Bessel interpolation formula

A formula which is defined as half the sum of the Gauss formula (cf. Gauss interpolation formula) for forward interpolation on the nodes

$$x _ {0} ,\ x _ {0} + h,\ x _ {0} - h \dots x _ {0} + nh,\ x _ {0} - nh,\ x _ {0} + (n + 1) h ,$$

at the point $x = x _ {0} + th$:

$$\tag{1 } G _ {2n + 2 } (x _ {0} + th) = \ f _ {0} + f _ {1/2} ^ {1} t + f _ {0} ^ {2} \frac{t (t - 1) }{2!} + \dots +$$

$$+ f _ {1/2} ^ {2n + 1 } \frac{t (t ^ {2} - 1) \dots (t ^ {2} - n ^ {2} ) }{(2n + 1)! } ,$$

and the Gauss formula of the same order for backward interpolation with respect to the node $x _ {1} = x _ {0} + h$, i.e. with respect to the population of nodes

$$x _ {0} + h, x _ {0} ,\ x _ {0} + 2h,\ x _ {0} - h \dots x _ {0} + (n + 1) h,\ x _ {0} - nh:$$

$$\tag{2 } G _ {2n + 2 } (x _ {0} + th) = f _ {1} + f _ {1/2} ^ {1} (t - 1) + f _ {1} ^ {2} \frac{t (t - 1) }{2!} + \dots +$$

$$+ f _ {1/2} ^ {2n + 1 } \frac{t (t ^ {2} - 1) \dots [t ^ {2} - (n - 1) ^ {2} ] (t - n) (t - n - 1) }{(2n + 1)! } .$$

Putting

$$f _ {1/2} ^ {2k} = \ \frac{(f _ {0} ^ {2k} + f _ {1} ^ {2k} ) }{2} ,$$

Bessel's interpolation formula assumes the form ([1], [2]):

$$\tag{3 } B _ {2n + 2 } (x _ {0} + th) =$$

$$= \ f _ {1/2} + f _ {1/2} ^ {1} \left ( t - { \frac{1}{2} } \right ) + f _ {1/2} ^ {2} \frac{t (t - 1) }{2!} + \dots +$$

$$+ f _ {1/2} ^ {2n} \frac{t (t ^ {2} - 1) \dots [t ^ {2} - (n - 1) ^ {2} ] (t - n) }{(2n)!} +$$

$$+ f _ {1/2} ^ {2n + 1 } \frac{t (t ^ {2} - 1) \dots [t ^ {2} - (n - 1) ^ {2} ] (t - n) (t - 1/2) }{(2n + 1)! } .$$

Bessel's interpolation formula has certain advantages over Gauss' formulas (1), (2); in particular, if the interpolation is at the middle of the segment, i.e. at $t = 1/2$, all coefficients at the differences of odd orders vanish. If the last term on the right-hand side of (3) is omitted, the polynomial $B _ {2n + 1 } (x _ {0} + th)$, which is not a proper interpolation polynomial (it coincides with $f(x)$ only in the $2n$ nodes $x _ {0} - (n - 1)h \dots x _ {0} + nh$), represents a better estimate of the residual term (cf. Interpolation formula) than the interpolation polynomial of the same degree. Thus, for instance, if $x = x _ {0} + th \in (x _ {0} , x _ {1} )$, the estimate of the last term using the polynomial which is most frequently employed

$$B _ {3} (x _ {0} + th) = \ f _ {1/2} + f _ {1/2} ^ {1} \left ( t - { \frac{1}{2} } \right ) + f _ {1/2} ^ {2} \frac{t (t - 1) }{2} ,$$

written with respect to the nodes $x _ {0} - h, x _ {0} , x _ {0} + h, x _ {0} + 2h$, is almost 8 times better than that of the interpolation polynomial written with respect to the nodes $x _ {0} - h, x _ {0} , x _ {0} + h$ or $x _ {0} , x _ {0} + h, x _ {0} + 2h$([2]).

References

[1]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , 1 , Pergamon (1973) (Translated from Russian)
[2]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)

References

[a1]	M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1970)
[a2]	F.B. Hildebrand, "Introduction to numerical analysis" , Addison-Wesley (1956)