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

the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

low complexity cubing and cube root computation over f_ 3 m in polynomial basis we present low complexity formulae for the computation of cubing and cube root over if3m constructed using special classes of irreducible trinomials tetranomials and pentanomials we show that for all those special classes of polynomials field cubing and field cube root operation have the same computational complexity when implemented in hardware or software platforms as one of the main applications of these two field arithmetic operations lies in pairing based cryptography we also give in this paper a selection of irreducible polynomials that lead to low cost field cubing and field cube root computations for supersingular elliptic curves defined over if3m where m is a prime number in the pairing based cryptographic range of interest namely m 47 541

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

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

efficient inversion of the galerkin matrix of general second order elliptic operators with nonsmooth coefficients this article deals with the efficient approximate inversion of finite element stiffness matrices of general second order elliptic operators with l coefficients it will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices h matrices furthermore numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity

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

a direct solver for elliptic pdes in three dimensions based on hierarchical merging of poincare steklov operators a numerical method for variable coefficient elliptic pdes on three dimensional domains is described the method is designed for problems with smooth solutions and is based on a multidomain spectral collocation discretization scheme the resulting system of linear equations can very efficiently be solved using a nested dissection style direct as opposed to iterative solver this makes the scheme particularly well suited to solving problems for which iterative solvers struggle in particular for problems with oscillatory solutions a principal feature of the scheme is that once the solution operator has been constructed the actual solve is extremely fast an upper bound on the asymptotic cost of the build stage of o n 4 3 is proved for the case where the pde is held fixed as n increases the solve stage has close to linear complexity the scheme requires a relatively large amount of storage per degree of freedom but since it is a high order scheme a small number of degrees of freedom is sufficient to achieve high accuracy the method is presented for the case where there is no body load present but it can with little difficulty be generalized to the non homogeneous case numerical experiments demonstrate that the scheme is capable of solving helmholtz type equations on a domain of size 20 20 20 wavelengths to three correct digits on a modest personal workstation with n 2 10 6

an efficient optimal normal basis type ii multiplier this paper presents a new parallel multiplier for the galois field gf 2 sup m whose elements are represented using the optimal normal basis of type ii the proposed multiplier requires 1 5 m sup 2 m xor gates as compared to 2 m sup 2 m xor gates required by the massey omura multiplier the time complexities of the proposed and the massey omura multipliers are similar

efficient structured multifrontal factorization for general large sparse matrices rank structures provide an opportunity to develop new efficient numerical methods for practical problems when the off diagonal blocks of certain dense intermediate matrices have small numerical

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

a new representation of elements of finite fields gf 2 sup m yielding small complexity arithmetic circuits let f sub 2 denote the binary field and f sub 2m an algebraic extension of degree m 1 over f sub 2 traditionally elements of f sub 2m are either represented as powers of a primitive element of f sub 2m together with 0 or by an expansion in a basis of the vector space f sub 2m over f sub 2 we propose a new representation based on an isomorphism from f sub 2m into the residue polynomial ring module x sup n 1 the new representation simultaneously satisfies the properties of various traditional representations which leads in some cases to architectures of parallel in parallel out arithmetic circuits adder multiplier exponentiator inverter squarer divider with average to small complexity we show that the implementation of all the arithmetic circuits designed for the new representation on an integrated circuit sometimes has smaller complexity than the implementation of all the arithmetic circuits designed for other representations in addition we derive a serial multiplier for the field f sub 2m which comprises the least number of gates of all the serial multipliers known to the author when m 1 is a prime such that 2 is primitive in the field z sub m 1

efficient preconditioning of hp fem matrix sequences with slowly varying coefficients an application to topology optimization abstract we previously introduced a preconditioner that has proven effective for h p fem discretizations of various challenging elliptic and hyperbolic problems the construction is inspired by standard nested dissection and relies on the assumption that the schur complements can be approximated to high precision by hierarchically semi separable matrices the preconditioner is built as an approximate l d m t factorization through a divide and conquer approach this implies an enhanced flexibility which allows to handle unstructured geometric meshes anisotropies and discontinuities we build on our previous numerical experiments and develop a preconditioner update strategy that allows us to handle matrix sequences arising from problems with slowly varying coefficients we investigate the performance of the preconditioner along with the update strategy in context of topology optimization of an acoustic cavity

bit parallel polynomial basis multiplier for new classes of finite fields in this paper three small classes of finite fields gf 2m are found for which low complexity bit parallel multipliers are proposed the proposed multipliers have lower complexities compared to those based on the irreducible pentanomials it is also shown that there does not always exist an irreducible all one polynomial equally spaced polynomial or trinomial for the new classes of fields

a preconditioned finite element solution of the coupled pressure temperature equations used to model trace gas sensors quartz enhanced photoacoustic spectroscopy qepas is a technique for detecting trace gases which relies on a quartz tuning fork resonator to amplify and measure the weak acoustic pressure waves th

improving the run time and quality of nested dissection ordering when performing sparse matrix factorization the ordering of matrix rows and columns has a dramatic impact on the factorization time this paper describes an approach to the reordering problem that produces significantly better orderings than prior methods the algorithm is a hybrid of nested dissection and minimum degree ordering and combines an assortment of different algorithmic advances new or improved algorithms are described for graph compression multilevel partitioning and separator improvement when these techniques are combined the resulting orderings average 39 better than minimum degree over a suite of test matrices while requiring roughly 2 7 times the run time of liu s multiple minimum degree

on a new representation of elements of finite fields gf 2 sup m yielding small complexity arithmetic circuits for original article see g drolet ibid vol 47 no 9 p 938 946 sept 1998 we characterize the smallest n with gf 2 x x sup n 1 containing an isomorphic copy of gf 2 sup m this characterization shows that the representation of finite fields described in a previous issue of the ieee transactions on computers is not optimal as claimed the representation considered there can often be improved significantly

rationality minimality and uniqueness of representation of matrix formal power series abstract in some multivariate time series models a matrix power series is involved these models can be identified as rational models if these series correspond to a matrix rational function moreover it is necessary to answer some questions about minimality and uniqueness of representation the main results of this paper fall within the sphere of matrix pade approximation on the basis of formal power series matrix rational functions of arbitrary dimensions are characterized furthermore we study certain minimality types that are global minimum degrees and row minimum degrees in addition given that the rational representation of the function for the same pair of degrees need not be unique we have obtained conditions to study the uniqueness of said representation and also to find a canonical unique representation moreover we consider an application to special series which is associated with time series models such series leads to new theoretical results relating to matrix pade approximation finally we comment on some illustrative examples

modified redundant representation for designing arithmetic circuits with small complexity we give a modified redundant representation for designing arithmetic circuits with small complexity using our modified redundant representation we improve many of the complexity values significantly our method works for any finite field we also give some applications in cryptography

systematic design of original and modified mastrovito multipliers for general irreducible polynomials this paper considers the design of bit parallel dedicated finite field multipliers using standard basis an explicit algorithm is proposed for efficient construction of mastrovito product matrix based on which we present a systematic design of mastrovito multiplier applicable to gf 2 sup m generated by an arbitrary irreducible polynomial this design effectively exploits the spatial correlation of elements in mastrovito product matrix to reduce the complexity using a similar methodology we propose a systematic design of modified mastrovito multiplier which is suitable for gf 2 sup m generated by high hamming weight irreducible polynomials for both original and modified mastrovito multipliers the developed multiplier architectures are highly modular which is desirable for vlsi hardware implementation applying the proposed algorithm and design approach we study the mastrovito multipliers for several special irreducible polynomials such as trinomial and equally spaced polynomial and the obtained complexity results match the best known results moreover we have discovered several new special irreducible polynomials which also lead to low complexity mastrovito multipliers

