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Lagrange Interpolation and Hermite Interpolation Along the Algebraic Manifold
　 Polynomial interpolation is an important subject of computational mathematics.The study of interpolation with multivariate polynomial has developed rapidly in the recent twenty years.In the research of multivariate interpolation, the well-posedness problem is the first problem. Thus there are two ways for the research of multivariate interpolation: one way is to construct the properly posed set of nodes (or PPSN, for short) for a given space of interpolating polynomials ( see [1]-[7]). The other way is to find out the proper space of interpolating polynomials for a given set of interpolation nodes, specially to determine the interpolation space of minimal degree ( see de Boor and A.Ron [8],T.Sauer[9][10]). The former is the main research aspect of this paper.In 1948, J.Radon[1] gave the Straight Line-Superposition Process for constructing the PPSN for bivariate polynomial space. Then Liang[2] deduced the Superposition Interpolation Process for constructing the PPSN for R~2( including Conic-Superposition Process). They changed the well-posedness problem into a geometrical problem so that we can use algebraic and geometrical theories to study multivariate polynomial problem.In 1998, Liang and Lii[12] posed the concept of PPSN along an algebraic curve without multiple factors and gave the method of constructing PPSN along it by the intersection between a line and an algebraic curve of degree k.In 2003, Liang and Cui [13] researched Lagrange interpolation in K~3.They posed the concepts of sufficient intersection of algebraic surfaces and PPSN for Lagrange inerpolation along an algebraic surface and along a space algrbraic curve, and deduced a general method of constructing PPSN along a space algebraic curve.In 2004, Liang and Zhang[14] researched Lagrange interpolation along the sufficiently intersected algebraic manifold in R~n.They proved the existence of the PPSN of arbitrary degree along sufficiently intersected algebraic manifold and deduced a general method of constructing PPSN along the algebraic manifold, that is, Superposition Interpolation Process.Our research is the continuation and impovement of the previous work. In this paper we research depply the Hermite interpoaltion along an algrbraic manifold in n-dimensional space. First, we give the description of the Hermite interpolation problem. Then we prove the Superposition Interpolation Process for Hermite interpolation along the algebraic manifold. They includ the results of Lagrange interpolation as particular cases so we regard our research in this paper is a continuation and impovement for the research of multivariate polynomial interpolation.Part 1.Hermite interpolation along the algebraic manifold.In this paper we manily consider polynomial interpolation in real space and we denote P_m(R~n) the real space of all n-variate polynomials of total degree≤m,i.e.where |α|=α_1+…+α_n,α_1,…,α_n denotes nonnegative integer.Before giving the Hermite interpolation problem, we introduce some notations.Let A_m-{Q_i:Q_i=(x_(i,1),…,x_(i,n),1≤i≤M) denote a set of M mutually distinct points in R~n.Let Z~n denotes the set of integral points in R~n andLetτ_i=(τ_(i,1),…,τ_(i,n)) denote n unit vectors of linearly independent andλ_i=λ(Q_i) denote a positive integer corresponding to the point Q_i,whereτ_(i,1),…,τ_(i,n).And letdenote the set of index corresponding to the point Q_i.For each K_(i,j)∈I(Q_i) we define its order by O(K_(i,j))=(?)k_(i,j,ι).For K_(i,j) and K_(i,v) in I(Q_i),we draw a directed line which parallels to axis between them and stipulate the descendent direction is pointing to lower order from higher order if the distance between them is 1.Thus we get a oriented graph of I(Q_i) and denote it by g(Q_i).Joining the directed segments of length 1 according to the descendent direction, we can get a descendent path. And the length of the path is the number of the directed segments of length 1 which joint the descendent path.If for each point K_(i,j)∈I(Q_i) one can find a descendent path in g(Q_i) whose length is O(K_(i,j)) and takes origin as its destination, then we call I(Q_i) a tree indexing set. We define the