on the invertible cellular automata 150 over f p in this work the reversibility problem for cellular automata with rule number 150 over the finite field f p is tackled it is shown that when null boundary conditions are stated the reversibility is independent of the state set f p and it appears when the number of cells of the cellular space satisfies some conditions furthermore the explicit expressions of the inverse cellular automata are computed

a computational algorithm for special nth order pentadiagonal toeplitz determinants abstract in this short note we present a fast and reliable algorithm for evaluating special n th order pentadiagonal toeplitz determinants in linear time the algorithm is suited for implementation using computer algebra systems cas such as macsyma and maple two illustrative examples are given

fast block diagonalization of k tridiagonal matrices abstract in the present paper we give a fast algorithm for block diagonalization of k tridiagonalmatrices the block diagonalization provides us with some useful results e g another der ivation of a very recent result on generalized k fibonacci numbers in m e a el mikkawy t sogabe a new family of k fibonacci numbers appl math comput 215 2010 4456 4461 ef cient symbolic algorithm for computing the matrix determinant 2011 elsevier inc all rights reserved 1 introduction and objectiveslet t k n be the k tridiagonal matrix of order n n i e t k n d 1 0 0 a 1 0 00 d 2 0 0 a 2 0 0 00 d n k a n k b k 1 0 00 b k 2 0 0 0 0 d n 1 00 0 b n 0 0 d n 0bbbbbbbbbbbbbbbbbbb 1ccccccccccccccccccca 1 for example t 3 7 is given as follows t 3 7 d 1 00a 1 00 00 d 2 00a 2 0000d 3 00a 3 0b 4 00d 4 00a 4 0 b 5 00d 5 0000b 6 00d 6 0000b 7 00d 7 0bbbbbbbbbbb 1ccccccccccca 2 0096 3003 see front matter 2011 elsevier inc all rights reserved doi 10 1016 j amc 2011 08 014

an averaging theorem for hamiltonian dynamical systems in the thermodynamic limit it is shown how to perform some steps of perturbation theory if one assumes a measure theoretic point of view i e if one renounces to control the evolution of the single trajectories and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase space for a system of coupled rotators estimates uniform in n for finite specific energy can be obtained in quite a direct way this is achieved by making reference not to the sup norm but rather following koopman and von neumann to the much weaker l2 norm

on the inverse of a general tridiagonal matrix in the current paper a new efficient computational algorithm to find the inverse of a general tridiagonal matrix is presented the algorithm is suited for implementation using computer algebra systems such as maple mathematica matlab and macsyma symbolic and numeric examples are given

a fast elementary algorithm for computing the determinant of toeplitz matrices in recent years a number of fast algorithms for computing the determinant of a toeplitz matrix were developed the fastest algorithm we know so far is of order k 2 log n k 3 where n is the number of rows of the toeplitz matrix and k is the bandwidth size this is possible because such a determinant can be expressed as the determinant of certain parts of the n th power of a related k i k companion matrix in this paper we give a new elementary proof of this fact and provide various examples we give symbolic formulas for the determinants of toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when k is small

comments on a note on a three term recurrence for a tridiagonal matrix abstract in el mikkawy a note on a three term recurrence for a tridiagonal matrix appl math comput 139 2003 503 511 the author obtained a two term recurrence for computing the determinant of a tridiagonal matrix in this note we show that this is an special case of a recurrence for block tridiagonal matrix

on contact anosov flows exponential decay of correlations for co 4 contact anosov flows is established this implies in particular exponential decay of correlations for all smooth geodesic flows in strictly negative curvature

the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

a new computational algorithm for solving periodic tri diagonal linear systems in this article we present a new efficient computational algorithm for solving periodic tri diagonal linear systems the implementation of the algorithm using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an example is given in order to illustrate the algorithm

determinants of block tridiagonal matrices an identity is proven that evaluates the determinant of a block tridiagonal matrix with or without corners as the determinant of the associated transfer matrix or a submatrix of it

on the inverse and determinant of general bordered tridiagonal matrices in the current paper we present a symbolic algorithm that never suffers from breakdown for computing the inverse of general bordered tridiagonal matrices meanwhile an explicit formula of evaluating the determinant of a bordered tridiagonal matrix is also derived the implementation of the proposed algorithm using computer algebra systems cass is straightforward some numerical examples are given in order to demonstrate the performance and validity of the presented results

