| 释义 |
Finite DifferenceThe finite difference is the discrete analog of the Derivative. The finite Forward Difference of a function is defined as
 | (1) |
 | (2) |
 , then the notation
 | (3) |
 th Forward Difference would then be written as  , and similarly, the thBackward Difference as  .
However, when is viewed as a discretization of the continuous function  , then the finite difference issometimes written
 | (4) |
 denotes Convolution and  is the odd Impulse Pair. The finite difference operator cantherefore be written
 | (5) |
An  th Power has a constant th finite difference. For example, take  and make a Difference Table,
 | (6) |
 column is the constant 6.
Finite difference formulas can be very useful for extrapolating a finite amount of data in an attempt to find the generalterm. Specifically, if a function  is known at only a few discrete values  , 1, 2, ... and it is desired todetermine the analytical form of  , the following procedure can be used if is assumed to be aPolynomial function. Denote the th value in the Sequence of interest by  . Then define  as theForward Difference  ,  as the second Forward Difference  , etc., constructing a table as follows Continue computing  ,  , etc., until a 0 value is obtained. Then the Polynomial function giving thevalues is given by
 | (8) |
 ,  , etc., is used, this beautiful equation is called Newton'sForward Difference Formula. To see a particular example, consider a Sequence with first few values of 1, 19, 143, 607,1789, 4211, and 8539. The difference table is then given byReading off the first number in each row gives  ,  ,  ,  ,  . Plugging thesein gives the equation
 | (9) |
 , and indeed fits the original data exactly!
Beyer (1987) gives formulas for the derivatives
 | (10) |
 | (11) |
Finite differences lead to Difference Equations, finite analogs of DifferentialEquations. In fact, Umbral Calculus displays many elegant analogs of well-knownidentities for continuous functions. Common finite difference schemes for Partial Differential Equations include the so-called Crank-Nicholson, Du Fort-Frankel, and Laasonen methods. See also Backward Difference, Bessel's Finite Difference Formula, Difference Equation, DifferenceTable, Everett's Formula, Forward Difference, Gauss's Backward Formula, Gauss's Forward Formula,Interpolation, Jackson's Difference Fan, Newton's Backward Difference Formula, Newton-CotesFormulas, Newton's Divided Difference Interpolation Formula, Newton's Forward Difference Formula,Quotient-Difference Table, Steffenson's Formula, Stirling's Finite Difference Formula, UmbralCalculus
References
 Finite Difference EquationsAbramowitz, M. and Stegun, C. A. (Eds.). ``Differences.'' §25.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 429-515, 1987. Boole, G. and Moulton, J. F. A Treatise on the Calculus of Finite Differences, 2nd rev. ed. New York: Dover, 1960. Conway, J. H. and Guy, R. K. ``Newton's Useful Little Formula.'' In The Book of Numbers. New York: Springer-Verlag, pp. 81-83, 1996. Iyanaga, S. and Kawada, Y. (Eds.). ``Interpolation.'' Appendix A, Table 21 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1482-1483, 1980. Jordan, K. Calculus of Finite Differences, 2nd ed. New York: Chelsea, 1950. Levy, H. and Lessman, F. Finite Difference Equations. New York: Dover, 1992. Milne-Thomson, L. M. The Calculus of Finite Differences. London: Macmillan, 1951. Richardson, C. H. An Introduction to the Calculus of Finite Differences. New York: Van Nostrand, 1954. Spiegel, M. Calculus of Finite Differences and Differential Equations. New York: McGraw-Hill, 1971.
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