代数流形上的Lagrange插值和Hermite插值
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摘要
多元Lagrange插值问题和多元Hermite插值问题是当前多元逼近研究领域中的两个热门课题,尤其是代数流形上的插值问题,越来越得到广泛的关注。本文正是针对这两个课题展开了研究,并取得了相应的研究成果。
文章在绪论中简要地介绍了目前多元多项式插值的研究近况。在此基础上,我们深入研究了高维空间中代数流形上的Hermite插值问题。首先给出了问题的提法,然后证明了代数流形上Hermite插值的叠加插值法。它包含代数流形上的Lagrange插值问题的相应结果作为特例,因此是对代数流形上的多项式插值问题研究的继续和深入。
针对三维空间中一种特殊的代数流形-球面,具体讨论了其上的Lagrange插值和Hermite插值问题。并将代数流形上Hermite插值的叠加插值法具体化,得到了一些球面上比较实用的插值格式,它们包含了之前相关文献中给出的插值格式做为特例,因此是对现有的球面上插值格式的一个扩充和发展。文中还给出了一些具体的插值格式的例子以及一些数值算例。
作为插值理论的一个简单应用,文章最后研究了一类特殊的无网格方法一有限点方法。对于二维和三维散乱数据,建立了一阶数值方向微商的近似公式。对于二阶数值微商的近似公式,给出了相应的稳态邻点集的证明。
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). If O(K_(i,j))=O(K_(i,u)),then sort the order of {K_(i,j)} according to its components,that is, if there hasι,0≤ι≤n-1 such thatthen K_(i,j) So we can get a order of the indexing set {K_(i,j)},that is,Because the descendent paths in the oriented graph of I(Q_i) have origin as their's destinations we get K_(i.0)=(0,0,…,0).And we suppose K_(i,j) sort its order from the higher order to the lower order.
We consider the following set of interpolating functionals: whereandα_u~((i,j))∈R.
Now we give the description of this kind of Hermite interpolation problem in the space P_m(R~n).
Problem 1.We consider the set of interpolating functionals just as (1) where(?)λ(Q_i)=e_m~((n)).For any given set of real numbers (?),we want to find p_n(X)∈P_m(R~n) such that
Definition 1 If there always exists a unique polynomial p_n(X)∈P_m(R~n) such that (2) is satisfied for any given real numbers (?),then we call Problem 1 well posed and call the set of interpolating functionals in (1) a properly posed set of interpolating functionals of degree m for Hermite interpolation in the space P_m(R~n) and A_m= {Q_i,1≤i≤M} is called the corresponding set of nodes.
In some practical problem people often need to consider the interpolation along the algebraic manifold. So we give the Hermite interpolation problem along the sufficiently intersected algebraic manifold. We suppose s(1≤s≤n) algebraic hypersur-faces without multiple factors f_1(X)=0,…,f_s(X)=0 of degree k_1,…,k_s respectively,sufficiently intersect at the algebraic manifold V_(n-s)=v(f_1,…,f_s) in R~n.Let U_m(V_(n-s))={Q_i,1≤i≤M} be a set of M mutually distinct points on the algebraic manifold V_(n-s).The definition of the grade of U_m(V_(n-s)) is G(U_m)=(?){G(Q_i)}.
Now we give the description of this kind of Hermite interpolation problem along the algebraic manifold V_(n-s).
Problem 2.We consider the set of interpolating functionals just as (1) and(?)λ(Q_i)=e_m~((n))(k_1,…,k_s).For any given set of real numbers (?),we want tofind p_n(X)∈P_m(R~n) such that
Definition 2 If there always exists a polynomial p_m(X)∈P_m(R~n) such that (3)is satisfied for any given set of real numbers (?),then we call Problem 2 well posed and call the set of interpolating functionals in (1) a properly posed set of interpolating functionab of degree m for Hermite interpolation along the algebraic manifold V_(n-s) and U_m(V_(n-s)) is called the corresponding set of nodes.
The following theorem is the corresponding superposition interpolation process for this kind of Hermite interpolation along an algebraic manifold.
