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Use of a user-defined function sq(x) in Microsoft Excel. The named variables x & y are identified in the Name Manager. The function sq is introduced using the Visual Basic editor supplied with Excel. Subroutine in Excel calculates the square of named column variable x read from the spreadsheet, and writes it into the named column variable y.
A spreadsheet's concatenation ("&") function can be used to assemble complex text strings in a single cell (in this example, XML code for an SVG "circle" element). This concatenation is a variation of the chaining of formulas, for which spreadsheets are commonly used. The ability to chain formulas together is what gives a spreadsheet its power.
The DFA can be constructed explicitly and then run on the resulting input string one symbol at a time. Constructing the DFA for a regular expression of size m has the time and memory cost of O(2 m), but it can be run on a string of size n in time O(n). Note that the size of the expression is the size after abbreviations, such as numeric ...
Quicksort applied to a list of n elements, again assumed to be all different and initially in random order. This popular sorting algorithm has an average-case performance of O(n log(n)), which contributes to making it a very fast algorithm in practice. But given a worst-case input, its performance degrades to O(n 2).
An n-bit LUT can encode any n-input Boolean function by storing the truth table of the function in the LUT. This is an efficient way of encoding Boolean logic functions, and LUTs with 4-6 bits of input are in fact the key component of modern field-programmable gate arrays (FPGAs) which provide reconfigurable hardware logic capabilities.
A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v 1, …, v n have free occurrences, then A preceded by ∀v 1 ⋯ ∀v n is a universal closure of A.
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.
Formally, if one expands () (), the terms are precisely (), where is either 0 or 1, accordingly as whether is included in the product or not, and k is the number of that are included, so the total number of factors in the product is n (counting with multiplicity k) – as there are n binary choices (include or x), there are terms ...