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Bessel's Finite Difference Formula

An interpolation formula, sometimes known as the Newton-Bessel formula, given by

f_p=f_0+pdelta_(1/2)+B_2(delta_0^2+delta_1^2)+B_3delta_(1/2)^3+B_4(delta_0^4+delta_1^4)+B_5delta_(1/2)^5+...,
(1)

for p in [0,1], where delta is the central difference and

B_(2n)=1/2G_(2n)
(2)
=1/2(E_(2n)+F_(2n))
(3)
B_(2n+1)=G_(2n+1)-1/2G_(2n)
(4)
=1/2(F_(2n)-E_(2n))
(5)
E_(2n)=G_(2n)-G_(2n+1)
(6)
=B_(2n)-B_(2n+1)
(7)
F_(2n)=G_(2n+1)
(8)
=B_(2n)+B_(2n+1),
(9)

where G_k are the coefficients from Gauss's backward formula and Gauss's forward formula and E_k and F_k are the coefficients from Everett's formula. The B_ks also satisfy

B_(2n)(p)=B_(2n)(q)
(10)
B_(2n+1)(p)=-B_(2n+1)(q),
(11)

for

q=1-p.
(12)

See also

Everett's Formula

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972.Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 90-91, 1990.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987.Whittaker, E. T. and Robinson, G. "The Newton-Bessel Formula." §24 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 39-40, 1967.

Referenced on Wolfram|Alpha

Bessel's Finite Difference Formula

Cite this as:

Weisstein, Eric W. "Bessel's Finite Difference Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BesselsFiniteDifferenceFormula.html

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