Theorem 1 Let f_1(X)=0,…,f_s(X)=0 be s(1≤s≤n) be algebraic hypersur-faces without multiple factors of degree k_1,…,k_s,respectively, and they sufficiently intersect at an algebraic manifold V_(n-s)=v(f_1,…,f_s) in R~n.Let L be a PPSIF of degree m+rk_s for Hermite interpolation along the algebraic manifold V_(n-s) and U_(m+rk_s)(V_(n-s)) be the corresponding set of nodes where r is the grade of U_(m+rk_s)(V_(n-s)). Let (?) be a PPSIF of degree m for Hermite interpolation along the algebraic manifold V_(n-s+1)-v(f_1,…,f_(s-1)) and U_m(V_(n-s+1)) be the corresponding set of nodes. Suppose f_s(X)=0 does not pass through any point in U_m(V_(n-s+1)).Then L∪(?) must be a properly posed set of interpolating functionals of degree m + rk along the algebraic manifold V_(n-s+1) and U_(m+rk_s)(V_(n-s))∪U_m(V_(n-s+1)) be the corresponding set of nodes.
Using this theorem we can get a series of properly posed set of interpolating functionals of higher degree from those of lower degree along the algebraic manifold.
We deeply research a kind of algebraic manifold in R~3,that is the sphere. We study Lagrange interpolation and Hermite interpolation on the unit sphere, that is to say, we make a concrete practice for the superposition interpolation process for Hermite interpolation along the algebraic manifold. And we get some schemes for interpolation on the sphere which include the corresponding results in the previous conferences.
Part 2. Lagrange interpolation on the unit sphere.
In this section, we research Lagrange interpolation on the unit spherein the polynomial space of degree 3Π_n~3={(?)α_(i,j,k)x~iy~jz~k,α_(i,j,k)∈R}.LetΠ_n(S~2) denote the space of spherical polynomials of degree n, that is, the restriction of polynomials degree n in three variables to S~2.
It is well known that dimΠ_n~3=(?),dimΠ_n(S~2)=(n+1)~2.The following is the definition of Lagrange interpolation problem on the unit sphere. Definition 3 Let n be a natural number and suppose a set of points U_n={Q_i}_(i=1)~((n+1)~2) include (n+1)~2 mutually distinct points on the unit sphere S~2.If there exists a polynomial p_n(X)∈Π_n~3 such thatfor any given set of real numbers {f_i}_(i=1)~((n+1)~2).Then we call the set of points U_n a PPSN of degree n on S~2.
For Lagrange interpolation on the unit sphere, it is more convenient to work with spherical coordinates:
It is corresponding to a latitude on the sphere for any given latitudinal angleφ(0≤φ≤π).Letλ≤n+1 be a positive integer and we choose arbitrarily a set consisting ofλlatitudinal angles and denote it byΦ_λ={φ_ι|1≤ι≤λ,0≤φ_1<φ_2<…<φ_λ≤π},moreover, we denote theλlatitudes corresponding to theλlatitudinal angles by the set of latitudes L(Φ_λ).After computation, we obtain the dimension of the space spanned by the set of latitudes L(Φ_λ),that is,λ(2n+2-λ). Now we give a description of the interpolation problem along the set of latitudesL(Φ_λ).
Problem 3: Let n and j be nonnegative integers. Supposeιandλ≤n+1 are positive integers and a set of latitudes L(Φ_λ) correspond toλlatitudinal angles 0≤φ_1<φ_2<…<φ_λ≤πon the unit sphere. Suppose {Q_(ι,j):,0≤j≤2n+1-λ,1≤ι≤λ} denotes the set of mutually distinct points on the latitude z=cosφ_ι,1≤ι≤λ.We want to seek a polynomial p_n(X)∈Π_n~3 such that
Definition 4 If there always exists a polynomial p_n(X)∈Π_n~3 such that (4) is true for any given real numbers {f_(ι,j)},then we call that Problem 3 is well-posed and the set of points a properly posed set of nodes of degree n along the set of latitudes L(Φ_λ).
Especially, we need to impose an additional symmetry whenλis an even number such asλ=2h where h andιare positive integers, that is,φ_(2h+1-ι)=π-φ_1,φ_ι∈(0,(?)),1≤ι≤h.
We denote the set of longitudinal angles byΘ_α~(n,λ)(α∈[0,2)) as following sinceλis a positive integer:
1.Ifλ=1,then
2.Ifλ>2 then
Then 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 Suppose the axis of a set of coaxial circumferences C whose total multiplicity isλisι.Choose a set of points A_i~n={Q_(i,j),1≤j≤m_j} on each circumference C_iand write A_n~λ=∪_(i=1)~κA_i~n which must satisfy (?)(?)δ_(i,j)=dim(C)=λ(2n+2-λ). We consider the following set of interpolating functionals:whereβ_1,β_2,…,β_k∈R.
