A Symbolic Algorithm for Polynomial Interpolation with Integral Conditions

Home , Polynomial interpolation

Appl. Math. Inf. Sci. 12, No. 5, 995-1001 (2018) 995 Applied Mathematics & Information Sciences An International Journal

http://dx.doi.org/10.18576/amis/120512

Srinivasarao Thota Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Ethiopia

Received: 2 Jun. 2018, Revised: 27 Jul. 2018, Accepted: 10 Aug. 2018 Published online: 1 Sep 2018

Abstract: This paper presents a new symbolic algorithm for polynomial interpolation with integral conditions at arbitrary points. For expressing the integral conditions in the present algorithm, we employ the algebra of integro-differential operators. We also present another algorithm for computing the polynomial interpolation with Stieltjes conditions (combination of general, differential and integral conditions) as a quotient of two determinants. Error due to the formulation of a given function by the proposed interpolation is discussed and its symbolic formulation is presented. This algorithm helps OR would help to implement the manual calculations in commercial packages such as Maple, Mathematica, Matlab, Singular, etc. Certain numerical examples are presented to verify the proposed algorithms.

Keywords: Polynomial interpolation, integral conditions, error estimation, integro-differential operators

1 Introduction problem. In Section 3, we discuss various formulation of the error estimation. Selected examples are discussed to demonstrate the proposed algorithm. In science and engineering, researchers often come up with data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent 1.1 Interpolation problem variable. It is often required to interpolate the value of We give, first, the general form of the interpolation that function for an intermediate value of the independent problem as follows [3,5,6,7]: Suppose S is a normed variable. There exist many interpolation techniques in the linear space. For a finite linearly independent set Θ ⊂ S literature for general functional values, see for of bounded functionals and associated values example, [1,2,3]. The purpose of this paper is to develop Ω = {αθ : θ ∈ Θ}⊂ R, the interpolation problem is to an algorithm to construct a polynomial interpolation with find a f (x) ∈ S such that a finite set of integral conditions alone as well as the s combination of general, differential and integral Θ( fs)= Ω, i.e. θ fs = αθ , θ ∈ Θ. (1) conditions, so-called Stieltjes conditions [4], via e integro-differential operators. Here s is called the order of the interpolating function e e The paper is organized as follows: In Section 1.1, we fs(x). To describe the polynomial interpolation, let present the definitions and basic concepts of the S = K[x] be a polynomial ring over a field K, where polynomial interpolation, in Section 1.2, we recall the Ke = Q,R or C. One can observe that the interpolation algebra of integro-differential operators and the operator problem given in equation (1) may have many solutions if representation of integral and Stieltjes there is no restriction on the dimension of the problem. functionals/conditions in terms of integro-differential But we want a single interpolate polynomial which must operators. Proposed symbolic algorithm for the satisfies the given conditions. Hence, for the unique polynomial interpolation is presented in Section 2, and solution of the problem, we must have finite dimensional Section 2.1 provides the condition of existence and subspace Θ of S having the dimension equal to the uniqueness of the solution of a given interpolation number of conditions.

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Definition 1.We call the pair (Θ,Ω) a polynomial operator E : S → K is a K-linear map, also called as interpolation problem, where Θ is a finite linearly character. In the above standard example S = C∞(R), independent set of functionals with associated values evaluate f at the initial point of the integral, i.e. Ω ⊂ K. E f (x) = f (0). It is shown in [8, Section 3] that the evaluation E = 1 − AD is multiplicative linear functional Definition 2.A polynomial interpolation problem (Θ,Ω) (character), it means that E fg = (E f )(Eg). The evaluation is called regular with respected to Θ if (Θ,Ω) has a allows to formulate the initial value problems, see [8,11]. unique solution for each choice of values of Ω ⊂ K such For treating boundary value problems, we need another Θ Ω that ( fs)= . Otherwise, it is called singular. character Ec : S → K, the evaluation operator at various ∗ points c ∈ R, i.e. Ec : f 7→ f (c). Let Φ ⊆ S be a set of The following proposition gives the regularity test in e all multiplicative linear functionals including E. terms of linear algebra. Definition 4.[8] Let (S ,D,A) be an ordinary Proposition 1. [8,7] Let M = {m0,...,mt−1} be a basis integro-differential algebra over K and Φ ⊆ S ∗. The M S for , a finite dimensional subspace of , and integro-differential operators S [D,A] are defined as the Θ θ θ S ∗ θ = { 0,..., s−1} ⊂ with i linearly independent. K-algebra generated by the symbols D and A, the Then the following statements are equivalent: functions f ∈ S and the characters (functionals) Ec ∈ Φ, (i)The polynomial interpolation problem is regular for modulo the Noetherian and confluent rewrite system M with respected to Θ. given in Table 1. (ii)t = s, and the evaluation matrix,