grade of I(Q_i) by G(Q_i)=(?){O(K_(i,j))}.Let I(Q_i) be a tree indexing set and for each K_(i,j)∈I(Q_i) we define the set of downstream ponits by B(K_(i,j)) which includ all ponits in the descendent paths from K_(i,j) to origin (not includ the point of K_(ij)).For each point Q_i we have the assumption that the conrresponding I(Q_i) is a tree indexing set. And we sort the order of the indexing set {K_(i,j)} according to the following rules.Rule 1.1). If O(K_(i,j))2 thenThen we choose the set of points U_n~λalong the A latitudes as following:1. Ifλis a odd number, thenU_n~λ={Q_(ι,j)|Q_(ι,j)=(sinφ_ιsinθ_j,sinφ_ιcosθ_j,cosφ_ι),θ_j∈Θ_α~(n,λ),0≤j≤2n+1-λ,1≤ι≤λ}2. If A is an even number,thenU_n~λ={Q_(ι,j)|Q_(ι,j)=(sinφ_ιsinθ_j,sinφ_ιcosθ_j,cosφ_ι),θ_j∈Θ_0~(n,λ),0≤j≤2n+1-λ,1≤ι≤h}∪{Q_(ι,j)|Q_(ι,j)=(sinφ_ιsinθ_j,sinφ_ιcosθ_j,cosφ_ι),θ_j∈Θ_1~(n,λ),0≤j≤2n+1-λ,h+1≤ι≤λ}Now we give the result of the interpolation problem along the set of latitudes L(Φ_λ).Theorem 2 Let n be a nonnegative integer and supposeλ≤n+1 is a positive integer. Then the set of points U_n~λgiven as above is a properly posed set of nodes of degree n for Lagrange interpolation along the set of latitudes L(Φ_λ) in the spaceΠ_n~3.The following theorem is the reification of the superposition process for interpolation on the sphere.Theorem 3 Let V_n={Q_i}_(i=1)~((n+1)~2) be a properly posed set of nodes of degree n of the spaceΠ_n(S~2) and suppose a set of latitudes L(Φ_λ) consisting ofλlatitudes does not pass through any point in V_n.Let U_(n+λ)={Q_i}_(i=1)~(λ(2n+2+λ) be a properly posed set ofnodes of degree n+λalong L(Φ_λ).Then V_n∪U_(n+λ) must be a properly posed set of nodes of degree n+λon the sphere S~2.Moreover, for n-2,Theorem 3 contains 8 sets of 16 points being a properlyposed set of nodes on the sphere, n = 4 gives 16 sets of 25 points,…….In general,the number of differently properly posed set of nodes for each degree is a powder series, that is, 2~n.Proposition 1 Let M_n denote the number of properly posed set of nodes in Theorem 3.Then M_n=2~n.We give some schemes for Lagrange interpoltation on the sphere and numerical examples and a algorithm to compute the polynomial for interpolation on the sphere.We will use the following transform of rotation axes to obtain an extension of our work in this paper.whereθ_0∈[0,2π],φ_∈[0,π].We rotate the set of latitudes L(Φ_λ) by means of above transform and get a set of parallel circumferences (?)(Φ_λ).Then the corresponding set of points (?)_n~λ={(?)_(ι,j): ,0≤j≤2n+1-λ,1≤ι≤λ} is a properly posed set of nodes along the set of parallel circumferences (?)(Φ_λ).So we extend the interpolation problem along the set of latitudes L(Φ_λ) to the case of the set of parallel circumferences (?)(Φ_λ).The following theorem describes this case.Theorem 4 Let n be a nonnegative integer and supposeλ≤n+1 is a positive integer. Then the set of points (?)_n~λobtained by rotation transformation is a properly posed set of nodes for Lagrange interpolation along the set of parallel circumferences (?)(Φ_λ) in the spaceΠ_n~3.Accordingly, Theorem 3 will be extended to the following one.Theorem 5 Let V_n-{Q_i}_(i=1)~((n+1)~2) be a properly posed set of nodes of degree n of the spaceΠ_n(S~2) and supposeλparallel planes intersect the sphere at a set of parallel circumferences (?)(Φ_λ) which does not pass through any point in V_n.Let (?)_(n+λ)= {Q_i}_(i=1)~(λ(2n+2+λ)) be a properly posed set of nodes of degree n+λalong (?)(Φ_λ).Then V_n∪(?)_(n+λ) must be a properly posed set of nodes of degree n+λof the spaceΠ_(n+λ)(S~2).It is easy to see from above theorems that we get a class of properly posed set of nodes for Lagrange interpolation on the unit sphere by superposition interpolation process. The results obtained in this paper include not only the schemes in [56]-[58] and the schemes in [55] but also the schemes which get by superposition of the properly posed sets of nodesalong the set of parallel circumferences. Clearly, it is an extension and a development of predecessor\'s work in this field.Part 3. Hermite interpolation on the sphere.Suppose C_1,C_2,…,C_σdenoteσmutually distinct circumferences on the unit sphere and l_i is a unit vector which is along a diameter of the sphere and perpendicular to the circumference C_i.We denote the intersections of extension line of l_i and the sphere by S_i and N_i and prescribe them the south pole and the north pole determined by C_i,respectively.Suppose C_1,…,C_σare corresponding toσlocal coordinates which have origin as their centres of sphere and have l_i as their z-axis. And the spherical coordinates of C_i is ((?)