algebra

a new method for computing the matrix exponential operation based on vector valued rational approximations in this paper a new method for computing the action of the matrix exponential on a vector e a tb where a is a complex matrix and t is a positive real number is proposed our approach is based on vector valued rational approximation where the approximants are determined by the denominator polynomials whose coefficients are obtained by solving an inexpensive linear least squares problem no matrix multiplications or divisions but matrix vector products are required in the whole process a technique of scaling and recurrence enables our method to be more effective when the problem is for fixed a b and many values of t we also give a backward error analysis in exact arithmetic for the truncation errors to derive our new algorithm preliminary numerical results illustrate that the new algorithm performs well

hybrid hard decision iterative decoding of irregular low density parity check codes time invariant hybrid hscrti decoding of irregular low density parity check ldpc codes is studied focusing on hscrti algorithms with majority based mb binary message passing constituents we use density evolution de and finite length simulation to analyze the performance and the convergence properties of these algorithms over memoryless binary symmetric channels to apply de we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them a tight upper bound on the threshold of mb hscrti algorithms is derived and it is proven that the asymptotic error probability for these algorithms tends to zero at least exponentially with the number of iterations we devise optimal mb hscrti algorithms for irregular ldpc codes and show that these algorithms outperform gallager s algorithm a applied to optimized irregular ldpc codes we also show that compared to switch type algorithms such as gallager s algorithm b where a comparable improvement is obtained by switching between different mb algorithms mb hscrti algorithms are more robust and can better cope with unknown channel conditions and thus can be practically more attractive

vector rational interpolation algorithms of burlisch stoer neville form we present algorithms of the bulirsch stoer neville form for calculating the value of the vector rational interpolant the derivation parallels that of the bulirsch stoer algorithm for the usual scalar rational interpolation algorithm of stoer and bulirsch using the clifford algebra foundations for vector rational interpolants of roberts

a new approach to the rational interpolation problem the vector case we generalize our earlier results on rational interpolation which were given in van barel and bultheel this journal 1990 for the scalar case and in bultheel and van barel 1990 for the vector case when all the interpolation points coincide to the case of vector data given at arbitrary points that may coincide or not this is the vector valued newton pade problem we give a recursive algorithm which has the important advantage over other algorithms that we do not need a reordering of the given interpolation data to overcome a singularity in the interpolation table not even in the nonnormal vector case it also generates all the information needed to give all solutions of the problem

multivariate reciprocal differences for branched thiele continued fraction expansion etude des differences reciproques multivariables pour les developpements en fractions continues de thiele ramifiees

block based thiele like blending rational interpolation thiele type continued fraction interpolation may be the favoured nonlinear interpolation in the sense that it is constructed by means of the inverse differences which can be calculated recursively and produce useful intermediate results however thiele s interpolation is in fact a point based interpolation by which we mean that a new interpolating continued fraction with one more degree of its numerator or denominator polynomial is obtained by adding a tail to the current one or in other words this can be reshaped simply by adding a new support point to the current set of support points once at a time in this paper we introduce so called block based inverse differences to extend the point based thiele type interpolation to the block based thiele like blending rational interpolation the construction process may be outlined as follows first of all divide the original set of support points into some subsets blocks then construct each block by using whatever interpolation means linear or rational and finally assemble these blocks by thiele s method to shape the whole interpolation scheme clearly many flexible rational interpolation schemes can be obtained in this case including the classical thiele s continued fraction interpolation as its special case as an extension a bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method

vector valued rational interpolants over triangular grids abstract given in this paper are some results about vector valued rational interpolants over triangular grids by means of thiele type branched continued fractions and the samelson inverse characterization theorem and uniqueness theorem in a certain sense are obtained moreover numerical examples are provided to support other properties of vector valued rational interpolants such as boundary interpolation and duality and it is pointed out that all the results in the vector case can be transplanted to the matrix valued case by the technique of expansion of matrix into vectors

bivariate thiele type matrix valued rational interpolants a new method for the construction of bivariate matrix valued rational interpolants on a rectangular grid is introduced in this paper the rational interpolants are of the continued fraction form with scalar denominator in this respect the approach is essentially different from that of bose and basu 1980 where a rational matrix valued approximant with matrix valued numerator and denominator is used for the approximation of a bivariate matrix power series the matrix quotients are based on the generalized inverse for a matrix introduced by gu chuanqing and chen zhibing 1995 which is found to be effective in continued fraction interpolation a sufficient condition of existence is obtained some important conclusions such as characterisation and uniqueness are proven respectfully the inner connection between two type interpolating functions is investigated some examples are given so as to illustrate the results in the paper

