In this paper we define a new type of fixedpoint of recursive definitions and investigate some of its properties. This opt imal fixedpoint (which always uniquely exists) contains, in some sense, the maximal amount of “interesting” information which can be extracted from the recursive definition, and it may be strictly more defined than the program’s least fixedpoint. This fixedpoint can be the ...

We consider an explicit iterate-to-fixedpoint operator and derive associated rules for both forward and reverse mode algorithmic differentiation. Like other AD transformation rules, these are exact and efficient. In this case, they generate code which itself invokes the iterate-to-fixedpoint operator. Loops which iterate until a variable changes less than some tolerance should be regarded as ap...

Fixedpoint theorems play a fundamental role in denotational semantics of programming languages. In 1955, Tarski [6] and Davis [l] showed that a lattice L is complete if and only if every monotonic function f: L + L has a fixedpoint. Since then, other fixedpoint properties are considered, by imposing either the function to be o-continuous or the fixedpoint to be the least fixedpoint. The key que...

OurmainpurposeofthispaperistointroducethemodifiedMannandIshikawaiteratesforfindingacommonattractivepoint of a finite family multivalued nonexpansive mappings in the setting uniformly convex Banach spaces. Weobtain necessary and sufficient conditions to guarantee strong convergence proposed algorithms withoutclosedness domain such mappings. Moreover, we derive some consequences from our main res...

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...

I give short and constructive proofs of Tarski’s fixedpoint theorem, and of Zhou’s extension of Tarski’s fixed-point theorem to set-valued maps.

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.

This paper presents a fixedpoint approach to inductive definitions. Instead of using a syntactic test such as “strictly positive,” the approach lets definitions involve any operators that have been proved monotone. It is conceptually simple, which has allowed the easy implementation of mutual recursion and iterated definitions. It also handles coinductive definitions: simply replace the least f...