Problem 5: For any given real numbers {f_(i,j,k)},we want to find p_n(X)∈Π_n~3such that
Definition 6 If there always exists a polynomial p_n(X)∈Π_n~3 such that (9) is satisfied for any given real numbers {f_(i,j,k),then we call Problem 5 well posed and call the set of interpolating functionals in (8) a properly posed set of interpolating functionals of degree n along the set of coaxial circumferences and A_n~## is called the corresponding set of nodes.
We have the following results for Hermite interpolation problem along the set of coaxial circumferences.
Case 1.Letλ=2h+1.
Supposeιis the axis of the set of coaxial circumferences C={C_i,1≤i≤κ}. Divide the equator corresponding to the axisιinto 2n+2-λequal shares and denote the great circle by (?)_j(1≤j≤2n+2-λ) which pass through the j-th equidistant point on the equator, the south pole and the north pole. Then we write the set of equidistant points by A_i~n={Q_(i,j),1 We call it Class I Hermite interpolation of degree n along the set of coaxial circumferences C.
Case 2.Letλ=2h.Letιbe the axis of set of coaxial circumferences C= {C_i,1≤i≤κ}.Especially, we need to impose an additional symmetry, that is, each C_i consisting of two coaxal circumferences,C_i={C_(i,1),C_(i,2)},where C_(i,1),C_(i,2)are symmetrical about the equator determined byι.Then we call the sum of multiplicities of C_(i,1),C_(i,2),1≤i≤κthe total multiplicity of C.
Divide the equator determined byιinto 2(2n+2-λ) equal shares and write the great circle (?)_j which pass through the j-th equidistant points on the equator, the south pole and the north pole where 1≤j≤2(2n+2-λ).We write the set of equidistant points (?)_i~n={(?)_(i,j),1≤j≤2(2n+2-λ)} in which the point (?)_(i,j) is the intersection of C_i and C_j.Write C_n~λ=∪_(i=1)~κC_i~n.Then writeAnd we letδ_(i,j)=δ_i,(?)2δ_i=λand (?)(?)δ_(i,j)=λ(2n+2-λ),m_i=2n+2-λ,1≤i≤κ.
We call it ClassⅡHermite interpolation of degree n along the set of coaxial circumferences C.
Case 3. Suppose the total multiplicity of a set of coaxial circumferences C= {C_i,1≤i≤κ} isλ.For a given nonnegative integer n, consider all possible cases ofκ≤λ≤n+1 onκandλ.For each case ofκandλ,let n_1 be a nonnegative integer and n_2,…,n_κbeκ-1 positive integers and 0≤n_1 We call it ClassⅢHermite interpolation of degree n along the set of coaxialcircumferences C.
We proved the following theorem.
Theorem 6 ClassⅠ,ClassⅡand ClassⅢHermite interpolations of degree n along the set of coaxial circumferences C are all well posed.
For H-class Hermite interpolation of degree n on the unit sphere, we give a useful superposition interpolation process, that is, the Circumferences Set-Superposition Process.
Theorem 7 Let L~n be a properly posed set of interpolating functionals of degree n on the sphere and let V_n be the corresponding set of nodes. Suppose a set of coaxial circumferences C whose total multiplicity is A does not pass through any points in V_n. Let (?)~(n+λ,λ) be a ClassⅠ,or ClassⅡ,or ClassⅢproperly posed set of interpolating functionals of degree n+λalong C and let A_(n+λ)~λbe the corresponding set of nodes. Then L~n∪(?)~(n+λ,λ) must be a properly posed set of interpolating functionals of degree n+λon the sphere and V_n∪A_(n+λ)~λbe the corresponding set of nodes.
In this way, we can get a series of properly posed set of interpolating functionals of higher degree from those of lower degree on the sphere by repeatedly using the Circumferences Set-Superposition Process.
Part 4. Some applications for interpolation in finite point method.
The finite point method is a meshless method which gets its approximate function by using moving least square and solves equations by using matching point menthod, that is to say, to solve partial differential equation by differential method on the heterocentric set of points. The finite point method have some advantages such as directly adding the conditions on the boundary and not need to compute the integral in the interior of region.
