Finite Difference

The finite difference is the discrete analog of the derivative. The finite forward difference of a function  is defined as

(1)

and the finite backward difference as

(2)

The forward finite difference is implemented in the Wolfram Language as DifferenceDelta[f, i].

If the values are tabulated at spacings , then the notation

(3)

is used. The th forward difference would then be written as , and similarly, the th backward difference as .

However, when  is viewed as a discretization of the continuous function , then the finite difference is sometimes written

			(4)
			(5)

where  denotes convolution and  is the odd impulse pair. The finite difference operator can therefore be written

(6)

An th power has a constant th finite difference. For example, take  and make a difference table,

(7)

The  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 general term. Specifically, if a function  is known at only a few discrete values , 1, 2, ... and it is desired to determine the analytical form of , the following procedure can be used if  is assumed to be a polynomial function. Denote the th value in the sequence of interest by . Then define  as the forward difference ,  as the second forward difference , etc., constructing a table as follows

	(8)
	(9)
	(10)
	(11)

Continue computing , , etc., until a 0 value is obtained. Then the polynomial function giving the values  is given by

			(12)
			(13)

When the notation , , etc., is used, this beautiful equation is called Newton's forward 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 by

(14)

Reading off the first number in each row gives , , , , . Plugging these in gives the equation

			(15)
			(16)

which indeed fits the original data exactly.

Formulas for the derivatives are given by

			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)

(Beyer 1987, pp. 449-451; Zwillinger 1995, p. 705).

Formulas for integrals of finite differences

(28)

are given by Beyer (1987, pp. 455-456).

Finite differences lead to difference equations, finite analogs of differential equations. In fact, umbral calculus displays many elegant analogs of well-known identities for continuous functions. Common finite difference schemes for partial differential equations include the so-called Crank-Nicolson, Du Fort-Frankel, and Laasonen methods.

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Fornberg, B. "Calculation of Weights in Finite Difference Formulas." SIAM Rev. 40, 685-691, 1998.

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, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965.

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.

Stirling, J. Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium. London, 1730. English translation by Holliday, J. The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series. 1749.

Tweddle, C. James Stirling: A Sketch of His Life and Works Along with his Scientific Correspondence. Oxford, England: Oxford University Press, pp. 30-45, 1922.

Weisstein, E. W. "Books about Finite Difference Equations." http://www.ericweisstein.com/encyclopedias/books/FiniteDifferenceEquations.html.

Zwillinger, D. (Ed.). "Difference Equations" and "Numerical Differentiation." §3.9 and 8.3.2 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 228-235 and 705-705, 1995.