_i,(?)_i).Choose a set of points on C_i and denote it by V_i~n={Q_(i,j),1≤j≤m_i} and write V_n=∪_(i=1)~σV_i~n$.Further more, letδ_(i,j) denote the multiplicity of each point Q_(i,j) and assume (?)(?)δ_(i,j)=(n+1)~2.We consider the following set of interpolating functionals:whereα_1,α_2,…,α_k∈R.We call this kind of set of interpolating functionals L~n a H-class set of interpolation functionals of degree n on the sphere.Now we give the description of this kind of Hermite interpolation problem on the sphere and we call it H-class Hermite interpolation problem on the sphere.Problem 4: For any given real numbers {f_(i,j,k)},we want to find p_n(X)∈Π_n(S~2) such that Definition 5 If there always exists a polynomial p_n(x,y,z)∈Π_n(S~2),such that (7) is satisfied for any given real numbers {f_(i,j,k)},then we call Problem 4 well posed and call the set of interpolating functionals in (6) a H-class properly posed set of interpolating functionals of degree n on S~2 and V_n is called the corresponding set of nodes.In order to resolve the H-class Hermite interpolation problem on the sphere, we study Hermite interpolation along a set of coaxial circumferences.Let -1:(?)(?),the algebraic area of the triangle consistingof the points i, j, k; (?):(?)(? :(?)(?),the algebraic volume of thetetrahedron consisting of the points i,j, k,m;1. The case of two dimensionWe discuss the finit point method by using the directional differential quotient and directional difference quotient. We can get the interpolating polynomial using our result in previous sections. Then we can get the directional differential quotient and set up the relational expression between the directional differential quotient such that the results have a compact format. In this way we obtain the formula of two point in the numerical direction which is the foundation of constructing some kinds of computational schemes for partial differential equation on the heterocentric points.As shown in Figure 1, the neighbours of point 0:(x_0,y_0) are poin 1 and 2 and their dirctions l_1,l_2 are not parallel. Let another direction 1 and we want to approximatively compute (?) by using the values on points 0,1,2, that is, p_0,p_1,p_2. This is a basic problem for directional differential quotient. Choosing a point 3 in the direction l and write the distance of (?) byΔι.Just as we know the choosing of point 3 is arbitrary because it is auxiliary. The order of magnitude ofΔιmust be equal to those ofΔι_1 andΔι_2 for the computational stability. Theorem 8 The formula for directional differential quotient of first order with two points in the direction ofιis as follows:We disperse the gradient operator and divergence operator by using the differential coefficient in the numerical direction because these operators are often used in practice. We disperse these operators using two point formula.Bacause some reasons we can not give the second order approximative formula for directional differential quotient. But we have finished some fundamental work such as to prove the well-posedness for interpolation on some points.2. The case of three dimensionAs shown in Figure 2, the neighbours of point 0:(x_0,y_0,z_0) are poin 1, 2 and 3 and their dirctions l_1,l_2,l_3 are not parallel. Let another direction l and we want to approximatively compute (?) by using the values on points 0,1,2,3, that is, p_0,p_1,p_2,p_3.This is a basic problem for directional differential quotient. Choosing a point 4 in the direction 1 and write the distance of (?) byΔι.Just as we know the choosing of point 4 is arbitrary because it is auxiliary. The order of magnitude ofΔιmust be equal to those ofΔι_i,i=1,2,3 for the computational stability. Theorem 9 The formula for directional differential quotient of first order with three points in the direction ofιis as follows:By the same way we disperse some basic differential operators using the derived formula for differential coefficient.