Fixed-point theoremIn mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
Recursive data typeIn computer programming languages, a recursive data type (also known as a recursively-defined, inductively-defined or inductive data type) is a data type for values that may contain other values of the same type. Data of recursive types are usually viewed as directed graphs. An important application of recursion in computer science is in defining dynamic data structures such as Lists and Trees. Recursive data structures can dynamically grow to an arbitrarily large size in response to runtime requirements; in contrast, a static array's size requirements must be set at compile time.
Floating-point unitA floating-point unit (FPU, colloquially a math coprocessor) is a part of a computer system specially designed to carry out operations on floating-point numbers. Typical operations are addition, subtraction, multiplication, division, and square root. Some FPUs can also perform various transcendental functions such as exponential or trigonometric calculations, but the accuracy can be very low, so that some systems prefer to compute these functions in software.
Algebraic data typeIn computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite type, i.e., a type formed by combining other types. Two common classes of algebraic types are product types (i.e., tuples and records) and sum types (i.e., tagged or disjoint unions, coproduct types or variant types). The values of a product type typically contain several values, called fields. All values of that type have the same combination of field types.
Machine epsilonMachine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subject of computational science. The quantity is also called macheps and it has the symbols Greek epsilon . There are two prevailing definitions. In numerical analysis, machine epsilon is dependent on the type of rounding used and is also called unit roundoff, which has the symbol bold Roman u.
Truncation errorIn numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. A summation series for is given by an infinite series such as In reality, we can only use a finite number of these terms as it would take an infinite amount of computational time to make use of all of them. So let's suppose we use only three terms of the series, then In this case, the truncation error is Example A: Given the following infinite series, find the truncation error for x = 0.
Arbitrary-precision arithmeticIn computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision.
Off-by-one errorAn off-by-one error or off-by-one bug (known by acronyms OBOE, OBO, OB1 and OBOB) is a logic error that involves a numerical value incorrectly bigger or smaller by one. It often occurs in computer programming when a loop iterates one time too many or too few. Such problem arises, for instance, when a programmer writes non-strict inequality (≤) in a terminating condition where strict inequality (
Kakutani fixed-point theoremIn mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces.
Array (data type)In computer science, array is a data type that represents a collection of elements (values or variables), each selected by one or more indices (identifying keys) that can be computed at run time during program execution. Such a collection is usually called an array variable or array value. By analogy with the mathematical concepts vector and matrix, array types with one and two indices are often called vector type and matrix type, respectively. More generally, a multidimensional array type can be called a tensor type, by analogy with the physical concept, tensor.
Discretization errorIn numerical analysis, computational physics, and simulation, discretization error is the error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a lattice. Discretization error can usually be reduced by using a more finely spaced lattice, with an increased computational cost. Discretization error is the principal source of error in methods of finite differences and the pseudo-spectral method of computational physics.
Binary numberA binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.