a note on a fast numerical algorithm for the determinant of a pentadiagonal matrix abstract a fast numerical algorithm for evaluating the determinant of a pentadiagonal matrix has been recently proposed t sogabe a fast numerical algorithm for the determinant of a pentadiagonal matrix appl math comput 196 2008 835 841 the algorithm whose cost is 14 n 28 where n 3 denotes the size of the matrix is composed of two steps first transform a pentadiagonal matrix into sparse hessenberg form second run a numerical algorithm for computing the determinant of the sparse hessenberg matrix in this note it is shown that we have an algorithm with the cost of 13 n 24 by applying twice the idea of the sparse hessenberg transformation to a pentadiagonal matrix

factorization of hessenberg matrices abstract the analysis of any hessenberg matrix as a product of a companion matrix and a triangular matrix is presented in this paper the factors are explicitly given on terms of the entries of the hessenberg matrix and further various factorizations are undertaken applications to comrade and confederate matrices as well as for the corresponding block cases are included

a fast algorithm for evaluating n th order tri diagonal determinants the cost of all existing algorithms for evaluating the nth order determinants numerical analysis 7th edition brooks cole publishing pacific grove ca 2001 is at most o n3 in the current article we present a new efficient computational algorithm for evaluating the nth order tri diagonal determinants with cost o n only the algorithm is suited for implementation using computer algebra systems such as maple and macsyma some examples are given to illustrate the algorithm

non symbolic algorithms for the inversion of tridiagonal matrices a representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices it enables us to introduce through the linear recurrence relations satisfied by such determinants a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix reduced as well as unreduced the numerical approach is preserved here without invoking the symbolic computation for tridiagonal diagonally dominant matrices a scaling transformation on the recurrences allows us to give another algorithm to avoid overflow and underflow

a novel elementary algorithm for solving comrade linear systems abstract recently three computational algorithms for solving comrade linear systems have been proposed by karawia 2013 2007 6 5 and by sogabe 2008 10 in this paper we derive a novel elementary algorithm for solving comrade linear systems based on matrix reordering and the sherman morrison woodbury formula an efficient method of evaluating the determinant of a comrade matrix is also discussed

on linear time solvers for comrade linear systems abstract a linear time solver with total flop count 10 n 11 and 4 n 2 of memory space is introduced for solving determined comrade linear systems of degree n this algorithm is based on a suitable factorization satisfied for every matrix coefficient it performs faster than other specialized linear time solvers proposed recently contrary speedup its stability and that of those specialized from the literature for large order comrade systems is not guaranteed to overcome such weakness an adapted solver is then proposed running in linear time and 7 n o 1 of memory space it takes advantage of the inexpensive givens reduction of the transpose of any comrade matrix and covering also linear systems involving matrices with comrade structure the results are thoroughly illustrated with proper numerical comparisons

a reliable givens lu approach for solving opposite bordered tridiagonal linear systems abstract a fast and reliable numerical solver is proposed to handle determined opposite bordered tridiagonal linear systems a x b a precise sequence of givens rotations on the nonsingular matrix coefficient a allows us to work with a simpler linear system this equivalent system which admits always a trivial lu factorization can be solved in linear time for large order matrices of use in applications the main advantage of this solver with respect to other special algorithms proposed is its stability and risk free performance as it is checked with comparative examples its extension to manage opposite bordered tridiagonal matrix equations is straightforward

the accuracy of solutions to triangular systems triangular systems play a fundamental role in matrix computations it has been prominently stated in the literature but is perhaps not widely appreciated that solutions to triangular systems are usually computed to high accuracy higher than the traditional condition numbers for linear systems suggest this phenomenon is investigated by use of condition numbers appropriate to the componentwise backward error analysis of triangular systems results of wilkinson are unified and extended among the conclusions are that the conditioning of a triangular system depends on the right hand side as well as the coefficient matrix that use of pivoting in lu qr and cholesky factorisations can greatly improve the conditioning of a resulting triangular system and that a triangular matrix may be much more or less ill conditioned than its transpose

a series expansion for the time autocorrelation of dynamical variables we present here a general iterative formula which gives a formal series expansion for the time autocorrelation of smooth dynamical variables for all hamiltonian systems endowed with an invariant measure we add some criteria theoretical in nature which enable one to decide whether the decay of the correlations is exponentially fast or not one of these criteria is implemented numerically for the case of the fermi pasta ulam system and we find indications which might suggest a sub exponential decay of the time autocorrelation of a relevant dynamical variable

oscillation of solutions of neutral partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of neutral partial functional differential equations of the form t p t t u x t i 1 l i t u x t i a t u x t k 1 s a k t u x t k t q x t u x t j 1 m q j x t f j u x t j x t 0 g where is a bounded domain inrnwith a piecewise smooth boundary and is the laplacian in the euclideann spacern

