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1. Finite Difference
Finite difference. There are two subfields of mathematics that concern themselves with finite differences. One is a finite analogue to differential calculus.
http://www.fact-index.com/f/fi/finite_difference.html

Extractions: Main Page See live article Alphabetical index There are two subfields of mathematics that concern themselves with finite differences . One is a finite analogue to differential calculus . See also difference operator The other is a branch of numerical analysis that aims at approximate solution of partial differential equations. The approach taken by finite difference methods for partial differential equations is to approximate differential operators such as u'(x) by a difference operator such as (u(x+h)-u(x))/h for some small but finite h. Doing this substitution for a large enough number of points in the domain of definition (for instance 0,h,2h,...,1 in the case of the unit interval ) gives a system of equations that can be solved algebraically. The error between this approximate solution and the true solution is determined by the truncation error that is made by going from a differential operator to a difference operator. The term "truncation error" reflects the fact that a difference operator can be viewed as a finite part of the infinite Taylor series of the differential operator.

2. 39: Difference And Functional Equations
Here are the AMS and Goettingen resource pages for area 39. Selected topics at this site. Introductory remarks to the calculus of finite differences.
http://www.math.niu.edu/~rusin/known-math/index/39-XX.html

Extractions: POINTERS: Texts Software Web links Selected topics here Functional equations are those in which a function is sought which is to satisfy certain relations among its values at all points. For example, we may look for functions satisfying f(x*y)=f(x)+f(y) and enquire whether the logarithm function f(x)=log(x) is the only solution. (It's not.) In some cases the nature of the answer is different when we insist that the functional equation hold for all real x, or all complex x, or only those in certain domains, for example. A special case involves difference equations, that is, equations comparing f(x) - f(x-1), for example, with some expression involving x and f(x). In some ways these are discrete analogues of differential equations; thus we face similar questions of existence and uniqueness of solutions, global behaviour, and computational stability. When the focus of a functional equation is on continuity of functions and a domain is specified, this becomes a question of

3. 1-D Finite Differences
1D finite differences. The finite difference approach is the most popular discretization technique, owing to its simplicity. Finite
http://www-ncce.ceg.uiuc.edu/tutorials/pde/html/node2.html

Extractions: Next: 1-D Examples Up: No Title Previous: General concepts The finite difference approach is the most popular discretization technique, owing to its simplicity. Finite difference approximations of derivatives are obtained by using truncated Taylor series. Consider the following Taylor expansions The first order derivative is given by the following approximations: ): Forward Difference ): Backward Difference By substracting ( ) : Centered Difference An approximation for the second order derivative is obtained by adding ( The terms and indicate the remainders which are truncated ( truncation error ) to obtain the approximate derivatives. The centered difference approximation given by ( ) is more precise than the forward difference ( ) or the backward difference ( ) because the truncation error is of higher order, a consequence of cancellation of terms of the expansions when taking the difference between ( ) and ( ). Since the centered difference involves both neighboring points, there is more balanced information on the local behavior of the function.

4. Finite Differences For 1-D Parabolic Equations
next up previous Next Stability analysis Up 1D Examples Previous finite differences for Poisson s. finite differences for 1-D Parabolic Equations.
http://www-ncce.ceg.uiuc.edu/tutorials/pde/html/node5.html

Extractions: Next: Stability analysis Up: 1-D Examples Previous: Finite Differences for Poisson's We consider here the 1-D diffusion equation which is discretized in space and time with uniform mesh intervals and timestep . A simple approach is to discretize the time derivative with a forward difference as The solution is known at time and a new solution must be found at time . Starting from the initial condition at , the time evoultion is constructed after each timestep either explicitly , by direct evaluation of an expression obtained from the discretized equation, or implicitly , when solution of a system of equations is necessary. An explicit approach is readily obtained by substituting the space derivative with the 3-point finite difference evaluated at the current timestep. The algorithm, written for a generic point of the discretization, is A fairly general implicit scheme is obtained by discretizing the space derivative with a weighted average of the finite difference approximation at and When , the scheme is fully implicit. The classic Crank-Nicholson scheme is obtained when