look up table based large finite field multiplication in memory constrained cryptosystems many cryptographic systems use multiplication in the finite field gf 2 sup n for their underlying computations in the recent past a number of look up table based algorithms have been proposed for the software implementation of gf 2 sup n multiplication look up table based algorithms can provide speed advantages but they either require a large memory space or do not fully utilize the resources of the processor on which the software is executed in this work an algorithm for gf 2 sup n multiplication is proposed which can alleviate this problem in each iteration of the proposed algorithm a group of bits of one of the input operands are examined and two look up tables are accessed the groupsize determines the table sizes but does not affect the utilization of the processor resources it can be used for both software and hardware realizations and is particularly suitable for implementations in memory constrained environment such as smart cards and embedded cryptosystems

low complexity bit parallel canonical and normal basis multipliers for a class of finite fields we present a new low complexity bit parallel canonical basis multiplier for the field gf 2m generated by an all one polynomial the proposed canonical basis multiplier requires m sup 2 1 xor gates and m sup 2 and gates we also extend this canonical basis multiplier to obtain a new bit parallel normal basis multiplier

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

homogenization of parabolic equations with a continuum of space and time scales this paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available namely we consider divergence form linear parabolic operators in omega subset mathbb r n with l infty omega times 0 t coefficients it appears that the inverse operator maps the unit ball of l 2 omega times 0 t into a space of functions which at small time and space scales are close in h 1 norm to a functional space of dimension n it follows that once one has solved these equations at least n times it is possible to homogenize them both in space and in time reducing the number of operation counts necessary to obtain further solutions in practice we show under a cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in l 2 instead of h 1 with euclidean coordinates if the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the parabolic equation

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

on the inherent space complexity of fast parallel multipliers for gf 2 sup m a lower bound to the number of and gates used in parallel multipliers for gf 2 sup m under the condition that time complexity be minimum is determined in particular the exact minimum number of and gates for primitive normal bases and optimal normal bases of type ii multipliers is evaluated this result indirectly suggests that space complexity is essentially a quadratic function of m when time complexity is kept minimum

new low complexity bit parallel finite field multipliers using weakly dual bases new structures of bit parallel weakly dual basis wdb multipliers over the binary ground field are proposed an upper bound on the size complexity of bit parallel multiplier using an arbitrary generating polynomial is given when the generating polynomial is an irreducible trinomial x sup m x sup k 1 1 spl les k spl les m 2 the structure of the proposed bit parallel multiplier requires only m sup 2 two input and gates and at most m sup 2 1 xor gates the time delay is no greater than t sub a log sub 2 m 2 t sub x where t sub a and t sub x are the time delays of an and gate and an xor gate respectively

mathcal h h matrix approximability of the inverses of fem matrices we study the question of approximability for the inverse of the fem stiffness matrix for scalar second order elliptic boundary value problems by blockwise low rank matrices such as those given by the mathcal h h matrix format introduced by hackbusch computing 62 2 89 108 1999 we show that exponential convergence in the local block rank r r can be achieved we also show that exponentially accurate lu lu decompositions in the mathcal h h matrix format are possible for the stiffness matrices arising in the fem our analysis avoids any coupling of the block rank r r to the mesh width h h we also cover fairly general boundary conditions of mixed dirichlet neumann robin boundary conditions

a redundant representation of gf q n for designing arithmetic circuits generalizing a construction of silverman 1999 we describe a redundant representation of finite fields gf qn where computations in gf qn are realized through computations in a suitable residue class algebra our focus is on fields of characteristic 2 and we show that the representation discussed here can in particular be used for designing a highly regular multiplication circuit for gf qn

generalised mersenne numbers revisited generalised mersenne numbers gmns were defined by solinas in 1999 and feature in the nist fips 186 2 and secg standards for use in elliptic curve cryptography their form is such that modular reduction is extremely efficient thus making them an attractive choice for modular multiplication implementation however the issue of residue multiplication efficiency seems to have been overlooked asymptotically using a cyclic rather than a linear convolution residue multiplication modulo a mersenne number is twice as fast as integer multiplication this property does not hold for prime gmns unless they are of mersenne s form in this work we exploit an alternative generalisation of mersenne numbers for which an analogue of the above property and hence the same efficiency ratio holds even at bitlengths for which schoolbook multiplication is optimal while also maintaining very efficient reduction moreover our proposed primes are abundant at any bitlength whereas gmns are extremely rare our multiplication and reduction algorithms can also be easily parallelised making our arithmetic particularly suitable for hardware implementation furthermore the field representation we propose also naturally protects against side channel attacks including timing attacks simple power analysis and differential power analysis which is essential in many cryptographic scenarios in constrast to gmns