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

extensive adiabatic invariants for nonlinear chains we look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit considering the quadratic part of the klein gordon hamiltonian by a linear change of variables we transform it into a sum of two parts in involution at variance with the usual method of introducing normal modes our constructive procedure allows us to exploit the complete resonance while keeping the extensive nature of the system next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear schrodinger model the fluctuations of this quantity are controlled via gibbs measure estimates independent of the system size for a large set of initial data at low specific energy finally by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory with fluctuation much smaller than expected according to standard statistical estimates

bounding prefix transposition distance for strings and permutations a transposition is an operation that exchanges two adjacent substrings transpositions over permutations the sequences with no repeated symbols are related to genome rearrangements if one of the substrings is restricted to a prefix then it is called a prefix transposition the prefix transposition distance between a pair of strings permutations is the minimum number of prefix transpositions required to transform a given string permutation into another given string permutation for a permutation of length n we improve the current prefix transposition distance upper bound of n log 8n to n log 9 2n for arbitrary strings we prove new prefix transposition distance upper and lower bounds for binary strings of length n we show that n 2 is an upper bound we show that the prefix transposition distance problem is np complete for binary strings

on a constant diagonals matrix in this paper we consider a constant diagonals matrix the matrix was discussed in witula and slota r witula d slota on computing the determinants and inverses of some special type of tridiagonal and constant diagonals matrices appl math comput 189 1 2007 514 527 the authors gave some results on determinant and the inverse of the matrix for some special cases we give lu factorization and then compute determinant of the matrix we determine eigenvalues of the matrix further we obtain some relationships between permanent of the matrix and terms of a certain recurrence relation

a generalized symbolic thomas algorithm for the solution of opposite bordered tridiagonal linear systems in the current paper we present a generalized symbolic thomas algorithm that never suffers from breakdown for solving the opposite bordered tridiagonal obt linear systems the algorithm uses a fill in matrix factorization and can solve an obt linear system in o n operations meanwhile an efficient method of evaluating the determinant of an opposite bordered tridiagonal matrix is derived the computational costs of the proposed algorithms are also discussed moreover three numerical examples are provided in order to demonstrate the performance and effectiveness of our algorithms and their competitiveness with some already existing algorithms all of the experiments are performed on a computer with the aid of programs written in matlab

two algorithms for solving comrade linear systems in this paper we present an efficient computational and symbolic algorithms for solving comrade linear systems the implementation of the algorithms using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an examples are given in order to illustrate the algorithms

tighter upper bound for sorting permutations with prefix transpositions permutations are sequences where each symbol in the given alphabet appears exactly once a transposition is an operation that exchanges two adjacent sublists in a permutation if one of these sublists is restricted to be a prefix then one obtains a prefix transposition transpositions over permutations have applications in genome rearrangements and computer interconnection networks the set of permutations with n symbols derived from the alphabet 0 1 n 1 form a symmetric group that we denote by p n the prefix transposition distance between two permutations p n and p n is the minimum number of prefix transpositions also called moves needed to transform into a prefix transposition can be modeled by a permutation operation a permutation is type of sequence and it is also an operation the generators for prefix transposition operation are a subset of permutation operation as a result transforming an arbitrary p n to an arbitrary p n is equivalent to sorting some p n thus upper bound for transforming any p n into any p n with prefix transpositions is simply the upper bound to sort any permutation p n with moves in the article that establishes the best known upper bound for prefix transposition distance over p n as n log 9 2 n explicit proofs were not given for some cases in this article we provide the missing proofs validating the stated upper bound furthermore we establish an upper bound of n log 7 2 n for prefix transposition distance over p n

on the inverse of a general pentadiagonal matrix in this paper employing the general doolittle factorization an efficient algorithm is developed to find the inverse of a general pentadiagonal matrix which is suitable for implementation using computer algebra systems software such as matlab and maple examples are given to illustrate the efficiency of the algorithm

relaxation times for hamiltonian systems usually the relaxation times of a gas are estimated in the frame of the boltzmann equation in this paper instead we deal with the relaxation problem in the frame of the dynamical theory of hamiltonian systems in which the definition itself of a relaxation time is an open question we introduce a lower bound for the relaxation time and give a general theorem for estimating it then we give an application to a concrete model of an interacting gas in which the lower bound turns out to be of the order of magnitude of the relaxation times observed in dilute gases