5. Finite Difference - Encyclopedia Article About Finite Difference. Free Access, N
encyclopedia article about Finite difference. There are two subfields of mathematics that concern themselves with finite differences.
http://encyclopedia.thefreedictionary.com/Finite difference

Extractions: Dictionaries: General Computing Medical Legal Encyclopedia Word: Word Starts with Ends with Definition There are two subfields of mathematics that concern themselves with finite differences . One is a finite analogue to differential calculus Calculus is a branch of mathematics, developed from algebra and geometry. Calculus focuses on rates of change (within functions), such as accelerations, curves, and slopes. The development of calculus is credited to Archimedes, Leibniz and Newton; lesser credit is given to Barrow, Descartes, de Fermat, Huygens, and Wallis. Fundamental to calculus are derivatives, integrals, and limits. One of the primary motives for the development of modern calculus was to solve the so-called "tangent line problem". Click the link for more information. . See also difference operator In mathematics, a difference operator maps a function f x ) to another function f x + a f x + b The forward difference operator occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of the derivative, except in discrete circumstances. Difference equations can often be solved with techniques very similar to those for solving differential equations. Analogously we can have the backward difference operator

6. Halfbakery: Finite Differences
finite differences Is it already used as an indexing method? (0), So word is sponsored by the numbers 4 (its length) and 28 (its 4th finite difference).
http://www.halfbakery.com/idea/finite_20differences

Extractions: I don't think so. The number of possible final differences is very low and there are a lot of four letter words. That technique seems anyway about the same as assigning codes to a limited number of dictionary entries and just using those. If you wanted to avoid losing information, you'd have to represent all the differences as well as the initial value which would, I think, actually take an extra bit per character for any range of codes. You might be able to identify fancy patterns using differences and represent those but then you'll be trying to do something that any ordinary archiving utility already does pretty well.

7. The Finite Differences Framework
The finite differences framework. An example of finite difference model. The BlackScholes equation can be written in the above form as.
http://www.quantlib.org/html/findiff.html

Extractions: http://quantlib.org This framework (corresponding to the ql/FiniteDifferences directory) contains basic building blocks for the numerical solution of a generic differential equation where is a differential operator in ``space'', i.e., one which does not contain partial derivatives in but can otherwise contain any derivative in any other variable of the problem. Writing the equation in the above form allows us to implement separately the discretization of the differential operator and the time scheme used for the evolution of the solution. The QuantLib::FiniteDifferenceModel class acts as a glue for such two steps-which are outlined in the following sections-and provides the interface of the resulting finite difference model for the end user. Furthermore, it provides the possibility of checking and operating on the solution array at each step-which is typically used to apply an exercise condition for an option. This is also outlined in a section below. The discretization of the differential operator depends on the discretization chosen for the solution of the given equation.

8. APPLICATIONS OF NONSTANDARD FINITE DIFFERENCE SCHEMES
Chen); Application of Nonstandard finite differences to Solve the Wave Equation and Maxwell s Equations (JB Cole); Nonstandard Discretization
http://www.worldscientific.com/books/mathematics/4272.html

Extractions: This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.

9. MathGuide: Finite Differences And Functional Equations
MathGuide finite differences and functional equations (2 records). 1. Electronic Journal of Differential Equations EJDE. Subject
http://www.mathguide.de/cgi-bin/ssgfi/anzeige.pl?db=math&sc=39

10. Finite Differences
finite differences. The method Brennan and Schwartz (1978) is one of the first finance applications of finite differences. Section 14.7
http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node11.html

Extractions: Finite Differences The method of choice for any engineer given a differential equation to solve is to numerically approximate it using a finite difference scheme, which is to approximate the continous differential equation with a discrete difference equation, and solve this difference equation. In the following we implement the two schemes described in chapter 14.7 of Hull (1993) , the implicit finite differences and the the explicit finite differences Brennan and Schwartz (1978) is one of the first finance applications of finite differences. Section 14.7 of Hull (1993) has a short introduction to finite differences. Wilmott et al. (1994) is an exhaustive source on option pricing from the perspective of solving partial differential equations. For European options we do not need to use the finite difference scheme, but we show how one would find the european price for comparison purposes. We show the case of an explitit finite difference scheme. This is an alternative to the implicit finite difference scheme The explicit version is faster, but a problem with the explicit version is that it may not converge. The following follows the discussion of finite differences starting on page 356 of Hull (1993) American Options.

