Multiplication of n-digit integers by long multiplication requires O(n2) operations and can be timeconsuming. In 1970 A. Schönhage and V. Strassen published an algorithm capable of performing the task with only O(n log(n)) arithmetic operations over C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixedpo...

This article describes some of our recent work in the development of computer architectures for efficient execution of artificial neural network algorithms. Our earlier system, the Ring Array Processor (RAP), was a multiprocessor based on commercial DSPs with a low-latency ring interconnection scheme. We have used the RAP to simulate variable precision arithmetic to guide us in the design of ar...

In embedded systems, efficient implementations of numerical algorithms typically use the fixed-point arithmetic rather than the standardized and costly floating-point arithmetic. But, fixed-point programmers face two difficulties: First, writing fixed-point codes is tedious and error prone. Second, the low dynamic range of fixed-point numbers leads to the persistent belief that fixedpoint compu...

A grove theory is a Lawvere algebraic theory T for which each hom-set T (n, p) is a commutative monoid; composition on the right distrbutes over all finite sums: ( ∑ i∈F fi) ·h = ∑ i∈F fi ·h. A matrix theory is a grove theory in which composition on the left and right distributes over finite sums. A matrix theory M is isomorphic to a theory of all matrices over the semiring S = M(1, 1). Example...

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo-Fraenkel set theory. Aczel’s Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel’s solution and substitution lemma...

̈ find x ∈ R s. t. x i ≥ 0, Fi(x)≥ 0, x iFi(x) = 0, for i = 1, · · · ,n, where F(x) : R → R is assumed to be continuously differentiable. There have been extensive studies for complementarity problems, see for example [5] and a recent review paper [7]. The NCP was introduced by Cottle [4] in his Ph.D. thesis in 1964. At the beginning, it was recognized that the NCP is a special case of a variati...

Signal processing theory and practice are enabling and driving forces behind multimedia devices, communications systems, and even such diverse fields as automotive and medical systems. Over 90% of the signal processing systems on the market used fixed-point arithmetic because of the cost, power, and area savings that fixed-point systems provide. However, most colleges and universities do not te...