Catastrophic Cancellation

	Catastrophic Cancellation In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_1 = 253.51\,\text long and the other L_2 = 252.49\,\text long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be \tilde L_1 = 254\,\text and \tilde L_2 = 252\,\text. These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 0.2% of the true lengths, , L_1 - \tilde L_1, /, L_1, < 0.2\%. However, if the ''approximate'' lengths are subtracted, the difference will be $\tilde L_1 - \tilde L_2 = 254\,\text - 252\,\text = 2\,\text$ , even though the true difference between the lengths is $L_1 - L_2 = 253.51\,\text - 252.49\,\text = 1.02\,\text$ . The difference of the approximations, $2 ... [...More Info...] [...Related Items...]$ 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]
picture info	Wall Stud Wall studs are framing components in timber or steel-framed walls, that run between the top and bottom plates. It is a fundamental element in frame building. The majority non-masonry buildings rely on wall studs, with wood being the most common and least-expensive material used for studs. Studs are positioned perpendicular to the wall they’re forming to give strength and create space for wires, pipes and insulation. Studs are sandwiched between two horizontal boards called top and bottom plates. These boards are nailed or screwed to the top and bottom ends of the studs, forming the complete wall frame. Studs are usually spaced 16 in. or 24 in. apart. Etymology ''Stud'' is an ancient word related to similar words in Old English, Old Norse, Middle High German, and Old Teutonic generally meaning ''prop'' or ''support''."Stud". def. 1. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) © Oxford University Press 2009 Other historical words with similar meaning are ... [...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]
picture info	Floating-point Arithmetic In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, 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 Binary number, base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Sterbenz Lemma In floating-point arithmetic, the Sterbenz lemma or Sterbenz's lemma is a theorem giving conditions under which floating-point differences are computed exactly. It is named after Pat H. Sterbenz, who published a variant of it in 1974. The Sterbenz lemma applies to IEEE 754, the most widely used floating-point number system in computers. Proof Let \beta be the radix of the floating-point system and p the precision. Consider several easy cases first: * If x is zero then x - y = -y, and if y is zero then x - y = x, so the result is trivial because floating-point negation is always exact. * If x = y the result is zero and thus exact. * If x < 0 then we must also have $y/2 \leq x < 0$ so $y < 0$ . In this case, $x - y = -(-x - -y)$ , so the result follows from the theorem restricted to $x, y \geq 0$ . * If $x \leq y$ , we can write $x - y = -(y - x)$ with $x/2 \leq y \leq 2 x$ , so the result follow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ill-conditioned 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Double-precision Floating-point Format 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]
picture info	Inverse Trigonometric Functions In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted Domain of a function, domains. Specifically, they are the inverses of the sine, cosine, tangent (trigonometry), tangent, cotangent, secant (trigonometry), secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. Notation Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of radians will correspond to an circular arc, arc whose length is , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	2Sum 2Sum is a floating-point algorithm for computing the exact round-off error in a floating-point addition operation. 2Sum and its variant Fast2Sum were first published by Ole Møller in 1965. Fast2Sum is often used implicitly in other algorithms such as compensated summation algorithms; Kahan's summation algorithm was published first in 1965, and Fast2Sum was later factored out of it by Dekker in 1971 for double-double arithmetic algorithms. The names ''2Sum'' and ''Fast2Sum'' appear to have been applied retroactively by Shewchuk in 1997. Algorithm Given two floating-point numbers a and b, 2Sum computes the floating-point sum s := a \oplus b rounded to nearest and the floating-point error t := a + b - (a \oplus b) so that s + t = a + b, where \oplus and \ominus respectively denote the addition and subtraction rounded to nearest. The error t is itself a floating-point number. :Inputs floating-point numbers a, b :Outputs rounded sum s = a \oplus b and exact error t = a + b - (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Well-conditioned 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]