11. Finite Differences
Up Financial Numerical Recipes. Previous Binomial approximation, dividends. Contents Index finite differences. The method of choice
http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node17.html

Extractions: Finite Differences The method of choice for any engineer given a differential equation to solve is to numerically approximate it using a finite difference scheme, which is to approximate the continous differential equation with a discrete difference equation, and solve this difference equation. In the following we implement the two schemes described in chapter 14.7 of Hull (1993) , the implicit finite differences and the the explicit finite differences

12. 1. Finite Difference Calculus
Use finite differences in place of infinitesimal differentials Given , let Remarks History opposite route; from finite differences to differentials
http://www.ap.univie.ac.at/users/ves/cp0102/dx/node3.html

13. Finite Differences
finite differences. 0. Preliminaries. This is a set of notes, mostly for myself, outlining Section 2.6 of 0, but with liberal inclusions from other sections.
http://www.paginar.net/matias/articles/finitediff.html

Extractions: This is a set of notes, mostly for myself, outlining Section 2.6 of [ ], but with liberal inclusions from other sections. Partly because HTML is not as flexible as longhand is, but especially because I prefer Dijkstra's to classical mathematical notation, these notes may not be immediately transparent to most people. I subscribe to his program for a calculational approach to Mathematics, and a linear, one-dimensional notation is ideal to that ends. In what follows, the application of any function to its argument is denoted by an infix dot, it having the highest binding power. Variable subscripting is a particular case of functional application, inasmuch countable sequences of objects are regarded as a function from the naturals N (which include the number zero, hence the numbering convention) to the underlying set of objects. For any commutative monoid M with identity element , the quantified expression i : P. i : E. i denotes, for any predicate P over M and expression E (possibly depending on i ) with range M , the repeated operation on every E.

14. Finite Difference Equations
Though forward and backward finite differences will generally converge, a simple combination of both schemes will reduce the magnitude of the offdiagonal
http://www.science.gmu.edu/~ccruz1/c80197/report/node15.html

Extractions: Next: Integration Solver Selection Up: Integration Scheme Previous: Integration Grid Definition Even though each of the equations only involves first order derivatives, simple backward or forward differences are not suitable for integration of three dimensional scalar fields. Gauss-Seidel requires a diagonally-dominant system for assured convergence. Though forward and backward finite differences will generally converge, a simple combination of both schemes will reduce the magnitude of the off-diagonal terms and thus guarantee convergence. In addition, this provides a set of difference equations which are symmetric with respect to grid traversal directions. First, some notational conventions: The free boundaries of the system (top and bottom ion boundaries and the daylight bottom and nighttime top flux boundaries) can easily be handled by extrapolating to virtual grid points beyond the boundaries and using the same difference equations. The virtual points, of course, are not integrated.

15. Advanced Finite Difference Methods For Financial Instrument Pricing
In particular, we approximate the onefactor and multi-factor Black Scholes equations by finite differences. Exponentially Fitted Finite Difference Schemes.
http://www.datasim.nl/education/coursedetails.asp?coursecategory=FE&coursecode=A

16. Workshop The Finite Difference Method In Instrument Pricing
The do s and don ts of finite differences; Displaying results in C++ and Excel. Systems of equations. Overview finite differences for Multifactor Models.
http://www.datasim.nl/education/coursedetails.asp?coursecategory=FE&coursecode=W

17. Finite Differences
finite differences. We begin our discussion of finite differences by examining column 3 in Pascal s Triangle 1, 4, 10, 20, 35, 56, and so on.
http://www.math.ilstu.edu/day/courses/old/305/contentfinitedifferences.html

