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The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π . However, it has some drawbacks (for example, it is computer memory -intensive) and therefore all record-breaking calculations for many years have used other ...
The structured program theorem, also called the Böhm–Jacopini theorem, [1] [2] is a result in programming language theory.It states that a class of control-flow graphs (historically called flowcharts in this context) can compute any computable function if it combines subprograms in only three specific ways (control structures).
Expressivity of calculi. Programming experience shows that certain problems are easier to solve in some languages than in others. This phenomenon calls for a more precise characterisation of the expressivity of calculi modeling computation than that afforded by the Church–Turing thesis. One way of doing this is to consider encodings between ...
The π-calculus belongs to the family of process calculi, mathematical formalisms for describing and analyzing properties of concurrent computation.In fact, the π-calculus, like the λ-calculus, is so minimal that it does not contain primitives such as numbers, booleans, data structures, variables, functions, or even the usual control flow statements (such as if-then-else, while).
An example of a programming language construct which may or may not terminate is a loop, as they can be run repeatedly. Loops implemented using a counter variable as typically found in data processing algorithms will usually terminate, demonstrated by the pseudocode example below:
The search procedure consists of choosing a range of parameter values for s, b, and m, evaluating the sums out to many digits, and then using an integer relation-finding algorithm (typically Helaman Ferguson's PSLQ algorithm) to find a sequence A that adds up those intermediate sums to a well-known constant or perhaps to zero.
A variant of the spigot approach uses an algorithm which can be used to compute a single arbitrary digit of the transcendental without computing the preceding digits: an example is the Bailey–Borwein–Plouffe formula, a digit extraction algorithm for π which produces base 16 digits. The inevitable truncation of the underlying infinite ...