By Michael T. Heideman, C.S. Burrus
This publication is meant to be a entire connection with multiplicative com plexity conception as utilized to electronic sign processing computations. even if a number of algorithms are incorporated to demonstrate the idea, I targeted extra at the enhance ment of the idea itself. Howie Johnson's infectious enthusiasm for designing effective DfT algorithms received me drawn to this topic. i'm thankful to Prof. Sid Burrus for encouraging and assisting me during this attempt. i'd additionally wish to thank Henrik Sorensen and Doug Jones for lots of stimulating discussions. lowe a good debt to Shmuel Winograd, who, nearly singlehandedly, supplied many of the key theoretical effects that ended in this current paintings. His monograph, mathematics Complexity o/Computations, brought me to the mechanism at the back of the proofs of theorems in multiplicative complexity. allowing me to come to his past papers and savour the beauty of his equipment for deriving the idea. the second one key paintings that inspired me used to be the paper through Louis Auslander and Winograd on multiplicative complexity of semilinear structures outlined by means of polynomials. After examining this paper, it was once transparent to me that this concept may be utilized to many impor tant computational difficulties. those impacts might be simply discerned within the current work.
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Additional info for Multiplicative Complexity, Convolution, and the DFT
M) where the set of ordered pairs (t,m) is ordered lexicographically. We will show that for any (t,m) the system (tA(j»My can be modified to yield a smaller system of the same form requiring fewer mid steps than (tA(J»My. The difference between the number of mid steps required for this smaller system and the original system can be bounded and the lower bound is always greater than zero, hence this type of reduction of a system can be recursively applied to obtain a lower bound on the number of mid steps required to compute (tA(j»My.
R}y" where 4>"(rj is the t+t' x n+n' block diagonal matrix in which the upper left block is the tX n matrix Ill(r) and the lower right block is the (Xn' matrix 4>'(x'). The vector y" is the concatenation o/the entries ofy andy'. B(S'; G), but the conjecture has not been proven in generaL It is also conjectured that all minimal algorithms for computing S'" compute Sand S' separately. The combination of these two conjectures is referred to as the Direct Sum Conjecture [17,471 and is still an open problem.
The concepts of reduction and inflation mappings can be applied to the ring rij of an s x r 9l-matrix with PF(ri ), yielding an sn xrn F -matrix. The inflation mapping from sn xrn F -matrices to s x r 9l-matrices exists under conditions similar to those specified for the inflation mapping from G -matrices to K -matrices. 9i = F ® K. 3. Equivalence of Products with a Fixed Polynomial The previous two sections have provided an algebraic framework and valuable tools for the analysis of systems consisting of a union of products of a single polynomial by several polynomials.
Multiplicative Complexity, Convolution, and the DFT by Michael T. Heideman, C.S. Burrus