scholarly journals The General Purpose Analog Computer and Computable Analysis are Two Equivalent Paradigms of Analog Computation

2006 ◽  
pp. 631-643 ◽  
Author(s):  
Olivier Bournez ◽  
Manuel L. Campagnolo ◽  
Daniel S. Graça ◽  
Emmanuel Hainry
Keyword(s):  
Analog Computer ◽  
General Purpose ◽  
Computable Analysis ◽  
Analog Computation

Tracking computability of GPAC-generable functions

2020 ◽  
Author(s):  
Diogo PoÇas ◽  
Jeffery Zucker
Keyword(s):  
Physical Process ◽  
Data Streams ◽  
Physical System ◽  
Analog Computer ◽  
General Purpose ◽  
Digital Model ◽  
Computable Analysis ◽  
Analog Computation ◽  
Fréchet Spaces ◽  
General Data

Abstract Analog computation attempts to capture any type of computation, that can be realized by any type of physical system or physical process, including but not limited to computation over continuous measurable quantities. A pioneering model is the General Purpose Analog Computer (GPAC), initially presented by Shannon in 1941. The GPAC is capable of manipulating real-valued data streams; however, it has been shown to be strictly less powerful than other models of computation on the reals, such as computable analysis. In previous work, we proposed an extension of the Shannon GPAC, denoted LGPAC, designed to overcome its limitations. Not only is the LGPAC model capable of expressing computation over general data spaces $\mathcal{X}$, but it also directly incorporates approximating computations by means of a limit module. An important feature of this work is the generalisation of the framework of the computation theory from Banach to Fréchet spaces. In this paper, we compare the LGPAC with a digital model of computation based on effective representations (tracking computability). We establish general conditions under which LGPAC-generable functions are tracking computable.

Digital simulation of analog computation and Church's thesis

1989 ◽  
Vol 54 (3) ◽  
pp. 1011-1017 ◽  
Author(s):  
Lee A. Rubel
Keyword(s):  
Digital Simulation ◽  
Analog Computer ◽  
General Purpose ◽  
Algebraic Differential Equation ◽  
Analog Computation ◽  
Church’S Thesis ◽  
Black Boxes ◽  
Church's Thesis ◽  
Research Questions ◽  
The Given

Church's thesis, that all reasonable definitions of “computability” are equivalent, is not usually thought of in terms of computability by a continuous computer, of which the general-purpose analog computer (GPAC) is a prototype. Here we prove, under a hypothesis of determinism, that the analytic outputs of a C∞ GPAC are computable by a digital computer.In [POE, Theorems 5, 6, 7, and 8], Pour-El obtained some related results. (The proof there of Theorem 7 depends on her Theorem 2, for which the proof in [POE] is incorrect, but for which a correct proof is given in [LIR]. Also, the proof in [POE] of Theorem 8 depends on the unproved assertion that a solution of an algebraic differential equation must be analytic on an open subset of its domain. However, this assertion was later proved in [BRR].) As in [POE], we reduce the problem to a problem about solutions of certain systems of algebraic differential equations (ADE's). If such a system is nonsingular (i.e. if the “separant” does not vanish along the given solution), then the argument is very easy (see [VSD] for an even simpler situation), so that the essential difficulties arise from singular systems. Our main tools in handling these difficulties are drawn from the excellent (and difficult) paper [DEL] by Denef and Lipshitz. The author especially wants to thank Leonard Lipshitz for his kind help in the preparation of the present paper.We emphasize here that our proof of the simulation result applies only to the GPAC as described below. The GPAC's form a natural subclass of the class of all analog computers, and are based on certain idealized components (“black boxes”), mostly associated with the technology of past decades. One can easily envisage other kinds of black boxes of an input-output character that would lead to different kinds of analog computers. (For example, one could incorporate delays, or spatial integrators in addition to the present temporal integrators, etc.) Whether digital simulation is possible for these “extended” analog computers poses a rich and challenging set of research questions.

Selected generalized integration techniques for analog computation

SIMULATION ◽  
1967 ◽  
Vol 9 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Arthur Hausner
Keyword(s):  
Differential Equations ◽  
Rational Functions ◽  
Analog Computer ◽  
Great Accuracy ◽  
Analog Computation ◽  
Major Disadvantage ◽  
Integration Techniques ◽  
High Degree

Generalized integration is a technique for generating ex plicit functions on an analog computer by solving the appropriate differential equations they satisfy. Setting up the solution of differential equations using the parametric technique is first reviewed. Two theorems regarding the capability of linear equipment in generating sums and products are stated, and their usefulness is illustrated with examples. Applications of the technique to generating high-degree oscillatory polynomials and rational functions (which require nonlinear equipment) are also described. The major advantage of the technique is achievement of great accuracy with minimum equipment in some cases. The major disadvantage is that, with time, errors may sometimes increase and may not be bounded.

