The differential analyzer. A new machine for solving differential equations

Bush, Vannevar

doi:10.1016/s0016-0032(31)90616-9

Cited by 264 publications

(88 citation statements)

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“…For example, the most well known example of analog model of computation is the General Purpose Analog Computer (GPAC) introduced by Claude Shannon in [2] as the idealization of an analog computer, the Differential Analyzer [3]. Shannon worked as an operator early in his career on these machines.…”

Section: Motivation 1 and Digressionmentioning

confidence: 99%

Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

Bournez

Graça

Pouly

2011

Mathematical Foundations of Computer Science 2011

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Abstract. In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of R n and C n , under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension to the complex plane.

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Section: Motivation 1 and Digressionmentioning

confidence: 99%

Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

Bournez

Graça

Pouly

2011

Mathematical Foundations of Computer Science 2011

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“…The GPAC is a mathematical model introduced by Claude Shannon of an earlier analog computer, the Differential Analyzer. The first general-purpose Differential Analyzer is generally attributed to Vannevar Bush [10]. Differential Analyzers have been used intensively up to the 1950's as computational machines to solve various problems from ballistic to aircraft design, before the era of the digital computer [18].…”

Section: Introductionmentioning

confidence: 99%

Turing Machines Can Be Efficiently Simulated by the General Purpose Analog Computer

Bournez¹,

Graça²,

Pouly³

2013

Lecture Notes in Computer Science

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Abstract. The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynomial reductions. However, the situation is less clear in what concerns models of computation using real numbers, and no analog of the Church-Turing thesis exists for this case. Recently it was shown that some models of computation with real numbers were equivalent from a computability perspective. In particular it was shown that Shannon's General Purpose Analog Computer (GPAC) is equivalent to Computable Analysis. However, little is known about what happens at a computational complexity level. In this paper we shed some light on the connections between this two models, from a computational complexity level, by showing that, modulo polynomial reductions, computations of Turing machines can be simulated by GPACs, without the need of using more (space) resources than those used in the original Turing computation, as long as we are talking about bounded computations. In other words, computations done by the GPAC are as space-efficient as computations done in the context of Computable Analysis.

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“…The GPAC was introduced in 1941 by Shannon [30] as a mathematical model of an analog device, the Differential Analyzer [5]. This device was one of the most popular analog computers in the 1930s and was intended to solve numerical problems, especially differential equations [3].…”

Section: Introductionmentioning

confidence: 99%

Some recent developments on Shannon's General Purpose Analog Computer

Graça

2004

Mathematical Logic Qtrly

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This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth continuous time dynamical systems. As a consequence, we conclude that some of these extensions achieve Turing universality. Finally, it is shown that if we introduce a new notion of computability for the GPAC, based on ideas from computable analysis, then one can compute transcendentally transcendental functions such as the Gamma function or Riemann's Zeta function.

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The differential analyzer. A new machine for solving differential equations

Cited by 264 publications

References 0 publications

Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

Turing Machines Can Be Efficiently Simulated by the General Purpose Analog Computer

Some recent developments on Shannon's General Purpose Analog Computer

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