Second-order Linear Recurrence Research Articles

Enumerating the Rationals from Left to Right

The rationals can be enumerated using Stern-Brocot sequences 풮ℬn, Calkin-Wilf sequences 풞풲n, or Farey sequences ℱn. We show that the rationals in 풮ℬn, 풞풲n, and ℱn can be computed using very similar second-order linear recurrence relations. (This, incidentally, obviates the need to compute previous sequences 풮ℬi, 풞풲i, ℱi with i < n.)

Reduce Computation Steps Can Increase the Efficiency of Computation Algorithm

Problem statement: Lucas function is a special form of second-order linear recurrence relation. It is used in LUC cryptosystems. The performance of LUC cryptosystem depends on the size of public key, messages and two relatively primes. The increasing of size of these parameters will increase the computation time need to perform the LUC Cryptosystem computation. The efficiency means the quality to avoid wasted time. Approach: Therefore, the main theme of this study was to design and implement an improve version of computation algorithm. The efficiency of computation can skip some computations time for a computerized calculation. Smaller computation time means the algorithm is better and more efficient than the other algorithm. In this study, the technique on reducing redundant number of computations steps in LUC Cryptosystem was investigated. The use of two variables w and t were proposed in order to reduce some computations steps in LUC Cryptosystem computation. Results: The new technique showed a better computation time compared to the existing algorithm. It also reduced some redundant multiplications without sacrificed the security of LUC Cryptosystem. At the same time, it increases the efficiency of computation algorithm. Conclusion: The proposed algorithm showed better speed and efficiency by reducing some redundant computation steps. It can reduce up to 20% of computation efforts compare to the existing one.

Recursive sequences and polynomial congruences

We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer m. We show that the period of such a sequence with characteristic polynomial f can be expressed in terms of the order of!D xCh fi as a unit in the quotient ring ZmT!UDZmTxU=h fi. When mD p is prime, this order can be described in terms of the factorization of f in the polynomial ring Z pTxU. We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree k 5 in Z pTxU. This article grew out of an undergraduate research project, performed by the second author under the direction of the first, to determine if results about the periodicity of second-order linear homogeneous recurrence relations modulo positive integers could be extended to higher orders. We arrived, somewhat unexpectedly, at algorithms to determine the degrees of the irreducible factors of quintic and smaller degree polynomials modulo prime numbers. The algebraic properties of certain finite rings, particularly automorphisms of those rings, provided the connection between these two topics. To illustrate some of the ideas in this article, we begin with the famous example of the Fibonacci sequence, defined by FnD Fn 1C Fn 2 with F0D 0 and F1D 1. If, for some positive integer m, we replace each Fn by its remainder on division by m, we obtain a new sequence of integers. For example, the Fibonacci sequence modulo mD 10 begins 0; 1; 1; 2; 3; 5; 8; 3; 1; 4; 5; 9; 4; 3; 7; 0; 7; 7; 4; 1; 5; 6; 1; 7; 8; 5;:::; with the n-th term simply the last digit of Fn. We can also view such a sequence as having terms in ZmDZ=hmi, the ring of integers modulo m. This has the advantage

Stability Classes of Second-Order Linear Recurrences Modulo $2^k$ II

Stability Classes of Second-Order Linear Recurrences Modulo $2^k$ II

Generalizing the Combinatorics of Binomial Coefficients via -Nomials

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On Certain Large Random Hermitian Jacobi Matrices With Applications to Wireless Communications

In this paper we study the spectrum of certain large random Hermitian <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Jacobi</i> matrices. These matrices are known to describe certain communication setups. In particular, we are interested in an uplink cellular channel which models mobile users experiencing a soft-handoff situation under joint multicell decoding. Considering rather general fading statistics we provide a closed-form expression for the per-cell sum-rate of this channel in high signal-to-noise ratio (SNR), when an intra-cell time-division multiple-access (TDMA) protocol is employed. Since the matrices of interest are <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tridiagonal</i> , their eigenvectors can be considered as sequences with second-order linear recurrence. Therefore, the problem is reduced to the study of the exponential growth of products of two-by-two matrices. For the case where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> users are simultaneously active in each cell, we obtain a series of lower and upper bound on the high-SNR power offset of the per-cell sum-rate, which are considerably tighter than previously known bounds.

Second-Order Linear Recurrences of Composite Numbers

Second-Order Linear Recurrences of Composite Numbers

Factorizations and representations of the backward second-order linear recurrences

We show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second-order linear recurrences. Also considering how to the negatively subscripted terms of second-order linear recurrences can be connected to Chebyshev polynomials by determinants of these matrices, we give factorizations and representations of these numbers.

Second-order linear recurrences with restricted coefficients and the constant (1/3)^1/3

This paper provides bounds for second-order linear recurrences with restricted coefficients. It is determined that whenever the coefficients of the associated monic equation are less than the constant (1/3)1/3 , all solutions tend to zero at an exponential rate. This constant is optimal. Explicit inequalities are also provided, and some residue class structure is revealed. Mathematics subject classification (2000): 39A10, 11B37, 65Q05.

SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem

We first introduce a second-order Krylov subspace $\mathcal{G}_n$(A,B;u) based on a pair of square matrices A and B and a vector u. The subspace is spanned by a sequence of vectors defined via a second-order linear homogeneous recurrence relation with coefficient matrices A and B and an initial vector u. It generalizes the well-known Krylov subspace $\mathcal{K}_n$(A;v), which is spanned by a sequence of vectors defined via a first-order linear homogeneous recurrence relation with a single coefficient matrix A and an initial vector v. Then we present a second-order Arnoldi (SOAR) procedure for generating an orthonormal basis of $\mathcal{G}_n$(A,B;u). By applying the standard Rayleigh--Ritzorthogonal projection technique, we derive an SOAR method for solving a large-scale quadratic eigenvalue problem (QEP). This method is applied to the QEP directly. Hence it preserves essential structures and properties of the QEP. Numerical examples demonstrate that the SOAR method outperforms convergence behaviors of the Krylov subspace--based Arnoldi method applied to the linearized QEP.

Second-order Krylov subspace and arnoldi procedure

We report our recent work on a second-order Krylov subspace and the corresponding second-order Arnoldi procedure for generating its orthonormal basis. The second-order Krylov subspace is spanned by a sequence of vectors defined via a second-order linear homogeneous recurrence relation with coefficient matrices A and B and an initial vector u. It generalizes the well-known Krylov subspace K n (A; v), which is spanned by a sequence of vectors defined via a first-order linear homogeneous recurrence relation with a single coefficient matrix A and an initial vector v. The applications are shown for the solution of quadratic eigenvalue problems and dimension reduction of second-order dynamical systems. The new approaches preserve essential structures and properties of the quadratic eigenvalue problem and second-order system, and demonstrate superior numerical results over the common approaches based on linearization of these second-order problems.

Second-Order Linear Recurrences of Composite Numbers

Second-Order Linear Recurrences of Composite Numbers

On Second-Order Linear Recurrence Sequences: Wall and Wyler Revisited

On Second-Order Linear Recurrence Sequences: Wall and Wyler Revisited

Algorithmic Manipulation of Second-Order Linear Recurrences

Algorithmic Manipulation of Second-Order Linear Recurrences

On the Existence of Couples of Second-Order Linear Recurrences with Reciprocal Representation Properties for Their Fibonacci Sequences

On the Existence of Couples of Second-Order Linear Recurrences with Reciprocal Representation Properties for Their Fibonacci Sequences

Solving Linear Recurrences with Loop Raking

We present a variation of the partition method for solving linear recurrences that is well-suited to vector multiprocessors. The algorithm fully utilizes both vector and multiprocessor capabilities, and reduces the number of memory accesses and temporary memory requirements as compared to the more commonly used version of the partition method. Our variation uses a general loop restructuring technique called loop raking. We describe an implementation of this technique on the CRAY Y-MP C90, and present performance results for first- and second-order linear recurrences. On a single processor of the C90 our implementations are up to 7.3 times faster than the corresponding optimized library routines in SCILIB, an optimized mathematical library supplied by Cray Research. On four processors, we gain an additional speedup of at least 3.7.

A chained-matrices approach for parallel computation of continued fractions and its applications

A chained-matrices approach for parallel computing thenth convergent of continued fractions is presented. The resulting algorithm computes the entire prefix values of any continued fraction inO(logn) time on the EREW PRAM model or a network withO(n/logn) processors connected by the cube-connectedcycles, binary tree, perfect shuffle, or hypercube. It can be applied to approximate the transcendental numbers, such as π ande, inO(logm) time by usingO(m/logm) processors for a result withm-digit precision. We also use it to costoptimally solve the second-order linear recurrence, the polynomial evaluation, the recurrence of vector norm, the general class of recurrence equation defined by Kogge and Stone (1973), and the generalmth order linear recurrence. It is easy to implement because there are only some matrix multiplications and a division operation involved.

Convergence of solutions of PSL (2, R)-recurrences with parabolic limit

The aim of this paper is the investigation of the convergence of the solutions { Z n } of a sequence of Möbius-transformations with parabolic limit. It is shown that either lim n→∞ Z n exists for all solutions or it exists for none of them. In the first part of the paper a description for the behaviour of solutions in the boundary region (between converging and non-converging type) is given with the aid of a certain class of renormalizations. A generalization of this idea is used in the second part to derive a necessary and sufficient condition for convergence in terms of the coefficients of the Möbius-transformations. Lastly, an application to second-order linear recurrences is given.

An O(log n) algorithm for computing periodic continued fractions and its applications

An O(log n) algorithm for computing the nth convergents of periodic continued fractions is presented. It can be applied to solve the second-order linear recurrences with constant coeffients very efficiently. We also use it to approximate the quadratic numbers in O(log m) time for a result with m-digit precision. Our method can be thought as a generalization of the matrix-vector approach for computing the Fibonacci numbers. It is easy to implement because there are only some matrix multiplications and a division operation involved.

Convergence Acceleration for the Numerical Solution of Second-Order Linear Recurrence Relations

The concept of modification used for accelerating the convergence of ordinary continued fractions is adapted to the case of the Gautschi–Aggarwal–Burgmeier algorithm for the computation of nondominant solutions of nonhomogeneous second-order linear recurrence relations.