A Large-Scale General-Purpose Electric Analog Computer

1948 ◽  
Vol 67 (1) ◽  
pp. 664-673 ◽  
Author(s):  
E. L. Harder ◽  
G. D. McCann
Keyword(s):  
Large Scale ◽  
Analog Computer ◽  
General Purpose ◽  
Electric Analog

Implementation of hybrid integrators on a general-purpose analog/logic computer

SIMULATION ◽  
1969 ◽  
Vol 12 (4) ◽  
pp. 201-204
Author(s):  
K. Søe Højberg
Keyword(s):  
Pulse Width ◽  
Analog Computer ◽  
General Purpose ◽  
Reference Voltage ◽  
Digital Output ◽  
Analog Input ◽  
Frequency Converters ◽  
Computer Equipment ◽  
Analog Logic

Two types of continuously working hybrid integrators with analog input and digital output have been realized by means of standard analog computer equipment. The basic units are voltage-to-frequency converters. The in tegral is measured in quanta determined by the reference voltage and a pulse width (clock controlled) or the size of a capacitor.

Acoustic Ray Tracing on the General-Purpose Electronic-Analog Computer

1965 ◽  
Vol EC-14 (3) ◽  
pp. 443-455 ◽  
Author(s):  
A. I. Rubin ◽  
G. F. Graber
Keyword(s):  
Ray Tracing ◽  
Analog Computer ◽  
General Purpose

The Use of Analog Computers for Determining Surface Parameters Required for Prediction of Thermal Contact Conductance

1964 ◽  
Vol 86 (4) ◽  
pp. 543-550 ◽  
Author(s):  
J. J. Henry ◽  
H. Fenech
Keyword(s):  
Experimental Data ◽  
Unit Area ◽  
Thermal Contact ◽  
Analog Computer ◽  
General Purpose ◽  
Thermal Contact Conductance ◽  
Average Thickness ◽  
Contact Interface ◽  
Contact Conductance ◽  
Surface Profiles

The mathematical analysis of a thermal contact by Fenech and Rohsenow requires knowledge of certain parameters describing the geometry of the contact interface. These parameters are volume average thickness of the void above and below the plane of the contact, the number of contacts per unit area, and the ratio of the actual contact area to the total area. The authors outline a method for determining these parameters graphically. This paper describes a method for obtaining analog voltages of surface profiles of contacting surfaces and the application of a general purpose analog computer to determine the geometric parameters of contact as a function of contact pressure. The results of applying this method are combined with the analysis of Fenech and Rohsenow. The predicted contact conductance is found to agree well with experimental data at mean contact temperatures of 100, 200, and 300 F for load pressures of 100 to 20,000 psi.

scholarly journals On the functions generated by the general purpose analog computer

2017 ◽  
Vol 257 ◽  
pp. 34-57 ◽  
Author(s):  
Olivier Bournez ◽  
Daniel Graça ◽  
Amaury Pouly
Keyword(s):  
Analog Computer ◽  
General Purpose

Optimization of Analog Computer Linear System Dynamic Characteristics

SIMULATION ◽  
1964 ◽  
Vol 2 (1) ◽  
pp. R-2-R-30
Author(s):  
Charles H. Sengle ◽  
Edward M. Billinghurst
Keyword(s):  
Linear System ◽  
Transient Response ◽  
Dynamic Characteristics ◽  
Computing System ◽  
Analog Computer ◽  
General Purpose ◽  
System Dynamic ◽  
System Stability ◽  
Equivalent Circuits ◽  
Computational Accuracy

The characteristics of the linear computing elements of the general-purpose electronic differential an alyzer are discussed. Equivalent circuits are given for these elements. Criteria for optimization of the linear computing system in order to obtain maximum computational accuracy are given. Examples of the effects of optimization in improving system stability, transient response, and in increasing bandwidth of high- and medium-accuracy computation are shown.

Analyzing networks containing diodes by analog computation

SIMULATION ◽  
1966 ◽  
Vol 6 (6) ◽  
pp. 356-358 ◽  
Author(s):  
Arthur Hausner
Keyword(s):  
Simulation Method ◽  
High Gain ◽  
Analog Computer ◽  
Analog Computation ◽  
Direct Simulation ◽  
Electrical Networks ◽  
The Stability ◽  
Small Inductance

Larrowe's direct simulation method of analyzing electrical networks with an analog computer is extended to include ideal diodes. The technique is somewhat artificial; each diode is replaced by a small inductance that passes cur rent in one direction. The limit as the value of the induct ance approaches zero is simply a high-gain limited ampli fier that introduces feedback to preserve all current and voltage equations at all times. The stability of the method is briefly discussed and an example is given to show its effectiveness.

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