a modified massey omura parallel multiplier for a class of finite fields a massey omura parallel multiplier of finite fields gf 2 sup m contains m identical blocks whose inputs are cyclically shifted versions of one another it is shown that for fields gf 2 sup m generated by irreducible all one polynomials a portion of the block is independent of the input cyclic shift hence the multiplier contains redundancy by removing the redundancy a modified parallel multiplier is presented which is modular and has a lower circuit complexity

mathcal h matrix based second moment analysis for rough random fields and finite element discretizations we consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method the solution s two point correlation can efficiently be approximated by means of an mathcal h matrix in particular if the correlation length is rather short or the correlation kernel is nonsmooth since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the mathcal h matrix format we can solve the correspondent mathcal h matrix equation in essentially linear time by using the mathcal h matrix arithmetic numerical experiments for three dimensional finite element discretizations for several correlation lengths and different smoothness are provided they validate the presented method and demonstrate that the computation times do not increase for nonsmooth or shortly correlated data

parallel accelerated cyclic reduction preconditioner for three dimensional elliptic pdes with variable coefficients abstract we present a robust and scalable preconditioner for the solution of large scale linear systems that arise from the discretization of elliptic pdes amenable to rank compression the preconditioner is based on hierarchical low rank approximations and the cyclic reduction method the setup and application phases of the preconditioner achieve log linear complexity in memory footprint and number of operations and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment numerical experiments with linear systems that feature symmetry and nonsymmetry definiteness and indefiniteness constant and variable coefficients demonstrate the preconditioner applicability and robustness furthermore it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations and the tuning of the admissibility condition parameter together these parameters allow for optimization of the memory requirements and performance of the preconditioner

a tabular methodology to identify minimal row degrees for matrix pade approximants in this paper we propose a tabular procedure for identifying sets of minimal row degrees corresponding to matrix pade approximants the objective is to make it easier to interpret and properly apply them in various fields when considering formal matrix power series with kxm coefficients it is useful to examine the linearly dependent rows that are in the last k rows in certain hankel matrices corresponding to the coefficients of the series a list of properties and suitable examples are added not only by way of illustration but also because they are an original part of the method having been chosen to teach the procedure and to underscore certain precautions to be taken

h matrix approximation for the operator exponential with applications we develop a data sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly p positive operator 4

a new hardware architecture for operations in gf 2 sup n the efficient computation of the arithmetic operations in finite fields is closely related to the particular ways in which the field elements are presented the common field representations are a polynomial basis representation and a normal basis representation in this paper we introduce a nonconventional basis and present a new bit parallel multiplier which is as efficient as the modified massey omura multiplier using the type i optimal normal basis

bit parallel systolic multipliers for gf 2 sup m fields defined by all one and equally spaced polynomials two operations the cyclic shifting and the inner product are defined by the properties of irreducible all one polynomials an effective algorithm is proposed for computing multiplications over a class of fields gf 2 sup m using the two operations then two low complexity bit parallel systolic multipliers are presented based on the algorithm the first multiplier is composed of m 1 sup 2 identical cells each consisting of one 2 input and gate one 2 input xor gate and three 1 bit latches the other multiplier comprises of m 1 sup 2 identical cells and mxor gates each cell consists of one 2 input and gate one 2 input xor gate and four 1 bit latches each multiplier exhibits very low latency and propagation delay and is thus very fast moreover the architectures of the two multipliers can be applied in computing multiplications over the class of fields gf 2 sup m in which the elements are represented with the root of an irreducible equally spaced polynomial of degree m

concurrent error detection and correction in gaussian normal basis multiplier over gf 2 m fault based cryptanalysis has been developed to effectively break both private key and public key cryptosystems making robust finite field multiplication a very important research topic in recent years however no robust normal basis multiplier has been proposed in the literature therefore this investigation presents a semisystolic gaussian normal basis multiplier based on the proposed gaussian normal basis multiplier both concurrent error detection and correction capabilities can be easily achieved using time redundancy technology with no hardware modification