a more accurate one dimensional analysis and design of irregular ldpc codes we introduce a new one dimensional 1 d analysis of low density parity check ldpc codes on additive white gaussian noise channels which is significantly more accurate than similar 1 d methods our method assumes a gaussian distribution in message passing decoding only for messages from variable nodes to check nodes compared to existing work which makes a gaussian assumption both for messages from check nodes and from variable nodes our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence similar to previous work the problem of designing irregular ldpc codes reduces to a linear programming problem however our method allows irregular code design in a wider range of rates without any limit on the maximum variable node degree we use our method to design irregular ldpc codes with rates greater than 1 4 that perform within a few hundredths of a decibel from the shannon limit the designed codes perform almost as well as codes designed by density evolution

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

vector extrapolation methods applications and numerical comparison the present paper is a survey of the most popular vector extrapolation methods such as the reduced rank extrapolation rre the minimal polynomial extrapolation mpe the modified minimal polynomial extrapolation mmpe the vector e algorithm vea and the topological e algorithm tea using projectors we derive a different interpretation of these methods and give some theoretical results the second aim of this work is to give a numerical comparison of the vector extrapolation methods above when they are used for practical large problems such as linear and nonlinear systems of equations

solution of integral equations using generalised inverse function valued pade approximants i abstract neumann series solutions for the linear integral equation x 1 1 0 1 x y y d y converge for small values of we use generalised inverse function valued pade approximants to accelerate the convergence of the neumann series and to estimate the singular value of these estimates are found to compare very favourably with those from the fredholm determinant method and the pade approximant method

vector valued thiele werner type osculatory rational interpolants a method of solving vector valued osculatory rational interpolation problems is presented by using the samelson inverse of vectors the rational interpolants are constructed in the form of thiele werner type continued fraction expression its interpolation structures are analyzed and important characteristic properties are obtained

vector valued rational interpolants i it is shown explicitly that the samelson inverse may be used to define vector valued thiele type rational interpolants and the equivalence with claessens algorithm is established for the special case of vector valued pade approximants the general form of mccleod s theorem follows as a corollary

convergence theorems for rows of differential and algebraic hermite pade approximations abstract the authors investigate the asymptotic behaviour of hermite pade polynomials of latin type which are formed from differential or algebraic expressions involving a meromorphic function the degrees of the denominator polynomials are chosen to correspond to rows of the hermite pade table for special choices of denominator degrees these asymptotic are used to prove de montessus type theorems for hermite pade approximants formed by solving differential or algebraic equations

multivariate generalized inverse vector valued rational interpolants abstract bivariate rational interpolating functions of the type introduced in 9 1 are shown to have a natural extension to the case of rational interpolation of vector valued quantities using the formalism of graves morris 2 in this paper the convergence of stieltjes type branched vector valued continued fractions for two variable functions are constructed by using the samelson inverse based on them a kind of bivariate vector valued rational interpolating function is defined on plane grids sufficient conditions for existence characterisation and uniqueness for the interpolating functions are proved the results in the paper are illustrated with some examples

generalized inverse matrix pade approximation on the basis of scalar products a new type of generalized matrix inverse is used to define the generalized inverse matrix pade approximants gmpa gmpa is introduced on the basis of scalar product of matrices with the form of matrix numerator and scalar denominator it is different from the existing matrix pade approximants in that it does not need multiplication of matrices in the construction process some algebraic properties are discussed the representations of gmpa are provided with the following three forms i the explicit determinantal formulas for the denominator scalar polynomials and the numerator matrix polynomials ii e algorithm expression iii thiele type continued fraction expression the equivalence relations above three representations are proposed

the epsilon algorithm and related topics the epsilon algorithm is recommended as the best all purpose acceleration method for slowly converging sequences it exploits the numerical precision of the data to extrapolate the sequence to its limit we explain its connections with pade approximation and continued fractions which underpin its theoretical base then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector valued sequences and some related topics in this paper we consider the class of methods based on using generalised inverses of vectors and the formulation specifically includes the complex case wherever possible

