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 Numerical Analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding 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, 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 simulating living cells in medicine and ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Square Root Of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to abou ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Gaussian Elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855) although some special cases of the method—albeit presented without proof—were known to Chinese mathematicians as early as circa 179 AD. To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. (subtraction can be achieved by multiplying one row with -1 and addi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Lagrange Polynomial In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for indices j \neq m, the Lagrange basis for polynomials of ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Newton's Method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Linear Interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known points are given by the coordinates (x_0,y_0) and (x_1,y_1), the linear interpolant is the straight line between these points. For a value in the interval (x_0, x_1), the value along the straight line is given from the equation of slopes \frac = \frac, which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with . Solving this equation for , which is the unknown value at , gives \begin y &= y_0 + (x-x_0)\frac \\ &= \frac + \frac\\ &= \frac \\ &= \frac, \end which is the formula for linear interpolation in the interval (x_0,x_1). Outside this interval, the formula is identical to linear extrapolation. This formula can also be understood as a weighted average. The weights are inv ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Actuary An actuary is a business professional who deals with the measurement and management of risk and uncertainty. The name of the corresponding field is actuarial science. These risks can affect both sides of the balance sheet and require asset management, liability management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms. While the concept of insurance dates to antiquity, the concepts needed to scientifically measure and mitigate risks have their origins in the 17th century studies of probability and annuities. Actuaries of the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems to design and manage programs that control risk. The actual steps needed to become an actuary are usually country-specific; however, almost all processes share a rigorous schooling or examination structure and take many years ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Operations Research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decision-making. It is considered to be a subfield of mathematical sciences. The term management science is occasionally used as a synonym. Employing techniques from other mathematical sciences, such as modeling, statistics, and optimization, operations research arrives at optimal or near-optimal solutions to decision-making problems. Because of its emphasis on practical applications, operations research has overlap with many other disciplines, notably industrial engineering. Operations research is often concerned with determining the extreme values of some real-world objective: the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost). Originating in military efforts before World War II, its techniques have grown ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Derivative (finance) In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be used for a number of purposes, including insuring against price movements ( hedging), increasing exposure to price movements for speculation, or getting access to otherwise hard-to-trade assets or markets. Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter (off-exchange) or on an exchange such as the Chicago Mercantile Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges. Derivatives are one of t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Stock In finance, stock (also capital stock) consists of all the Share (finance), shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company is divided, or these shares considered together" "When a company issues shares or stocks ''especially AmE'', it makes them available for people to buy for the first time." (Especially in American English, the word "stocks" is also used to refer to shares.) A single share of the stock means fractional ownership of the corporation in proportion to the total number of shares. This typically entitles the shareholder (stockholder) to that fraction of the company's earnings, proceeds from liquidation of assets (after discharge of all Seniority (financial), senior claims such as secured and unsecured debt), or voting power, often dividing these up in proportion to the amount of money each stockholder has invested. Not all stock i ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Hedge Fund A hedge fund is a pooled investment fund that trades in relatively liquid assets and is able to make extensive use of more complex trading, portfolio-construction, and risk management techniques in an attempt to improve performance, such as short selling, leverage, and derivatives. Financial regulators generally restrict hedge fund marketing to institutional investors, high net worth individuals, and accredited investors. Hedge funds are considered alternative investments. Their ability to use leverage and more complex investment techniques distinguishes them from regulated investment funds available to the retail market, commonly known as mutual funds and ETFs. They are also considered distinct from private equity funds and other similar closed-end funds as hedge funds generally invest in relatively liquid assets and are usually open-ended. This means they typically allow investors to invest and withdraw capital periodically based on the fund's net asset value, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Partial Differential Equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the ex ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]