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Finite Difference

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

Deltaf_p=f_(p+1)-f_p,
(1)

and the finite backward difference as

del f_p=f_p-f_(p-1).
(2)

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

If the values are tabulated at spacings h, then the notation

f_p=f(x_0+ph)=f(x)
(3)

is used. The kth forward difference would then be written as Delta^kf_p, and similarly, the kth backward difference as del ^kf_p.

However, when f_p is viewed as a discretization of the continuous function f(x), then the finite difference is sometimes written

Deltaf(x)=f(x+1/2)-f(x-1/2)
(4)
=2AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x)*f(x),
(5)

where * denotes convolution and AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x) is the odd impulse pair. The finite difference operator can therefore be written

Delta^~=2AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25]*.
(6)

An nth power has a constant nth finite difference. For example, take n=3 and make a difference table,

x; 1; 2; 3; 4; 5x^3; 1; 8; 27; 64; 125Delta; 7; 19; 37; 61Delta^2; 12; 18; 24Delta^3; 6; 6Delta^4; 0.
(7)

The Delta^3 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 f(n) is known at only a few discrete values n=0, 1, 2, ... and it is desired to determine the analytical form of f, the following procedure can be used if f is assumed to be a polynomial function. Denote the nth value in the sequence of interest by a_n. Then define b_n as the forward difference Delta_n=a_(n+1)-a_n, c_n as the second forward difference Delta_n^2=b_(n+1)-b_n, etc., constructing a table as follows

a_0=f(0) a_1=f(1) a_2=f(2) ... a_p=f(p)
(8)
b_0=a_1-a_0 b_1=a_2-a_1 ... b_(p-1)=a_p-a_(p-1)
(9)
c_0=b_1-b_0 ... ...
(10)
...
(11)

Continue constructing the difference table through the last available row. The unique polynomial of degree at most p interpolating the p+1 values a_0, a_1, ..., a_p is then given by

f(n)=sum_(k=0)^(p)alpha_k(n; k)
(12)
=a_0+b_0n+(c_0n(n-1))/2+(d_0n(n-1)(n-2))/(2·3)+....
(13)

Here alpha_k=Delta^ka_0, so alpha_0=a_0, alpha_1=b_0, etc. This 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, 8539, .... The difference table is then given by

1 19 143 607 1789 4211 8539 18 124 464 1182 2422 4328 106 340 718 1240 1906 234 378 522 666 144 144 144 0 0
(14)

Reading off the first number in each row gives a_0=1, b_0=18, c_0=106, d_0=234, e_0=144. Plugging these in gives the equation

f(n)=1+18n+1/2106n(n-1)+1/6234n(n-1)(n-2)+1/(24)144n(n-1)(n-2)(n-3)
(15)
=6n^4+3n^3+2n^2+7n+1,
(16)

which indeed fits the original data exactly.

Formulas for the derivatives are given by

f^'(x_0)=1/h[f(x_0+h)-f(x_0)]-1/2hf^('')(xi)
(17)
=1/(2h)[-3f(x_0)+4f(x_0+h)-f(x_0+2h)]+1/3h^2f^((3))(xi)
(18)
=1/(2h)[f(x_0+h)-f(x_0-h)]-1/6h^2f^((3))(xi)
(19)
=1/(12h)(f_(-2)-8f_(-1)+8f_1-f_2)+1/(30)h^4f^((5))(xi)
(20)
=1/(12h)(-25f_0+48f_1-36f_2+16f_3-3f_4)+1/5h^4f^((5))(xi)
(21)
f^('')(x_0)=1/(h^2)(f_(-1)-2f_0+f_1)-1/(12)h^2f^((4))(xi)
(22)
=1/(h^2)(f_0-2f_1+f_2)+1/6h^2f^((4))(xi_1)-hf^((3))(xi_2)
(23)
f^((3))(x_0)=1/(h^3)(f_3-3f_2+3f_1-f_0)+O(h)
(24)
=1/(2h^3)(f_2-2f_1+2f_(-1)-f_(-2))+O(h^2)
(25)
f^((4))(x_0)=1/(h^4)(f_4-4f_3+6f_2-4f_1+f_0)+O(h)
(26)
=1/(h^4)(f_2-4f_1+6f_0-4f_(-1)+f_(-2))+O(h^2)
(27)

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

Formulas for integrals of finite differences

int_(x_0)^(x_n)f(x)dx=hint_0^nf_pdp
(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 Crank-Nicolson method, Du Fort-Frankel method, and Laasonen method.

See also

Backward Difference, Bessel's Finite Difference Formula, Derivative, Difference Equation, Difference Table, Everett's Formula, Finite Element Method, Forward Difference, Gauss's Backward Formula, Gauss's Forward Formula, Interpolation, Jackson's Difference Fan, Newton-Cotes Formulas, Newton's Backward Difference Formula, Newton's Divided Difference Interpolation Formula, Newton's Forward Difference Formula, Quotient-Difference Table, Recurrence Equation, Steffenson's Formula, Stirling's Finite Difference Formula, Umbral Calculus

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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, England: 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, England, 1730. English translation: The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series (Trans. J. Holliday). 1749.Tweedie, C. James Stirling: A Sketch of His Life and Works Along with His Scientific Correspondence. Oxford, England: Clarendon 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.

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Finite Difference

Cite this as:

Weisstein, Eric W. "Finite Difference." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FiniteDifference.html

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