Algorithms for Evaluating the Matrix Polynomial I+A+A2+…+AN-1 with Reduced Number of Matrix Multiplications

Kotaro MATSUMOTO  Kazuyoshi TAKAGI  Naofumi TAKAGI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E101-A   No.2   pp.467-471
Publication Date: 2018/02/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E101.A.467
Type of Manuscript: PAPER
Category: Algorithms and Data Structures
evaluation of matrix polynomial,  matrix multiplication,  

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The problem of evaluating the matrix polynomial I+A+A2+…+AN-1 with a reduced number of matrix multiplications has long been considered. Several algorithms have been proposed for this problem, which find a procedure requiring O(log N) matrix multiplications for a given N. Among them, the hybrid algorithm based on the double-base representation of N, i.e., using mixed radices 2 and 3, proposed by Dimitrov and Cooklev is most efficient. It has been suggested by them that the use of higher radices would not bring any more efficient algorithms. In this paper, we show that we can derive more efficient algorithms by using higher radices, and propose several efficient algorithms.