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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] |
Shmuel Gal
Shmuel Gal (; born 1940) is a mathematician and professor of statistics at the University of Haifa in Israel. He devised the Gal's accurate tables method for the computer evaluation of elementary functions. With Zvi Yehudai he developed in 1993 a new algorithm for sorting which is used by IBM. Gal has solved the Princess and monster game and made several significant contributions to the area of search games. He has been working on rendezvous problems with his collaborative colleagues Steve Alpern, Vic Baston, and John Howard.S. Gal and J. Howard (2005). Rendezvous-evasion search in two boxes, OPERATIONS RESEARCH. Gal received a Ph.D. in mathematics from the Hebrew University of Jerusalem The Hebrew University of Jerusalem (HUJI; ) is an Israeli public university, public research university based in Jerusalem. Co-founded by Albert Einstein and Chaim Weizmann in July 1918, the public university officially opened on 1 April 1925. .... His thesis advisor was Aryeh Dvoretz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lookup Table
In computer science, a lookup table (LUT) is an array data structure, array that replaces runtime (program lifecycle phase), runtime computation of a mathematical function (mathematics), function with a simpler array indexing operation, in a process termed as ''direct addressing''. The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying out an "expensive" computation or input/output operation. The tables may be precomputation, precalculated and stored in static memory allocation, static program storage, calculated (or prefetcher, "pre-fetched") as part of a program's initialization phase (''memoization''), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and, in some programming languages, may include pointer functions (or offsets to labels) to process the matching input. Fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling (statistics), sampling or experimentation, which represent the values of a function for a limited number of values of the Dependent and independent variables, independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the function approximation, approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their multiplicative inverse, reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding Inverse trigonometric functions, inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rounding Error
In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of numerical analysis is to estimate computation errors. Computation errors, also called numerical errors, include both truncation errors and roundoff errors. When a sequence of calculations with an input involving any roundoff error are made, errors may accumulate, sometimes dominating the calculation. In ill-conditioned problems, significant error may accumulate. In short, there are two major facets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table-maker's Dilemma
Rounding or rounding off is the process of adjusting a number to an approximate, more convenient value, often with a shorter or simpler representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression √2 with . Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement, or estimate; for example, a quantity that was computed as but is known to be accurate only to within a few hundred units is usually better stated as "about ". On the other hand, rounding of exact numbers will introduce some round-off error in the reported result. Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating-point representation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating Point
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469/200 = 12.345 = \! \underbrace_\text \! \times \! \underbrace_\text\!\!\!\!\!\!\!\overbrace^ However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rounding
Rounding or rounding off is the process of adjusting a number to an approximate, more convenient value, often with a shorter or simpler representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression √2 with . Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid false precision, misleadingly precise reporting of a computed number, measurement, or estimate; for example, a quantity that was computed as but is known to be accuracy and precision, accurate only to within a few hundred units is usually better stated as "about ". On the other hand, rounding of exact numbers will introduce some round-off error in the reported result. Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
