|
Machine Precision
Machine 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 \varepsilon. 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. However, by a less formal, but more widely-used definition, machine epsilon is independent of rounding method and may be equivalent to u or 2u. Values for standard hardware arithmetics The following table lists machine epsilon values for standard floating-point formats. Each format uses round-to-nearest. Formal definition ''Rounding'' is a procedure for choosing the representation of a real number in a floating point number system. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Approximation Error
The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur because of computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm). In the mathematics, mathematical field of numerical analysis, the numerical stability of an algorithm indicates how the error is propagated by the algorithm. Formal definition One commonly distinguishes between the relative error and the absolute error. Given some value ''v'' and its approximation ''v''approx, the absolute error is :\epsilon = , v-v_\text, \ , where the vertical bars denote the absolute value. If v \ne 0, the relative error is : \eta = \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNU Octave
GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License. History The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Full development was started by John W. Eaton in 1992. The first alpha release dates back to 4 January 1993 and on 17 February 1994 version 1.0 was released. Version 7.1.0 was released on Apr 6, 2022. The program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel was known for his ability to perform quick back-of-the-envelope calculations. Development history Developments In additio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Error
The approximation error in a data value is the discrepancy between an exact value and some '' approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur because of computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm). In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates how the error is propagated by the algorithm. Formal definition One commonly distinguishes between the relative error and the absolute error. Given some value ''v'' and its approximation ''v''approx, the absolute error is :\epsilon = , v-v_\text, \ , where the vertical bars denote the absolute value. If v \ne 0, the relative error is : \eta = \frac = \left, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number System
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called ''numerals''; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' that it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating Point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be represented as a base-ten floating-point number: 12.345 = \underbrace_\text \times \underbrace_\text\!\!\!\!\!\!^ In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. The term ''floating point'' refers to the fact that the number's radix point can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation. A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom. For this reason, floating-poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal128 Floating-point Format
In computing, decimal128 is a decimal floating-point computer numbering format that occupies 16 bytes (128 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Decimal128 supports 34 decimal digits of significand and an exponent range of −6143 to +6144, i.e. to . (Equivalently, to .) Therefore, decimal128 has the greatest range of values compared with other IEEE basic floating-point formats. Because the significand is not normalized, most values with less than 34 significant digits have multiple possible representations; , etc. Zero has possible representations ( if both signed zeros are included). Decimal128 floating point is a relatively new decimal floating-point format, formally introduced in the 2008 version of IEEE 754 as well as with ISO/IEC/IEEE 60559:2011. Representation of decimal128 values IEEE 754 allows two alternative representation metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal64 Floating-point Format
In computing, decimal64 is a decimal floating-point computer numbering format that occupies 8 bytes (64 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Decimal64 supports 16 decimal digits of significand and an exponent range of −383 to +384, i.e. to . (Equivalently, to .) In contrast, the corresponding binary format, which is the most commonly used type, has an approximate range of to . Because the significand is not normalized, most values with less than 16 significant digits have multiple possible representations; , etc. Zero has 768 possible representations (1536 if both signed zeros are included). Decimal64 floating point is a relatively new decimal floating-point format, formally introduced in the 2008 version of IEEE 754 as well as with ISO/IEC/IEEE 60559:2011. Representation of decimal64 values IEEE 754 allows two alternative representation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal32 Floating-point Format
In computing, decimal32 is a decimal floating-point computer numbering format that occupies 4 bytes (32 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Like the binary16 format, it is intended for memory saving storage. Decimal32 supports 7 decimal digits of significand and an exponent range of −95 to +96, i.e. to ±. (Equivalently, to .) Because the significand is not normalized (there is no implicit leading "1"), most values with less than 7 significant digits have multiple possible representations; , etc. Zero has 192 possible representations (384 when both signed zeros are included). Decimal32 floating point is a relatively new decimal floating-point format, formally introduced in the 2008 version of IEEE 754 as well as with ISO/IEC/IEEE 60559:2011. Representation of decimal32 values IEEE 754 allows two alternative representation methods for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadruple-precision Floating-point Format
In computing, quadruple precision (or quad precision) is a binary floating point–based computer number format that occupies 16 bytes (128 bits) with precision at least twice the 53-bit double precision. This 128-bit quadruple precision is designed not only for applications requiring results in higher than double precision, but also, as a primary function, to allow the computation of double precision results more reliably and accurately by minimising overflow and round-off errors in intermediate calculations and scratch variables. William Kahan, primary architect of the original IEEE-754 floating point standard noted, "For now the 10-byte Extended format is a tolerable compromise between the value of extra-precise arithmetic and the price of implementing it to run fast; very soon two more bytes of precision will become tolerable, and ultimately a 16-byte format ... That kind of gradual evolution towards wider precision was already in view when IEEE Standard 754 for Floating-P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Double
In C and related programming languages, long double refers to a floating-point data type that is often more precise than double precision though the language standard only requires it to be at least as precise as double. As with C's other floating-point types, it may not necessarily map to an IEEE format. long double in C History The long double type was present in the original 1989 C standard, but support was improved by the 1999 revision of the C standard, or C99, which extended the standard library to include functions operating on long double such as sinl() and strtold(). Long double constants are floating-point constants suffixed with "L" or "l" (lower-case L), e.g., 0.3333333333333333333333333333333333L or 3.1415926535897932384626433832795028L for quadruple precision. Without a suffix, the evaluation depends on FLT_EVAL_METHOD. Implementations On the x86 architecture, most C compilers implement long double as the 80-bit extended precision type supported by x86 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Precision
Extended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. In contrast to ''extended precision'', arbitrary-precision arithmetic refers to implementations of much larger numeric types (with a storage count that usually is not a power of two) using special software (or, rarely, hardware). Extended precision implementations There is a long history of extended floating-point formats reaching back nearly to the middle of the last century. Various manufacturers have used different formats for extended precision for different machines. In many cases the format of the extended precision is not quite the same as a scale-up of the ordinary single- and double-precision formats it is meant to extend. In a few cases the implementation was merely a software-based change i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
