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Trigonometric Tables
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices. Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of trigonometric functions is still used in computer graphics, where only modest accuracy may be required and speed is often paramount. Another important application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called ''twiddle fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Approximation
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbitrary-precision Arithmetic
In 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 potentially 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. Several modern programming languages have built-in support for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of bits related to the size of the processor register, these implementations typically use variable-length arrays of digits. Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. It should ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax Approximation Algorithm
A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. ... that minimizes maximum error. For example, given a function f defined on the interval ,b/math> and a degree bound n, a minimax polynomial approximation algorithm will find a polynomial p of degree at most n to minimize ::\max_, f(x)-p(x), . Polynomial approximations The Weierstrass approximation theorem states that every continuous function defined on a closed interval ,bcan be uniformly approximated as closely as desired by a polynomial function. For practical work it is often desirable to minimize the maximum absolute or relative error of a polynomial fit for any given number of ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Hardware
Computer hardware includes the physical parts of a computer, such as the central processing unit (CPU), random-access memory (RAM), motherboard, computer data storage, graphics card, sound card, and computer case. It includes external devices such as a Computer monitor, monitor, Computer mouse, mouse, Computer keyboard, keyboard, and Computer speakers, speakers. By contrast, software is a set of written instructions that can be stored and run by hardware. Hardware derived its name from the fact it is ''Hardness, hard'' or rigid with respect to changes, whereas software is ''soft'' because it is easy to change. Hardware is typically directed by the software to execute any command or Instruction (computing), instruction. A combination of hardware and software forms a usable computing system, although Digital electronics, other systems exist with only hardware. History Early computing devices were more complicated than the ancient abacus date to the seventeenth century. French ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Operator
In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag operator''. Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator. Definition Functions of a real variable The shift operator (where ) takes a fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CORDIC
CORDIC, short for coordinate rotation digital computer, is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, and exponentials and logarithms In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ... with arbitrary base, typically converging with one digit (or bit) per iteration. CORDIC is therefore also an example of digit-by-digit algorithms. The original system is sometimes referred to as Volder's algorithm. CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available (e.g. in simple microcontrollers and field-programmable gate arrays or FPGAs), as the only operations they require are addition, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardware Multiplier
A binary multiplier is an electronic circuit used in digital electronics, such as a computer, to multiply two binary numbers. A variety of computer arithmetic techniques can be used to implement a digital multiplier. Most techniques involve computing the set of ''partial products,'' which are then summed together using binary adders. This process is similar to long multiplication, except that it uses a base-2 ( binary) numeral system. History Between 1947 and 1949 Arthur Alec Robinson worked for English Electric, as a student apprentice, and then as a development engineer. Crucially during this period he studied for a PhD degree at the University of Manchester, where he worked on the design of the hardware multiplier for the early Mark 1 computer. However, until the late 1970s, most minicomputers did not have a multiply instruction, and so programmers used a "multiply routine" which repeatedly shifts and accumulates partial results, often written using loop unwinding. Main ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gal's Accurate Tables
Gal's accurate tables is a method devised by Shmuel Gal to provide accurate values of special functions using a lookup table and interpolation. It is a fast and efficient method for generating values of functions like the exponential or the trigonometric functions to within last-bit accuracy for almost all argument values without using extended precision arithmetic. The main idea in Gal's accurate tables is a different tabulation for the special function being computed. Commonly, the range is divided into several subranges, each with precomputed values and correction formulae. To compute the function, look up the closest point and compute a correction as a function of the distance. Gal's idea is to not precompute equally spaced values, but rather to perturb the points ''x'' so that both ''x'' and ''f''(''x'') are very nearly exactly representable in the chosen numeric format. By searching approximately 1000 values on either side of the desired value ''x'', a value can be found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