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

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

structure of parallel multipliers for a class of fields gf 2 m abstract this paper presents a configuration of parallel multipliers for gf 2 m based on canonical bases the possible parallel multipliers by the proposed configuration are limited to a class of fields gf 2 m however they can be constructed by o m 2 and gates and o m 2 eor gates with the structural modularity this is a desirable feature for the hardware implementation and their operation time is about log m t where m is the dimension of gf 2 m and t is the delay time of an eor gate in order to construct such parallel multipliers we define two types of polynomials of special form over gf 2 one is called all one polynomial denoted by aop and the other is called equally spaced polynomial denoted by esp furthermore we show a necessary and sufficient condition for esps to be irreducible over gf 2 and the uniqueness of the irreducible esps over gf 2 finally we propose the configuration of parallel multipliers for a class of fields gf 2 m based on irreducible aops and esps over gf 2

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

low complexity bit parallel multiplier for gf 2 sup m defined by all one polynomials using redundant representation this paper presents a new bit parallel multiplier for the finite field gf 2 sup m defined by an irreducible all one polynomial in order to reduce the complexity of the multiplier we introduce a redundant representation and use the well known multiplication method proposed by karatsuba the main idea is to combine the redundant representation and the karatsuba method to design an efficient bit parallel multiplier as a result the proposed multiplier requires about 25 percent fewer and xor gates than the previously proposed multipliers using an all one polynomial while it has almost the same time delay as the previously proposed ones

a fast direct solver for elliptic problems on general meshes in 2d we present a fast direct algorithm for solutions to linear systems arising from 2d elliptic equations we follow the approach in xia et al 2009 on combining the multifrontal method with hierarchical matrices we present a variant of that approach with additional hierarchical structure extend it to quasi uniform meshes and detail an adaptive decomposition procedure for general meshes linear time complexity is shown for a quasi regular grid and demonstrated via numerical results for the adaptive algorithm

error analysis and iterative solvers for navier stokes projection methods with standard and sparse grad div stabilization abstract this paper shows that use of a recently introduced sparse grad div stabilization can increase the accuracy of projection methods for solving the navier stokes equations sparse grad div stabilization has recently been introduced as an alternative to standard grad div stabilization which has a sparser matrix representation for both sparse and standard grad div stabilized projection methods we prove error estimates and provide numerical experiments which reveal that both stabilizations can cause a significant decrease in the error we then compare iterative solvers for the linear systems of equations arising from the use of both of the stabilizations a theoretical analysis of a simplified model problem as well as numerical tests show that iterative solvers perform better for systems arising from sparse grad div compared to standard grad div stabilized systems

low complexity bit parallel multipliers for a class of finite fields new implementations of bit parallel multipliers for a class of finite fields are proposed the class of finite fields is constructed with irreducible aops all one polynomials and esps equally spaced polynomials the size and time complexities of our proposed multipliers are lower than or equal to those of the previously proposed multipliers of the same class

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

bit parallel finite field multipliers for irreducible trinomials a new formulation for the canonical basis multiplication in the finite fields gf 2 sup m based on the use of a triangular basis and on the decomposition of a product matrix is presented from this algorithm a new method for multiplication named transpositional applicable to general irreducible polynomials is deduced the transpositional method is based on the computation of 1 cycles and 2 cycles given by a permutation defined by the coordinate of the product to be computed and by the cardinality of the field gf 2 sup m the obtained cycles define groups corresponding to subexpressions that can be shared among the different product coordinates this new multiplication method is applied to five types of irreducible trinomials these polynomials have been widely studied due to their low complexity implementations the theoretical complexity analysis of the corresponding bit parallel multipliers shows that the space complexities of our multipliers match the best results known to date for similar canonical gf 2 sup m multipliers the most important new result is the reduction in two of the five studied trinomials of the time complexity with respect to the best known results

