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Ridder's Method
In numerical analysis, Ridders' method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a continuous function f(x). The method is due to C. Ridders. Ridders' method is simpler than Muller's method or Brent's method but with similar performance. The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall order of convergence In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of co ... of the method is \sqrt. If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval ... [...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 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 ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root-finding Algorithm
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound). Solving an equation is the same as finding the roots of the function . Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most nume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
False Position Method
In mathematics, the ''regula falsi'', method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjusting the test value according to the outcome. This is sometimes also referred to as "guess and check". Versions of the method predate the advent of algebra and the use of equations. As an example, consider problem 26 in the Rhind papyrus, which asks for a solution of (written in modern notation) the equation . This is solved by false position. First, guess that to obtain, on the left, . This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply (currently set to 4) by 3 and substitute again to get , verifying that the solution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for positive i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Muller's Method
Muller's method is a root-finding algorithm, a numerical method for solving equations of the form ''f''(''x'') = 0. It was first presented by David E. Muller in 1956. Muller's method is based on the secant method, which constructs at every iteration a line through two points on the graph of ''f''. Instead, Muller's method uses three points, constructs the parabola through these three points, and takes the intersection of the ''x''-axis with the parabola to be the next approximation. Recurrence relation Muller's method is a recursive method which generates an approximation of the root ξ of ''f'' at each iteration. Starting with the three initial values ''x''0, ''x''−1 and ''x''−2, the first iteration calculates the first approximation ''x''1, the second iteration calculates the second approximation ''x''2, the third iteration calculates the third approximation ''x''3, etc. Hence the ''k''''th'' iteration generates approximation ''x''''k''. Each iteration takes as input the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brent's Method
In numerical analysis, Brent's method is a hybrid root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast-converging secant method or inverse quadratic interpolation if possible, but it falls back to the more robust bisection method if necessary. Brent's method is due to Richard Brent and builds on an earlier algorithm by Theodorus Dekker. Consequently, the method is also known as the Brent–Dekker method. Modern improvements on Brent's method include Chandrupatla's method, which is simpler and faster for functions that are flat around their roots; Ridders' method, which performs exponential interpolations instead of quadratic providing a simpler closed formula for the iterations; and the ITP method which is a hybrid between regula-falsi and bisection that achieves optimal worst-case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Recipes
''Numerical Recipes'' is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. In various editions, the books have been in print since 1986. The most recent edition was published in 2007. Overview The ''Numerical Recipes'' books cover a range of topics that include both classical numerical analysis ( interpolation, integration, linear algebra, differential equations, and so on), signal processing ( Fourier methods, filtering), statistical treatment of data, and a few topics in machine learning (hidden Markov model, support vector machines). The writing style is accessible and has an informal tone. The emphasis is on understanding the underlying basics of techniques, not on the refinements that may, in practice, be needed to achieve optimal performance and reliability. Few results are proved with any degree of rigor, although the ideas behind proofs are often sketched, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of convergence'' q \geq 1 and ''rate of convergence'' \mu if : \lim _ \frac=\mu. The rate of convergence \mu is also called the ''asymptotic error constant''. Note that this terminology is not standardized and some authors will use ''rate'' where this article uses ''order'' (e.g., ). In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence. Similar concepts are used for discretization methods. The solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
