Numerical Divided Difference Formula And Its Applications

This thesis gives an overview of the numerical divided differenceformula, and further studies its applications in numerical analysis.First of all, the first part provides some of the basic knowledge ondivided difference and Enumerative Combinatory used in this paper. Then,the second part is the basic facts about the numerical divided difference【1】and divided difference expansions, Then discussed the error bounds forthe remainder term in Floater' s divided difference expansions【2】, and offers an purely elementary proof for the estimation |R_p|≤k^p-k/k!(p-1-k)!h^p-k-1ω(f^(p-1),h)take advantage of the numerical divided difference formula. (Aprobabilistic approach was used by Jose A. Adelland C. Sanguesa【3】to givea better bound for the special case in which the data pointsaα_o,...,α_n areuniformly spaced).Next, the third part describes the application ofnumerical divided difference formula in numerical differentiation.Furthermore the remainder for the numerical differentiation formula thatthe nth derivative is approximated by an nth divided difference with super‖f⁽ⁿ⁾(x)-n!f[α_o,……,α_n]‖_∞≤n/24 h²â€–f⁽ⁿ⁺²⁾‖_∞h=max{|α₁₊₁-α₁|,0≤i≤n-1}convergence is studied, and the following estimation was established:Finally, in the last part the numerical divided difference formula isapplied to Hadamard finite-part integral, and the following approximateintegral formula is derived, H_p(f,ξ)≡sum from k=1,k≠c to mλ_kf[x_k(?)dx+λ_c/h^p+1sum from v=p+1 to n G^v sum from k=p+1 to v A_v^k(x_c-ξ/h)^k-p-1+sum from k=0 to p f^(k)(ξ)/k!*1/p-k[1/(α-ξ)^p-k-1/(b-ξ)^p-k]Which provide a solution to the Calculation of instability in Hadmardfinite-part integral when some of the integral nodes are too close tothe singular point.