The element z is called a transcendental for the class F if functions in F can be uniquely identified by their values at z. Conditions for the existence of transcendentals are discussed for certain classes of polynomials, multinomials, and rational functions.Of particular interest are those transcendentals having an exact representation in computer arithmetic.Algorithms are presented for reconstruction of the coefficients of a polynomial from its value at a transcendental.The theory is illustrated by application to polynomials, quadratic forms, and quadrature formulas.

Let Y be a family of sequences defined by linear difference equations of the form $y(n + 1) = P(n)y(n) + Q(n)$, where P and Q are restricted to be polynomials of limited degree, constant matrices, multinomials, and so forth. The central problem is to find a finite sample which uniquely identifies members of Y. It is shown that such a sample exists when P and Q are polynomials of limited degree. The sample size depends linearly on the degree limits. A similar result holds for systems of difference equations with P a constant matrix and Q a column vector with polynomial components. Testing procedures are also derived for the case where the coefficients of P and Q are multinomials in a vector parameter x, and y is considered to be a function of its initial value.

This paper considers informally the relationship between computer aided mathematical proof, formal algebraic languages, computation with transcendental numbers, and proof by sampling.

article Free AccessA Multiple Purpose Orthonormalizing Code and its Uses Authors: Philip Davis National Bureau of Standards, Washington, D. C. National Bureau of Standards, Washington, D. C.View Profile , Philip Rabinowitz National Bureau of Standards, Washington, D. C. National Bureau of Standards, Washington, D. C.View Profile Authors Info & Claims Journal of the ACMVolume 1Issue 4pp 183–191https://doi.org/10.1145/320783.320788Published:01 October 1954Publication History 20citation478DownloadsMetricsTotal Citations20Total Downloads478Last 12 Months12Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF