Algorithms For Tri-variate Vector Valued Rational Interpolation

There are many technological problems can be attributed to the discussion of the nonlinear problem.Vector valued rational interpolation as a powerful tool to solve nonlinear problems is concerned widely. After the development of the last 20 years, vector valued rational interpolation was developed from univariate to multivariate, and has formed a complete set of theories and methods. But for the research of more than brivariate vector valued rational interpolation, there are still some problems, such as how to construct multivariate vector valued rational interpolation function;the existence of Thiele-type vector valued rational interpolation hasn't been proved.This paper studies the problem of tri-variate symmetric form of vetor valued rational interpolation and tri-variate vector valued rational interpolation over discrete data set.Algorithms of tri-variate vector valued rational interpolation are constructed ,and continued fraction coefficient is set up by recursive algorithm,and then programs are given to compute continued fraction coefficient.The first chapter gives a brief introduction about vetor valued rational interpolation.In the second chapter ,the definition of univariable vetor valued rationl interpolation and Samelson inverse are given ,and based on this,the method of the construction of univariable Thiele-type vetor rational interpolation is introduced.In the third chapter.we have got vector form branched continued fraction,if we define the brivariate Samelson inverse and imitate univariable situation to constructbrivariate vetor valued rational interpolation. Take into account of context, we only introduce the bivariate symmetric form vector rational interpolation and brivariate vector valued rational interpolation over discrete data set.The fourth chapter is the prime part of this paper.It mainly constructs tri-variate vector valued rational interpolation question by using triple branched continued fraction and tri-variate Samelson inverse transformation. Tri-variate Symmetric Form Vector Valued Rational InterpolationThe construction of tri-variate symmetric form vector valued rational interpolationfunction and continued fraction coefficient recursive algorithm are given as followsLetΠ^n,n,n are interpolation lattice and V^n,n,n are the corresponding vector valuedDefinition 1 : letwhere m = l + 1,…,nbe called to Tri-variate symmetric vector valued branched continued fraction .Let each component of the vector V_ijk is finite,recursive algorithm is given as follows :step 1 (?)(x_i,y_j,z_k)∈Π^n,n,n,letA_ijk⁰=V_ijkstep 2对t = 0,1,…,n- 1Theorem 1 LetA_ijkⁿ (i = 0,1,…,n; j =0,1,…,n;k = 0,1,…n;) be defined as the above algorithm,tri-variate symmetric vector valued branched continued fraction R(x, y, z) satisfyR(x_i,y_j,z_k)=V_ijk (?)(x_i,y_j,z_k∈Π^n,n,nTri-variate Vetor Valued Rational Interpolation Over Discrete Data SetLet Gⁿ = {(x_i,y_i,z_i}|i=0,1…,n} be some discrete points in R³, V_i is associatewith a interpolation point (x_i,y_i,z_i), denoted by Vⁿ = {V_i = V{x_i,y_i,z_i)∈C^d , (x_i,y_i,z_i)∈Gⁿ}. Letu := {the number of different x_i in Gⁿ} - 1 , denoted by x_i, i = 0,1,…, u; v := {the number of different y_j in Gⁿ} - 1 , denoted by y_j, j = 0,1,…, v; w := {the number of different z_k in Gⁿ} - 1 , denoted by z_k, k = 0,1,…, w; u_i := {the number of different y_h in (x_i,y_h,z_l)} - 1 , i = 0,1,…, u u_ij := {the number of different {x_i,y_j,z_l) in Gⁿ}- 1 , i = 0,…, u, j = 0,…, vstep 1 for i- 0,1,…,u; j = 0,1,…,v; k = 0,1,…,w, letstep 2 for j =.0,1,…,v; k = 0,1,…,w; i =1,…,u; p = 1,…,i step 3 for i = 0,1,…,u; k = 0,1,…,ω; j=1,…,v; g = 1,…, jstep 4 for i = 0,1,…, u; j = 0,1,…,v; k=1,…,ω; r = 1,…,kUsing the data which got from the algorithm ,we construct branched continuedfractionwhere m(0),…, m(v) is the arrangement of 0,1,…, v ; n(0),…, n(ω) is the arrangement of 0,1,…,ω, for m(h), exist (x_p,y_m(h),z_k)∈Gⁿ ; for n(h) , so that (x_p,y_q,z_n(h))∈Gⁿ, and m(0) < m(1) <…< m(u_p); n(0) < n(1) <…< n(u_pq)Theorem 2:The coefficient of branched continued fraction R(x, y, x) is definedas algorithm,we haveR(x_l,y_l,z_l)=V_l (?)(x_l,y_l,z_l)∈GnTheorem 3(Characteristic Property) :(1)Let a_p,n,(up)=u_p,n(up),p = 0,1,…, u(2)for i = u_p - 1, u_p- 2,…,0, carries on the iterative computation according to the equation below(3)letα_u=a_u,0 for j = u - 1, u - 2,…, 0, carries on the iterative computation according to the equation belowα_j=max{1+α_j+1+2[a_j,0/2,2[α_j+1+1/2] + a_j,0}then R(x, y,z)∈[α₀/2[α₀/2]].In this paper,two recursive algorithms are constructed to compute tri-variate vetor valued rational interpolation.For the vetor valued rational interpolation over discrete data set,its characteristic property theorem is given with a recursive formula.Two programms are given to compute the continued fraction coefficient.