vlsi architectures for computing multiplications and inverses in gf 2 m finite field arithmetic logic is central in the implementation of reed solomon coders and in some cryptographic algorithms there is a need for good multiplication and inversion algorithms that can be easily realized on vlsi chips massey and omura 1 recently developed a new multiplication algorithm for galois fields based on a normal basis representation in this paper a pipeline structure is developed to realize the massey omura multiplier in the finite field gf 2m with the simple squaring property of the normal basis representation used together with this multiplier a pipeline architecture is also developed for computing inverse elements in gf 2m the designs developed for the massey omura multiplier and the computation of inverse elements are regular simple expandable and therefore naturally suitable for vlsi implementation

domain decomposition based mathcal h lu preconditioning hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an mathcal h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approximation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based mathcal h matrices this new approach yields mathcal h lu factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an mathcal h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decomposition based mathcal h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

a new finite field multiplier using redundant representation this article presents simple and highly regular architectures for finite field multipliers using a redundant representation the basic idea is to embed a finite field into a cyclotomic ring which is based on the elegant multiplicative structure of a cyclic group one important feature of our architectures is that they provide area time trade offs which enable us to implement the multipliers in a partial parallel hybrid fashion this hybrid architecture has great significance in its vlsi implementation in very large fields the squaring operation using the redundant representation is simply a permutation of the coordinates it is shown that when there is an optimal normal basis the proposed bit serial and hybrid multiplier architectures have very low space complexity constant multiplication is also considered and is shown to have an advantage in using the redundant representation

efficient multiplier architectures for galois fields gf 2 sup 4n this contribution introduces a new class of multipliers for finite fields gf 2 sup n sup 4 the architecture is based on a modified version of the karatsuba ofman algorithm koa by determining optimized field polynomials of degree four the last stage of the koa and the module reduction can be combined this saves computation and area in vlsi implementations the new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with koa based architectures with separate module reduction the new multipliers lead to highly modular architectures and are thus well suited for vlsi implementations three types of field polynomials are introduced and conditions for their existence are established for the small fields where n 2 3 8 which are of primary technical interest optimized field polynomials were determined by an exhaustive search for each field order exact space and time complexities are provided

a type of matrix pade approximant inspired by scalar component models in this paper we define a type of matrix pade approximant inspired by the identification stage of multivariate time series models considering scalar component models of course the formalization of certain properties in the matrix pade approximation framework can be applied to time series models and in other fields specifically we want to study matrix pade approximants as follows to find rational representations or rational approximations of a matrix formal power series with both matrix polynomials numerator and denominator satisfying three conditions a minimum row degrees for the numerator and denominator b an invertible denominator at the origin and c canonical representation without free parameters

an algorithm for computing a pade approximant with minimal degree denominator in this paper a new definition of a reduced pade approximant and an algorithm for its computation are proposed our approach is based on the investigation of the kernel structure of the toeplitz matrix it is shown that the reduced pade approximant always has nice properties which the classical pade approximant possesses only in the normal case the new algorithm allows us to avoid the appearance of froissart doublets induced by computer roundoff in the non normal pade table

a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases abstract this paper proposes a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases normal bases have the following useful property in the case that an element x in gf 2 m is represented by normal bases 2 k power operation of an element x in gf 2 m can be carried out by k times cyclic shift of its vector representation c c wang et al proposed an algorithm for computing multiplicative inverses using normal bases which requires m 2 multiplications in gf 2 m and m 1 cyclic shifts the fast algorithm proposed in this paper also uses normal bases and computes multiplicative inverses iterating multiplications in gf 2 m it requires at most 2 log 2 m 1 multiplications in gf 2 m and m 1 cyclic shifts which are much less than those required in the wang s method the same idea of the proposed fast algorithm is applicable to the general power operation in gf 2 m and the computation of multiplicative inverses in gf q m q 2 n

