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Kahan Summation
In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite- precision floating-point numbers, compared to the naive approach. This is done by keeping a separate ''running compensation'' (a variable to accumulate small errors), in effect extending the precision of the sum by the precision of the compensation variable. In particular, simply summing n numbers in sequence has a worst-case error that grows proportional to n, and a root mean square error that grows as \sqrt for random inputs (the roundoff errors form a random walk).. With compensated summation, using a compensation variable with sufficiently high precision the worst-case error bound is effectively independent of n, so a large number of values can be summed with an error that only depends on the floating-point precision of the result. The algorithm is attributed to William Kahan;. Ivo Babuška s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhäuser
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particularly: history of science, geosciences, computer science) and mathematics books and journals under the Birkhäuser imprint (with a leaf logo) sometimes called Birkhäuser Science. * Birkhäuser Verlag – an architecture and design publishing company was (re)created in 2010 when Springer sold its design and architecture segment to ACTAR. The resulting Spanish-Swiss company was then called ActarBirkhäuser. After a bankruptcy, in 2012 Birkhäuser Verlag was sold again, this time to De Gruyter. Additionally, the Reinach-based printer Birkhäuser+GBC operates independently of the above, being now owned by '' Basler Zeitung''. History The original Swiss publishers program focused on regional literature. In the 1920s the sons of Emil Bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Summation
In numerical analysis, pairwise summation, also called cascade summation, is a technique to sum a sequence of finite-arithmetic precision, precision floating-point numbers that substantially reduces the accumulated round-off error compared to naively accumulating the sum in sequence. Although there are other techniques such as Kahan summation that typically have even smaller round-off errors, pairwise summation is nearly as good (differing only by a logarithmic factor) while having much lower computational cost—it can be implemented so as to have nearly the same cost (and exactly the same number of arithmetic operations) as naive summation. In particular, pairwise summation of a sequence of ''n'' numbers ''xn'' works by recursion (computer science), recursively breaking the sequence into two halves, summing each half, and adding the two sums: a divide and conquer algorithm. Its worst-case roundoff errors grow Big O notation, asymptotically as at most ''O''(ε log ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeitschrift Für Angewandte Mathematik Und Mechanik
The ''Journal of Applied Mathematics and Mechanics'', also known as ''Zeitschrift für Angewandte Mathematik und Mechanik'' or ''ZAMM'' is a monthly peer-reviewed scientific journal dedicated to applied mathematics. It is published by Wiley-VCH on behalf of the Gesellschaft für Angewandte Mathematik und Mechanik. The editor-in-chief is Holm Altenbach ( Otto von Guericke University Magdeburg). According to the ''Journal Citation Reports'', the journal has a 2022 impact factor of 2.3. Publication history The journal's first issue appeared in 1921, published by the Verein Deutscher Ingenieure and edited by Richard von Mises. in the |
Non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between a positive and a negative zero. In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate binary aspects of mathematical or scientific objects, such as odd and even ( sign of a permutation), sense of orientation or rotation ( cw/ccw), one sided limits, and other concepts described in below. Sign of a number Numbers from various number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbitrary-precision
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] |
Backwards Stable
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context: one important context is numerical linear algebra, and another is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condition Number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given f(x) = y, one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. The condition number is derived from the theory of propagation of uncertainty, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Error
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation derived for it. This inherent error in approximation can be quantified and expressed in two principal ways: as an absolute error, which denotes the direct numerical magnitude of this discrepancy irrespective of the true value's scale, or as a relative error, which provides a scaled measure of the error by considering the absolute error in proportion to the exact data value, thus offering a context-dependent assessment of the error's significance. An approximation error can manifest due to a multitude of diverse reasons. Prominent among these are limitations related to computing machine precision, where digital systems cannot represent all real numbers with perfect accuracy, leading to unavoidable truncation or rounding. Another common source is inherent measurement error, stemming from the practical limitations of inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double-precision
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double precision may be chosen when the range or precision of single precision would be insufficient. In the IEEE 754 standard, the 64-bit base-2 format is officially referred to as binary64; it was called double in IEEE 754-1985. IEEE 754 specifies additional floating-point formats, including 32-bit base-2 ''single precision'' and, more recently, base-10 representations (decimal floating point). One of the first programming languages to provide floating-point data types was Fortran. Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the computer manufacturer and computer model, and upon decisions made by programming-language implementers. E.g., GW-BASIC's double-precision ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Precision
Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point number systems. 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, denoted here as ''rounding machine epsilon'' or the ''formal definition'' and ''interval machine epsilon'' or ''mainstream definition''. In the ''mainstream definition'', machine epsilon is independent of rounding method, and is defined simply as ''the difference between 1 and the next larger floating point number''. In the ''formal definition'', machine epsilon is dependent on the type of rounding used and is also called unit roundoff, which has the symbol bold Roman u. The two terms can generally be considered to differ by simply a factor of two, with the ''formal definition'' yi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Epsilon
Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point number systems. 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, denoted here as ''rounding machine epsilon'' or the ''formal definition'' and ''interval machine epsilon'' or ''mainstream definition''. In the ''mainstream definition'', machine epsilon is independent of rounding method, and is defined simply as ''the difference between 1 and the next larger floating point number''. In the ''formal definition'', machine epsilon is dependent on the type of rounding used and is also called unit roundoff, which has the symbol bold Roman u. The two terms can generally be considered to differ by simply a factor of two, with the ''formal definition'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
