This paper presents a new data representation known as Dual FiXedpoint (DFX), which employs a single bit exponent to select two different fixedpoint scalings. DFX provides a compromise between conventional fixed-point and floating-point representations. It has the implementation complexity similar to that of a fixed-point system together with the improved dynamic range offered by a floating-poi...

Since a number of signal processing algorithms requiring the EVD of a symmetric matrix are implemented on fixedpoint DSPs, there is considerable interest in matrix diagonalization algorithms that can be implemented in fixed-point arithmetic. We propose modifications to the Jacobi Cyclic Row algorithm that favor a fixed-point implementation. We then compare the eigenvalues and eigenvectors obtai...

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...

This correspondence considers error analysis of blockimplemented 2-D digital filters. Expressions for error bound and meansquare error for roundoff error accumulation are derived using fixedpoint arithmetic, and compared with the results obtained usingordinary 2-D difference equations.

A special final coalgebra theorem, in the style of Aczel’s [2], is proved within standard Zermelo-Fraenkel set theory. Aczel’s AntiFoundation 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 Lemmas ...

The classical method for constructing the least fixedpoint of a recursive definition is to generate a sequence of functions whose initial element is the totally undefined function and which converges to the desired least fixedpoint. This method, due to Kleene, cannot be generalized to allow the construction of other fixedpoints. In this paper we present an alternate definition of convergence an...

We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations that have right-hand sides that are polynomials in the state variables. In order to verify non-trivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required di...

We introduce Petruchio, a tool for computing Petri net translations of dynamic networks. To cater for unbounded architectures beyond the capabilities of existing implementations, the principle fixedpoint engine runs interleaved with coverability queries. We discuss algorithmic enhancements and provide experimental evidence that Petruchio copes with models of reasonable size.

I show that the general implicit-function problem (or parametrized fixedpoint problem) in one complex variable has an explicit series solution given by a trivial generalization of the Lagrange inversion formula. I give versions of this formula for both analytic functions and formal power series.