We give some notations in this section.
i:the label of the point (x_i,y_i) or (x_i,y_i,z_i).When we consider the especial point we often denotes it by "0";
p_i:the discrete value of unknown function p_i-p(x_i,y_i) or p_i=p(x_i,x_j,z_i);
Δι_i:the distance from point 0 to point i,Δι_i=(?) orΔι_i=(?)
Δp_i:the differential of unknown function p,Δp_i=p_i-p_0;
δp_i:the difference quotient of unkown function p,δp_i=Δp_i/Δι_i;
(?):the vector from the point A; to the point i;
l_i:the vector from the point k to the point i;
(i,j,k):sin(?);
(?):cos(?);
(i,j):(i,j,0),sin(?); (?):(?):cos(?);:(?)(?),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 dimension
We 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 dimension
As 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.
文章在绪论中简要地介绍了目前多元多项式插值的研究近况。在此基础上,我们深入研究了高维空间中代数流形上的Hermite插值问题。首先给出了问题的提法,然后证明了代数流形上Hermite插值的叠加插值法。它包含代数流形上的Lagrange插值问题的相应结果作为特例,因此是对代数流形上的多项式插值问题研究的继续和深入。
针对三维空间中一种特殊的代数流形-球面,具体讨论了其上的Lagrange插值和Hermite插值问题。并将代数流形上Hermite插值的叠加插值法具体化,得到了一些球面上比较实用的插值格式,它们包含了之前相关文献中给出的插值格式做为特例,因此是对现有的球面上插值格式的一个扩充和发展。文中还给出了一些具体的插值格式的例子以及一些数值算例。
作为插值理论的一个简单应用,文章最后研究了一类特殊的无网格方法一有限点方法。对于二维和三维散乱数据,建立了一阶数值方向微商的近似公式。对于二阶数值微商的近似公式,给出了相应的稳态邻点集的证明。
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))
We consider the following set of interpolating functionals: whereandα_u~((i,j))∈R.
Now we give the description of this kind of Hermite interpolation problem in the space P_m(R~n).
Problem 1.We consider the set of interpolating functionals just as (1) where(?)λ(Q_i)=e_m~((n)).For any given set of real numbers (?),we want to find p_n(X)∈P_m(R~n) such that
Definition 1 If there always exists a unique polynomial p_n(X)∈P_m(R~n) such that (2) is satisfied for any given real numbers (?),then we call Problem 1 well posed and call the set of interpolating functionals in (1) a properly posed set of interpolating functionals of degree m for Hermite interpolation in the space P_m(R~n) and A_m= {Q_i,1≤i≤M} is called the corresponding set of nodes.
In some practical problem people often need to consider the interpolation along the algebraic manifold. So we give the Hermite interpolation problem along the sufficiently intersected algebraic manifold. We suppose s(1≤s≤n) algebraic hypersur-faces without multiple factors f_1(X)=0,…,f_s(X)=0 of degree k_1,…,k_s respectively,sufficiently intersect at the algebraic manifold V_(n-s)=v(f_1,…,f_s) in R~n.Let U_m(V_(n-s))={Q_i,1≤i≤M} be a set of M mutually distinct points on the algebraic manifold V_(n-s).The definition of the grade of U_m(V_(n-s)) is G(U_m)=(?){G(Q_i)}.
Now we give the description of this kind of Hermite interpolation problem along the algebraic manifold V_(n-s).
Problem 2.We consider the set of interpolating functionals just as (1) and(?)λ(Q_i)=e_m~((n))(k_1,…,k_s).For any given set of real numbers (?),we want tofind p_n(X)∈P_m(R~n) such that
Definition 2 If there always exists a polynomial p_m(X)∈P_m(R~n) such that (3)is satisfied for any given set of real numbers (?),then we call Problem 2 well posed and call the set of interpolating functionals in (1) a properly posed set of interpolating functionab of degree m for Hermite interpolation along the algebraic manifold V_(n-s) and U_m(V_(n-s)) is called the corresponding set of nodes.
The following theorem is the corresponding superposition interpolation process for this kind of Hermite interpolation along an algebraic manifold.
