# On the Solution of Linear Recurrence Equations

@article{Akra1998OnTS,
title={On the Solution of Linear Recurrence Equations},
author={Mohamad A. Akra and Louay Bazzi},
journal={Computational Optimization and Applications},
year={1998},
volume={10},
pages={195-210}
}
• Published 1998
• Mathematics, Computer Science
• 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…
52 Citations
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
• Mathematics
• 2017
Divide-and-conquer recurrences of the form f(n) = f (⌊ n/2⌋ ) + f ( ⌈ n/2⌉ ) + g(n) (n⩾ 2), with g(n) and f(1) given, appear very frequently in the analysis of computer algorithms and related areas.
Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term
• Mathematics
ArXiv
• 2021
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
• Computer Science, Mathematics
SODA '11
• 2011
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.
Probabilistic Recurrence Relations for Work and Span of Parallel Algorithms
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
• Computer Science
ArXiv
• 2017
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
• Mathematics, Computer Science
SIAM J. Discret. Math.
• 2016
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 ⋆
• Computer Science, Mathematics
• 2019
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
• Computer Science, Mathematics
• 2018
An improvement of this factorization algorithm based on computations over derivatives of multilinear polynomials is described and preliminary experimental analysis is reported on.

## References

SHOWING 1-3 OF 3 REFERENCES
A general method for solving divide-and-conquer recurrences
• Computer Science
SIGA
• 1980
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.
Fundamental Algorithms, volume 1 of The Art of Computer Programming