On the Solution of Linear Recurrence Equations

  title={On the Solution of Linear Recurrence Equations},
  author={Mohamad A. Akra and Louay Bazzi},
  journal={Computational Optimization and Applications},
AbstractIn this article, we present a general solution for linear divide-and-conquer recurrences of the form $$u_n = \sum\limits_{i = 1}^k {a_i u} $$ ⌊ $$\frac{n}{{b_i }}$$ ⌋ + g(n) Our approach handles more cases than the Master method does {1}. We achieve this advantage by defining a new transform - the Order transform - which has useful properties for providing asymptotic answers (compared to other transforms which supply exact answers). This transform helps in mapping the sequence under… 
A Simple Master Theorem For Discrete Divide And Conquer Recurrences
The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: $$X_n = a_n+\sum_{j=1}^m b_j X_{\lfloor p_j n\rfloor},$$ where the $p_i$'s belong to $(0,1)$. The
Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half
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Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term
This paper gives an explicit expression —in terms of the binary decomposition of n— for the solution xn of a recurrence of this form, with given initial condition x1, when the independent term p(n) is a polynomial in ⌈n/2⌉ and ⌊n/ 2⌋.
A master theorem for discrete divide and conquer recurrences
Powerful techniques such as Dirichlet series, Mellin-Perron formula, and (extended) Tauberian theorems of Wiener-Ikehara are applied to provide a complete and precise solution to this basic computer science recurrence.
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A method for obtaining tail-bounds for random variables satisfying certain probabilistic recurrences that arise in the analysis of randomized parallel divide and conquer algorithms, and shows that in some cases, the work-recurrence can be bounded under simpler assumptions than Karp's.
Fast and Stable Pascal Matrix Algorithms
A family of fast and stable algorithms for multiplying and inverting Pascal matrices that run in O(n log^2 n) time and are closely related to De Casteljau's algorithm for B\'ezier curve evaluation are derived.
Notes on Better Master Theorems for Divide-and-Conquer Recurrences
A simple inductive proof of the Akra-Bazzi result is provided and the result is extended to handle variations of divide-and-conquer recurrences that commonly arise in practice.
Hitting Forbidden Minors: Approximation and Kernelization
A number of algorithmic results on the problem parameterized by p, using p to refer to the parameterized version of the problem, and polynomial kernels for the case when $\cal F$ only contains graph $\theta_c$ as a minor for a fixed integer $c$...
J an 2 01 9 On a Polytime Factorization Algorithm for Multilinear Polynomials over F 2 ⋆
An improvement of this factorization algorithm based on computations over derivatives of multilinear polynomials is described and preliminary experimental analysis is reported on.
On A Polytime Factorization Algorithm for Multilinear Polynomials Over F2
An improvement of this factorization algorithm based on computations over derivatives of multilinear polynomials is described and preliminary experimental analysis is reported on.


A general method for solving divide-and-conquer recurrences
A unifying method for solving recurrence relations of the form T(n) = kT(n/c) + f( n) is described that is both general in applicability and easy to apply.
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Fundamental Algorithms, volume 1 of The Art of Computer Programming
  • AddisonWesley,
  • 1968