a recursive computation scheme for multivariate rational interpolants we derive here a recursive computation scheme for the rational interpolation method introduced in 7 explicit formulas for these multivariate rational interpolants are repeated in 2 while the recursive algorithm is described in 3 a number of interesting special cases such as the univariate rational interpolation problem and the multivariate pade approximants introduced in 6 and 10 are dealt with in 4 for some of these rational approximants other recursive schemes were described previously finally 5 contains the numerical results the multivariate rational interpolants described here are compared with multivariate polynomial interpolants interpolating branched continued fractions introduced by cuyt and verdonk 8 interpolating branched continued fractions introduced by siemaszko 12 and several multivariate pade approximants 6 3 10 besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in ord

a worpitzky theorem for vector valued continued fractions generalizations of worpitzky s convergence theorem for ordinary continued fractions are proved for vector valued continued fractions defined by the samelson inverse for vectors our convergence proof is constructive and thus yields more refined truncation error estimates this work shows that some of our results are exact generalizations or improvements of the scalar ones

composite schemes for multivariate blending rational interpolation it is demonstrated that newton s interpolation polynomials and thiele s interpolating continued fractions can be incorporated to generate various interpolation schemes based on rectangular grids among them are two kinds of bivariate blending rational interpolants however blending rational interpolants strongly depend on the existence of so called blending differences which means that for some grids of data one may fail to find out the corresponding rational interpolants as a whole in this paper we offer a solution scheme by adopting composite interpolation over triangular sub grids characterization theorem is given error estimation is worked out and vector valued case as well as matrix valued case is discussed

threshold values and convergence properties of majority based algorithms for decoding regular low density parity check codes this work presents a detailed study of a family of binary message passing decoding algorithms for low density parity check ldpc codes referred to as majority based algorithms both gallager s algorithm a g sub a and the standard majority decoding algorithm belong to this family these algorithms which are in fact the building blocks of gallager s algorithm b g sub b work based on a generalized majority decision rule and are particularly attractive for their remarkably simple implementation we investigate the dynamics of these algorithms using density evolution and compute their noise threshold values for regular ldpc codes over the binary symmetric channel for certain ensembles of codes and certain orders of majority based algorithms we show that the threshold value can be characterized as the smallest positive root of a polynomial and thus can be determined analytically we also study the convergence properties of majority based algorithms including their convergence speed our analysis shows that the stand alone version of some of these algorithms provides significantly better performance and or convergence speed compared with g sub a in particular it is shown that for channel parameters below threshold while for g sub a the error probability converges to zero exponentially with iteration number this convergence for other majority based algorithms is super exponential

a fitzpatrick algorithm for multivariate rational interpolation abstract in this paper we first apply the fitzpatrick algorithm to osculatory rational interpolation then based on a fitzpatrick algorithm we present a neville like algorithm for cauchy interpolation with this algorithm we can determine the value of the interpolating function at a single point without computing the rational interpolating function

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

the simplified topological varepsilon algorithms for accelerating sequences in a vector space when a sequence of numbers is slowly converging it can be transformed into a new sequence which under some assumptions converges faster to the same limit one of the best known sequence transformations is the shanks transformation which can be recursively implemented by the varepsilon algorithm of wynn this transformation and this algorithm have been extended in two different ways to a sequence of elements of a topological vector space e in this paper we present new algorithms for implementing these topological shanks transformations they no longer require the manipulation of elements of the algebraic dual space e of e nor do they use the duality product inside the rules of the algorithms they need the storage of fewer elements of e and the stability is improved they also allow us to prove convergence and acceleration results for some types of sequences various applications involving sequences of vectors or matrices show the interest of the new algorithms

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

the construction of λµ b spline curves and its application to rotational surfaces in order to provide more flexible approaches for designers the b spline curves are constructed as a generalization of the traditional cubic uniform b spline curves possessing multiple local shape control parameters b spline curves not only inherit the properties of cubic uniform b spline curves but also exhibit better performance when adjusting its local shapes through two local shape control parameters particularly to adjust and control the shapes of rotational surfaces more elegantly the b spline rotational surfaces with two local shape parameters are presented and utilized a rotational surface is produced by combining b spline with a transfinite vector valued rational interpolation function further the properties of rotational surfaces as well as its applications in rotational surface designs are explored finally the modeling examples are supplied to illustrate the proposed method in admitting the easy control of the shape of a surface which suggest the much wider applications to the pattern design system of apparel cad cam

newton pade approximations for multivariate functions abstract we modify cuyt and verdonk s approach to multivariate newton pade approximations explicit formulas are given for coefficients which can be computed once for given system of nodes and the linear system of equations to find multivariate newton pade approximants can be written using simple formulas with those coefficients and the newton series of interpolated function