a new construction of massey omura parallel multiplier over gf 2 sup m the massey omura multiplier of gf 2 sup m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another in the past it was shown that for a class of finite fields defined by irreducible all one polynomials the parallel massey omura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed in this article it is shown that not only does this type of multiplier contain redundancy in that special class of finite fields but it also has redundancy in fields gf 2 sup m defined by any irreducible polynomial by removing the redundancy we propose a new architecture for the normal basis parallel multiplier which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original massey omura normal basis parallel multiplier the proposed multiplier structure is also modular and hence suitable for vlsi realization when applied to fields defined by the irreducible all one polynomials the multiplier s circuit complexity matches the best result available in the open literature

recursiveness in matrix rational interpolation problems abstract we consider the problem of computing solutions to a variety of matrix rational interpolation problems these include the partial realization problem for matrix power series and newton pade hermite pade simultaneous pade m pade and multipoint pade approximation problems along with their matrix generalizations a general recurrence relation is given for solving these problems unlike other previous recursive methods our recurrence works along arbitrary computational paths when restricted to specific paths the recurrence relation generalizes previous work of antoulas cabay and labahn beckermann van barel and bultheel and gutknecht along with others our results rely on the concept of extended m pade approximation introduced in this paper this is a natural generalization of the two point pade approximation problem extended to multiple interpolation points including infinity and matrix laurent and newton series by using module theoretic techniques we determine complete parameterizations of all solutions to this problem our recurrence relation then efficiently computes these parameterizations this recursion requires no conditions on the input data we also discuss the concept of duality which was shown to be of particular interest for a stable computation of those approximants finally we show the invariance of our approximation problem under linear transformations of the extended complex plane

aop arithmetic architectures over gf 2m this paper presents bit serial arithmetic architectures for gf 2 m based on an irreducible all one polynomial first modular multiplier and a squarer are designed with significantly less hardware complexity than previous architectures then two arithmetic architectures are proposed based on the modular multiplier and squarer they can be used as kernel architecture for modular exponentiations which is very important operation in the most of public key cryptosystem since the multipliers have low hardware requirements and regular structures they are suitable for vlsi implementation

hierarchical interpolative factorization for elliptic operators differential equations this paper introduces the hierarchical interpolative factorization for integral equations hif ie associated with elliptic problems in two and three dimensions this factorization takes the form of an approximate generalized lu decomposition that permits the efficient application of the discretized operator and its inverse hif ie is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features 1 a matrix factorization framework for sparsifying structured dense matrices and 2 a recursive dimensional reduction strategy to decrease the cost thus higher dimensional problems are effectively mapped to one dimension and we conjecture that constructing applying and inverting the factorization all have linear or quasilinear complexity numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method direct solver and preconditioner hif ie is compatible with geometric adaptivity and can handle both boundary and volume problems matlab codes are freely available 2015 wiley periodicals inc

look ahead methods for block hankel systems in this paper the updating formulas used by three look ahead methods for solving hankel systems are generalized to the square block case each of the original methods was described in the literature using the terminology of different but strongly related fields formal orthogonal polynomials pade approximants structured matrices this paper gives several of these connections generalized to the block case and shows that each viewpoint has its own merits

modular construction of low complexity parallel multipliers for a class of finite fields gf 2 sup m structures for parallel multipliers of a class of fields gf 2 sup m based on irreducible all one polynomials aop and equally spaced polynomials esp are presented the structures are simple and modular which is important for hardware realization relationships between an irreducible aop and the corresponding irreducible esps have been exploited to construct esp based multipliers of large fields by a regular expansion of the basic modules of the aop based multiplier of a small field some features of the structures also enable a fast implementation of squaring and multiplication algorithms and therefore make fast exponentiation and inversion possible it is shown that if for a certain degree an irreducible aop as well as an irreducible esp exist then from the complexity point of view it is advantageous to use the esp based parallel multiplier