on a two term recurrence for the determinant of a general matrix recently a two term recurrence for computing the determinant of a tridiagonal matrix has been found by el mikkawy m el mikkawy a note on a three term recurrence for a tridiagonal matrix appl math comput 139 2003 503 511 then the result has been extended to a block tridiagonal matrix by salkuyeh d k salkuyeh comments on a note on a three term recurrence for a tridiagonal matrix appl math comput 176 2006 442 444 in this note we show that the relation can be obtained for a general matrix and that as a by product we obtain a generalization of the detgtri algorithm m el mikkawy a fast algorithm for evaluating nth order tri diagonal determinants j comput appl math 166 2004 581 584

a fast and reliable algorithm for evaluating nth order pentadiagonal determinants abstract in the current article we present a fast and reliable algorithm for evaluating n th order pentadiagonal determinants in linear time it is a natural generalization of the detgtri algorithm m el mikkawy a fast algorithm for evaluating n th order tri diagonal determinants j comput appl math 166 2004 581 584 the algorithm is suited for implementation using computer algebra systems cas such as macsyma and maple some illustrative examples are given

a fast numerical algorithm for the determinant of a pentadiagonal matrix abstract recently a two term recurrence for the determinant of a general matrix has been found t sogabe on a two term recurrence for the determinant of a general matrix appl math comput 187 2007 785 788 and it leads to a natural generalization of the detgtri algorithm m el mikkawy a fast algorithm for evaluating nth order tridiagonal determinants j comput appl math 166 2004 581 584 for computing the determinant of a tridiagonal matrix in this paper we derive a fast numerical algorithm for computing the determinant of a pentadiagonal matrix from the generalization of the detgtri algorithm

a note on computing the inverse and the determinant of a pentadiagonal toeplitz matrix abstract a fast and effective numerical algorithm for computing the determinant of a pentadiagonal toeplitz matrix has been recently proposed e kilic m el mikkawy a computational algorithm for special nth order pentadiagonal toeplitz determinants appl math comput 199 2 2008 820 822 the complexity of the algorithm is 11n 17 in this paper we present an algorithm with the cost of 9 n 3 for calculating the determinant of a pentadiagonal toeplitz matrix the inverse of a pentadiagonal toeplitz matrix is also considered numerical examples are given to illustrate the effectiveness of our method

numerical algorithms for solving comrade linear systems based on tridiagonal solvers recently two efficient algorithms for solving comrade linear systems have been proposed by karawia a a karawia two algorithms for solving comrade linear systems appl math comput 189 2007 291 297 the two algorithms are based on the lu decomposition of the comrade matrix in this paper two algorithms are presented for solving the comrade linear systems based on the use of conventional fast tridiagonal solvers and an efficient way of evaluating the determinant of the comrade matrix is discussed

a simpler and faster 1 5 approximation algorithm for sorting by transpositions an important problem in genome rearrangements is sorting permutations by transpositions the complexity of the problem is still open and two rather complicated 1 5 approximation algorithms for sorting linear permutations are known bafna and pevzner 98 and christie 99 the fastest known algorithm is the quadratic algorithm of bafna and pevzner in this paper we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions hence all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations our main result is a new o n3 2 log n 1 5 approximation algorithm which is considerably simpler than the previous ones and whose analysis is significantly less involved

an averaging theorem for fpu in the thermodynamic limit consider an fpu chain composed of n gg 1 particles and endow the phase space with the gibbs measure corresponding to a small temperature beta 1 given a fixed k we construct k packets of normal modes whose energies are adiabatic invariants i e are approximately constant for times of order beta 1 a a 0 for initial data in a set of large measure furthermore the time autocorrelation function of the energy of each packet does not decay significantly for times of order beta the restrictions on the shape of the packets are very mild all estimates are uniform in the number n of particles and thus hold in the thermodynamic limit n rightarrow infty beta 0

reliable solution of tridiagonal systems of linear equations in this paper we present new formulas for characterizing the sensitivity of tridiagonal systems that are independent of the condition number of the underlying matrix we also introduce efficient algorithms for solving tridiagonal systems of linear equations which are stable and reliable namely stable in the backward sense and little sensitive to perturbations in the coefficients

generalized langevin equations for systems with local interactions we present a new method to approximate the mori zwanzig mz memory integral in generalized langevin equations describing the evolution of smooth observables in high dimensional nonlinear systems with local interactions building upon the faber operator series we recently developed for the orthogonal dynamics propagator and an exact combinatorial algorithm that allows us to compute memory kernels from first principles we demonstrate that the proposed method is effective in computing auto correlation functions intermediate scattering functions and other important statistical properties of the observable we also develop a new stochastic process representation of the mz fluctuation term for systems in statistical equilibrium numerical applications are presented for the fermi pasta ulam model and for random wave propagation in homegeneous media