Extractions: Another way to search for an explicit representation is to use the method of finite differences . Let us illustrate the method. To use the method of finite differences, generate a table that shows, in each row, the arithmetic difference between the two elements just above it in the previous row, where the first row contains the original sequence for which you seek an explicit representation. Here are the first few rows for the sequence we grabbed from Pascal's Triangle: differences (D3) Notice that the third-differences row is constant (i.e., all 1s). This is the signal we look for in an application of finite differences. If and when we reach a difference row that contains a constant value, we can write an explicit representation for the existing relationship, based on the data at hand. In fact, we can be more specific and say that the existing relationship is a polynomial whose order is equal to the row number of the row in which the constant difference first occurs. In our example, because the constant difference first occurred in the third row of differences, a third-degree, or cubic, polynomial can be found to represent the relationship, based on the ordered pairs we have. The next question: How do we find that polynomial representation?

18. First Steps In Numerical Analysis
STEP 29. NUMERICAL DIFFERENTIATION. finite differences. In Analysis, we are usually able to obtain the derivative of a function by
http://mpec.sc.mahidol.ac.th/numer/STEP29.HTM

Extractions: In Analysis, we are usually able to obtain the derivative of a function by the methods of elementary calculus. However, if a function is very complicated or known only from values in a table, it may be necessary to resort to numerical differentiation. Formulae for numerical differentiation may easily be obtained by differentiating interpolation polynomials. The essential idea is that the derivatives f', f", . . . of a function are represented by the derivatives P' n , P" n , . . . of the interpolating polynomial P n . For example, differentiating Newton's forward difference formula (cf. STEP 22 with respect to x gives formally, since , etc., : In particular, if we set x j If we set = 1/2, we have a relatively accurate formula at half-way points (without second differences): if we set = 1 in the formula for the second derivative, we find (without third differences): i.e., a formula for the second derivative at the next point. Note that, if one retains only one term, one arrives at the well-known formulae:

19. Finite Difference Equations -- From Eric Weisstein's Encyclopedia Of Scientific
A Treatise on the Calculus of finite differences, 2nd rev. ed. New York Dover, 1960. Fort, T. finite differences. Oxford, England Clarendon Press, 1948.
http://www.ericweisstein.com/encyclopedias/books/FiniteDifferenceEquations.html

Extractions: see also Difference Equations Agarwal, Ravi P. Difference Equations and Inequality: Theory, Methods, and Applications, 2nd ed., rev. exp. New York: Dekker, 2000. 971 p. \$?. Batchelder, Paul M. An Introduction to Linear Difference Equations. New York: Dover, 1967. 209 p. \$?. Bellman, Richard Ernest and Cooke, Kenneth L. Differential-Difference Equations. New York: Academic Press, 1963. 462 p. \$?. Boole, George and Moulton, John Fletcher. A Treatise on the Calculus of Finite Differences, 2nd rev. ed. New York: Dover, 1960. Brand, Louis. Differential and Difference Equations. New York: Wiley, 1966. 698 p. Forsythe, George Elmer and Wasow, Wolfgang Richard. Finite-Difference Methods for Partial Differential Equations. New York: Wiley, 1960. 444 p. Fort, T. Finite Differences. Oxford, England: Clarendon Press, 1948. Fulford, Glenn; Forrester, Peter; and Jones, Arthur. Modeling with Differential and Difference Equations. New York: Cambridge University Press, 1997. \$29.95. Goldberg, Samuel.

20. MATH 922 - INTRO TO FINITE DIFFERENCES
MATH 922 INTRO TO finite differences. The following *.m files were used to create the plots shown on the first day of classes
http://www.math.sfu.ca/mast/people/faculty/mkropins/math922/lectures/intro.html

Extractions: The following *.m files were used to create the plots shown on the first day of classes: The main driver file is finitediff.m The function is defined in func.m , and its derivatives are in funcp.m funcpp.m funcppp.m funcpppp.m The difference operators are found in dp.m dm.m , and d0.m MATH 992 Home Dr. Kropinski class list

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