a parallel shared memory implementation of a high order accurate solution technique for variable coefficient helmholtz problems abstract the recently developed hierarchical poincare steklov hps method is a high order discretization technique that comes with a direct solver results from previous papers demonstrate the method s ability to solve helmholtz problems to high accuracy without the so called pollution effect while the asymptotic scaling of the direct solver s computational cost is the same as the nested dissection method serial implementations of the solution technique are not practical for large scale numerical simulations this manuscript presents the first parallel implementation of the hps method specifically we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra this approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes performance results on a desktop computer resembling a large memory node are presented

a direct solver with o n complexity for variable coefficient elliptic pdes discretized via a high order composite spectral collocation method a numerical method for solving elliptic pdes with variable coefficients on two dimensional domains is presented the method is based on high order composite spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct as opposed to iterative solver that has optimal o n complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the laplace and the stokes equations numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of 10 10 or better for challenging problems such as highly oscillatory helmholtz problems and convection dominated convection diffusion equations in terms of speed it is demonstrated that a problem with a nonoscillatory solution that was discretized using 10 8 nodes can be solved in 115 minutes on a personal workstation with two quad core 3 3 ghz cpus since the solver is direct and the solution

efficient scalable algorithms for solving dense linear systems with hierarchically semiseparable structures hierarchically semiseparable hss matrix techniques are emerging in constructing superfast direct solvers for both dense and sparse linear systems here we develop a set of novel parallel algorithms for key hss operations that are used for solving large linear systems these are parallel rank revealing qr factorization hss constructions with hierarchical compression ulv hss factorization and hss solutions the hss tree based parallelism is fully exploited at the coarse level the textttblacs and textttscalapack libraries are used to facilitate the parallel dense kernel operations at the fine grained level we appply our new solvers for discretized helmholtz equations for multifrequency seismic imaging and iteratively solve time harmonic seismic inverse boundary value problems in particular we use the hss algorithms to solve the dense schur complement systems associated with the root separator of the separator tree obtained from nested dissection of the graph of discretized helmholtz equations we

multi core parallel robust structured multifrontal factorization method for large discretized pdes in this paper based on the current mainstream multi core architecture of parallel computer and the robust structured multifrontal factorization in brief rsmf method we propose a multi core parallelization of rsmf in brief mrsmf method mrsmf method parallelizes the nested dissection ordering symbolic decomposition and numerical decomposition of rsmf method which aims to implement these operations on the multi core parallel machine the multi core parallelization of symbolic decomposition and numerical decomposition are based on the binary elimination tree numerical experiments show that the mrsmf method is effective

a neumann neumann preconditioned iterative substructuring approach for computing solutions to poisson s equation with prescribed jumps on an embedded boundary in multifluid problems with surface tension the fluid pressure and its derivative are discontinuous at fluid interfaces we present a cartesian grid embedded boundary method for numerically resolving these discontinuities in which we use neumann neumann preconditioned iterative substructuring to solve the governing equations we validate this method by computing several well known poisson problems with discontinuous coefficients and we compare its performance to an approach based on simple iteration by analogy with the conjugate gradient method we hypothesize that the scaling of the neumann neumann preconditioned iterative substructuring is o h dlnh 1 where h is the cell size and d 2 3 is the dimensionality of the problem in contrast we show that the simple iterative procedure scales like o h d 1 and is slower by a factor of 4000 for a small i e 64x64 cell model calculation with physical parameters corresponding to a 1 5mm air bubble in water we present an analytical model to explain the scaling of this iterative procedure

a novel approach for bit serial ab2 multiplication in finite fields gf 2m this paper presents a new inner product ab 2 multiplication algorithm and effective hardware architecture for exponentiation in finite fields gf 2 m exponentiation is more efficiently implemented by applying ab 2 multiplication repeatedly rather than ab multiplication thus efficient ab 2 multiplication algorithms and simple architectures are the key to implementing exponentiation accordingly this paper proposes an efficient inner product multiplication algorithm based on an irreducible all one polynomial aop and simple architecture which has the same hardware equipment as fenn s ab multiplier the proposed bit serial multiplication algorithm and architecture are highly regular and simpler than those of previous works