a new recursive algorithm for inverting general tridiagonal and anti tridiagonal matrices in the current article the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti tridiagonal matrices an illustrative example is given

sorting permutations with transpositions in o n3 amortized time abstract the problem of sorting a permutation over alphabet 1 2 3 n with the minimum number of transpositions is known to be intractable that is the computation of transposition distance i e d t is hard such sorting is equivalent to determining the distance between any of the symmetric group sn over under the transposition operation it has applications in computer interconnection networks where a node is denoted by a permutation and genetics where a genome is modeled by a permutation the maximum distance between any pair of permutations in sn is the diameter i e diam s n of the corresponding cayley graph s n it corresponds to the maximum latency in the computer interconnection network modeled by s n and the maximum genetic dissimilarity between any pair in sn we call transpositions prefix transpositions suffix transpositions and prefix suffix transpositions as block moves we design an algorithm to compute d t in sn and compute diam s n with transpositions in o n n 3 time and o n n 2 space at an amortized time of o n 3 the same can be computed for the remaining block moves in o n 2 log n amortized time and o n n space

two algorithms for solving a general backward tridiagonal linear systems in this paper we present an efficient computational and symbolic algorithms for solving a backward tridiagonal linear systems the implementation of the algorithm using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an examples are given in order to illustrate the algorithms

a note on a fast breakdown free algorithm for computing the determinants and the permanents of k tridiagonal matrices k tridiagonal matrices have attracted much attention in recent years which are a generalization of tridiagonal matrices in this note a breakdown free numerical algorithm of o n is presented for computing the determinants and the permanents of k tridiagonal matrices even though the algorithm is not a symbolic algorithm it never suffers from breakdown furthermore it produces exact values when all entries of the k tridiagonal matrices are given in integer in addition the algorithm can be simplified for a general symmetric toeplitz case and it generates the kth powers of fibonacci pell and jacobsthal numbers for a certain symmetric toeplitz case

a note on a three term recurrence for a tridiagonal matrix in this article the author shows that under certain conditions a three term recurrence for a tridiagonal matrix becomes a two term recurrence using this new recurrence the possibility of the lu factorization of any tridiagonal matrix is now easy to investigate the positive definiteness of any real symmetric tridiagonal matrix is now easy to check an algorithm for solving any linear system with positive definite tridiagonal matrix is given some numerical examples are given

a numerical solver for general bordered tridiagonal matrix equations abstract to overcome several limitations of symbolic algorithms introduced recently for matrices of large order a fast numerical solver is proposed for the matrix linear equation a x b where the n n coefficient matrix a is a general nonsingular bordered tridiagonal matrix its sparse structure is preserved through partial givens reduction in particular the matrix inverse of a can be computed for a wide range of bordered tridiagonal linear systems a x b the solution is computed in linear time using back substitution and sherman morrison s formula numerical comparisons illustrate the results

new algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems abstract recently an efficient computational algorithm for solving periodic pentadiagonal linear systems has been proposed by karawia a a karawia a computational algorithm for solving periodic pentadiagonal linear systems appl math comput 174 2006 613 618 the algorithm is based on the lu factorization of the periodic pentadiagonal matrix in this paper new algorithms are presented for solving periodic pentadiagonal linear systems based on the use of any pentadiagonal linear solver in addition an efficient way of evaluating the determinant of a periodic pentadiagonal matrix is discussed the corresponding results in this paper can be readily obtained for solving periodic tridiagonal linear systems

on a homogeneous recurrence relation for the determinants of general pentadiagonal toeplitz matrices pentadiagonal toeplitz matrices frequently arise in many application areas and have been attracted much attention in recent years in this paper we present a numerical algorithm of o log n for computing the determinants of general pentadiagonal toeplitz matrices without imposing any restrictive conditions in addition we investigate some special pentadiagonal toeplitz determinants and their relations to well known number sequences and give a few identity formulas for the ordinary fibonacci sequences and the generalized k fibonacci sequences

explicit formula for the inverse of a tridiagonal matrix by backward continued fractions in this paper we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse for this purpose considering usual continued fraction we define backward continued fraction for a real number and give some basic results on backward continued fraction we give the relationships between the usual and backward continued fractions then we reobtain the lu factorization and determinant of a tridiagonal matrix furthermore we give an efficient and fast computing method to obtain the elements of the inverse of a tridiagonal matrix by backward continued fractions comparing the earlier result and our result on the elements of the inverse of a tridiagonal matrix it is seen that our method is more convenient efficient and fast