Theorem 1 Let f_1(X)=0,…,f_s(X)=0 be s(1≤s≤n) be algebraic hypersur-faces without multiple factors of degree k_1,…,k_s,respectively, and they sufficiently intersect at an algebraic manifold V_(n-s)=v(f_1,…,f_s) in R~n.Let L be a PPSIF of degree m+rk_s for Hermite interpolation along the algebraic manifold V_(n-s) and U_(m+rk_s)(V_(n-s)) be the corresponding set of nodes where r is the grade of U_(m+rk_s)(V_(n-s)). Let (?) be a PPSIF of degree m for Hermite interpolation along the algebraic manifold V_(n-s+1)-v(f_1,…,f_(s-1)) and U_m(V_(n-s+1)) be the corresponding set of nodes. Suppose f_s(X)=0 does not pass through any point in U_m(V_(n-s+1)).Then L∪(?) must be a properly posed set of interpolating functionals of degree m + rk along the algebraic manifold V_(n-s+1) and U_(m+rk_s)(V_(n-s))∪U_m(V_(n-s+1)) be the corresponding set of nodes.
Using this theorem we can get a series of properly posed set of interpolating functionals of higher degree from those of lower degree along the algebraic manifold.
We deeply research a kind of algebraic manifold in R~3,that is the sphere. We study Lagrange interpolation and Hermite interpolation on the unit sphere, that is to say, we make a concrete practice for the superposition interpolation process for Hermite interpolation along the algebraic manifold. And we get some schemes for interpolation on the sphere which include the corresponding results in the previous conferences.
Part 2. Lagrange interpolation on the unit sphere.
In this section, we research Lagrange interpolation on the unit spherein the polynomial space of degree 3Π_n~3={(?)α_(i,j,k)x~iy~jz~k,α_(i,j,k)∈R}.LetΠ_n(S~2) denote the space of spherical polynomials of degree n, that is, the restriction of polynomials degree n in three variables to S~2.
It is well known that dimΠ_n~3=(?),dimΠ_n(S~2)=(n+1)~2.The following is the definition of Lagrange interpolation problem on the unit sphere. Definition 3 Let n be a natural number and suppose a set of points U_n={Q_i}_(i=1)~((n+1)~2) include (n+1)~2 mutually distinct points on the unit sphere S~2.If there exists a polynomial p_n(X)∈Π_n~3 such thatfor any given set of real numbers {f_i}_(i=1)~((n+1)~2).Then we call the set of points U_n a PPSN of degree n on S~2.
For Lagrange interpolation on the unit sphere, it is more convenient to work with spherical coordinates:
It is corresponding to a latitude on the sphere for any given latitudinal angleφ(0≤φ≤π).Letλ≤n+1 be a positive integer and we choose arbitrarily a set consisting ofλlatitudinal angles and denote it byΦ_λ={φ_ι|1≤ι≤λ,0≤φ_1<φ_2<…<φ_λ≤π},moreover, we denote theλlatitudes corresponding to theλlatitudinal angles by the set of latitudes L(Φ_λ).After computation, we obtain the dimension of the space spanned by the set of latitudes L(Φ_λ),that is,λ(2n+2-λ). Now we give a description of the interpolation problem along the set of latitudesL(Φ_λ).
Problem 3: Let n and j be nonnegative integers. Supposeιandλ≤n+1 are positive integers and a set of latitudes L(Φ_λ) correspond toλlatitudinal angles 0≤φ_1<φ_2<…<φ_λ≤πon the unit sphere. Suppose {Q_(ι,j):,0≤j≤2n+1-λ,1≤ι≤λ} denotes the set of mutually distinct points on the latitude z=cosφ_ι,1≤ι≤λ.We want to seek a polynomial p_n(X)∈Π_n~3 such that
Definition 4 If there always exists a polynomial p_n(X)∈Π_n~3 such that (4) is true for any given real numbers {f_(ι,j)},then we call that Problem 3 is well-posed and the set of points a properly posed set of nodes of degree n along the set of latitudes L(Φ_λ).
Especially, we need to impose an additional symmetry whenλis an even number such asλ=2h where h andιare positive integers, that is,φ_(2h+1-ι)=π-φ_1,φ_ι∈(0,(?)),1≤ι≤h.
We denote the set of longitudinal angles byΘ_α~(n,λ)(α∈[0,2)) as following sinceλis a positive integer:
1.Ifλ=1,then
2.Ifλ>2 then
Then 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
Problem 5: For any given real numbers {f_(i,j,k)},we want to find p_n(X)∈Π_n~3such that
Definition 6 If there always exists a polynomial p_n(X)∈Π_n~3 such that (9) is satisfied for any given real numbers {f_(i,j,k),then we call Problem 5 well posed and call the set of interpolating functionals in (8) a properly posed set of interpolating functionals of degree n along the set of coaxial circumferences and A_n~## is called the corresponding set of nodes.
We have the following results for Hermite interpolation problem along the set of coaxial circumferences.
Case 1.Letλ=2h+1.
Supposeιis the axis of the set of coaxial circumferences C={C_i,1≤i≤κ}. Divide the equator corresponding to the axisιinto 2n+2-λequal shares and denote the great circle by (?)_j(1≤j≤2n+2-λ) which pass through the j-th equidistant point on the equator, the south pole and the north pole. Then we write the set of equidistant points by A_i~n={Q_(i,j),1
Case 2.Letλ=2h.Letιbe the axis of set of coaxial circumferences C= {C_i,1≤i≤κ}.Especially, we need to impose an additional symmetry, that is, each C_i consisting of two coaxal circumferences,C_i={C_(i,1),C_(i,2)},where C_(i,1),C_(i,2)are symmetrical about the equator determined byι.Then we call the sum of multiplicities of C_(i,1),C_(i,2),1≤i≤κthe total multiplicity of C.
Divide the equator determined byιinto 2(2n+2-λ) equal shares and write the great circle (?)_j which pass through the j-th equidistant points on the equator, the south pole and the north pole where 1≤j≤2(2n+2-λ).We write the set of equidistant points (?)_i~n={(?)_(i,j),1≤j≤2(2n+2-λ)} in which the point (?)_(i,j) is the intersection of C_i and C_j.Write C_n~λ=∪_(i=1)~κC_i~n.Then writeAnd we letδ_(i,j)=δ_i,(?)2δ_i=λand (?)(?)δ_(i,j)=λ(2n+2-λ),m_i=2n+2-λ,1≤i≤κ.
We call it ClassⅡHermite interpolation of degree n along the set of coaxial circumferences C.
Case 3. Suppose the total multiplicity of a set of coaxial circumferences C= {C_i,1≤i≤κ} isλ.For a given nonnegative integer n, consider all possible cases ofκ≤λ≤n+1 onκandλ.For each case ofκandλ,let n_1 be a nonnegative integer and n_2,…,n_κbeκ-1 positive integers and 0≤n_1
We proved the following theorem.
Theorem 6 ClassⅠ,ClassⅡand ClassⅢHermite interpolations of degree n along the set of coaxial circumferences C are all well posed.
For H-class Hermite interpolation of degree n on the unit sphere, we give a useful superposition interpolation process, that is, the Circumferences Set-Superposition Process.
Theorem 7 Let L~n be a properly posed set of interpolating functionals of degree n on the sphere and let V_n be the corresponding set of nodes. Suppose a set of coaxial circumferences C whose total multiplicity is A does not pass through any points in V_n. Let (?)~(n+λ,λ) be a ClassⅠ,or ClassⅡ,or ClassⅢproperly posed set of interpolating functionals of degree n+λalong C and let A_(n+λ)~λbe the corresponding set of nodes. Then L~n∪(?)~(n+λ,λ) must be a properly posed set of interpolating functionals of degree n+λon the sphere and V_n∪A_(n+λ)~λbe the corresponding set of nodes.
In this way, we can get a series of properly posed set of interpolating functionals of higher degree from those of lower degree on the sphere by repeatedly using the Circumferences Set-Superposition Process.
Part 4. Some applications for interpolation in finite point method.
The finite point method is a meshless method which gets its approximate function by using moving least square and solves equations by using matching point menthod, that is to say, to solve partial differential equation by differential method on the heterocentric set of points. The finite point method have some advantages such as directly adding the conditions on the boundary and not need to compute the integral in the interior of region.
We give some notations in this section.
i:the label of the point (x_i,y_i) or (x_i,y_i,z_i).When we consider the especial point we often denotes it by "0";
p_i:the discrete value of unknown function p_i-p(x_i,y_i) or p_i=p(x_i,x_j,z_i);
Δι_i:the distance from point 0 to point i,Δι_i=(?) orΔι_i=(?)
Δp_i:the differential of unknown function p,Δp_i=p_i-p_0;
δp_i:the difference quotient of unkown function p,δp_i=Δp_i/Δι_i;
(?):the vector from the point A; to the point i;
l_i:the vector from the point k to the point i;
(i,j,k):sin(?);
(?):cos(?);
(i,j):(i,j,0),sin(?); (?):(?):cos(?);
1. The case of two dimension
We 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 dimension
As 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.
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