Search results
Results from the WOW.Com Content Network
Prefix sums are trivial to compute in sequential models of computation, by using the formula y i = y i − 1 + x i to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort, [1] [2] and they form the basis of the scan higher-order function in functional programming languages.
The infinite series whose terms are the natural numbers 1 ... (−1), and they take the "lunatic asylum" line in his second ... = –1/12. Sum of Natural Numbers ...
This characterization states that a function is primitive recursive if and only if there is a natural number m such that the function can be computed by a Turing machine that always halts within A(m,n) or fewer steps, where n is the sum of the arguments of the primitive recursive function. [5]
For example, the sum of the first n natural numbers can be denoted as ∑ i = 1 n i {\displaystyle \sum _{i=1}^{n}i} For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result.
The base case of the recursion could in principle be the sum of only one (or zero) numbers, but to amortize the overhead of recursion, one would normally use a larger base case. The equivalent of pairwise summation is used in many fast Fourier transform (FFT) algorithms and is responsible for the logarithmic growth of roundoff errors in those ...
A natural number is either 1 or n+1, where n is a natural number. Similarly recursive definitions are often used to model the structure of expressions and statements in programming languages. Language designers often express grammars in a syntax such as Backus–Naur form ; here is such a grammar, for a simple language of arithmetic expressions ...
The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers.In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition are defined. [1]
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0) Each equation follows by definition [A1]; the first with a + b, the second with b. Now, for the induction. We assume the induction hypothesis, namely we assume that for some ...