θ0(m0) ··· θ0(mt−1) Θ . .. . Table 1: Rewrite rules for integro-differential operators M =  . . .  (2) ′ θs 1(m0) ··· θs 1(mt 1) f g → f · g D f → f D + f A f A → (A f )A − A(A f )  − − −    χφ → φ Dφ → 0 A f D → f − A f ′ − (E f )E is nonsingular. Denote the evaluation matrix ΘM by E for simplicity. φ f → (φ f )φ DA → 1 A f φ → (A f )φ (iii)S = M ⊕Θ ⊥.

The following lemma shows that every 1.2 Algebra of integro-differential operators integro-differential operator can be expresses as linear combination of monomials of the form f φAgψDi. In this section, we recall some basic concepts of integro- Lemma 1. [8] Every integro-differential operator in differential algebras and operators see, for example, [8,9, S [D,A] can be reduced to a linear combination of 10,11,12,13] for further details. In this section, K denotes monomials f φAgψDi, where i ≥ 0 and each of the field of characteristic zero. f ,φ,A,g,ψ may also be absent. S Definition 3. [8] Let be a commutative algebra over a Definition 5. [8] The elements of the right ideal |Φ)= S field K. The structure ( ,D,A) is called an Φ ·S [D,A] are called Stieltjes conditions over S , andthe S integro-differential algebra if ( ,D) is a commutative elements of the two-sided ideal (Φ) of S [D,A] generated differential algebra over K and the differential Baxter by Φ are called Stieltjes operators. axiom Since the rewrite system of Table 1 is Noetherian and (AD f )(ADg)+ AD( fg) = (AD f )g + f (ADg) confluent (see, for example, [8] for further details), every integro-differential operator has a unique normal form. S S S S holds. Where D : → and A : → are two maps Moreover, every monomial is either a differential operator such that D is a derivation and A is a K-linear right inverse or an integral operator or a Stieltjes operator, so the of D, i.e. DA = 1 (the identity map). The map A is called normal form of integro-differential operators can be an integral for D. An integro-differential algebra over K is expressed as a sum of differential, integral and Stieltjes called ordinary if Ker(D)= K. operators. The normal form of differential operators is as For example [8], we have S = C∞(R), the set of usual, the normal form of integral operators are itself and A smooth functions over the field of real numbers with linear combinations of terms of the form f g, and the x normal form of Stieltjes operators is of the following D = d and A = f 7→ f (ξ ) dξ . Here A is the right dx 0 form [8, Proposition 25] inverse of D, i.e. DA = 1, but AD = 1 − E. Indeed, R (AD) f (x)= f (x) − f (0)= f (x) − E f (x) 6= f (x). The operator E = 1 − AD, called the evaluation of S , i ∑ ∑ aφ,iφD + φA fφ (3) evaluates at the initial point of the integral. The evaluation φ∈Φ i∈N !

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φ φ S T σ α α T with a ,i ∈ K and f ∈ almost all zero. These where u = (a0,...,as−1) and = ( θ0 ,..., θs−1 ) . operators act on S as linear functions in S ∗. The Existence of the solution depends upon nature of the following proposition states the above fact. evaluation matrix E . As mentioned at beginning of Section 2, there exists a solution for (Θ,Ω) if and only if Proposition 2. [8] For an ordinary integro-differential E is regular. For uniqueness of the solution, suppose f (x) algebra S and characters Φ ⊆ S ∗, we have s ′ θ αθ θ ′ S [D,A] = S [D] ⊕ S [A] ⊕ (Φ), where S [D] is the and fs (x) are two solutions, then i fs(x)= i = i fs (x) ′ θ Θ differential operators and S [A] is the bimodule implies that fs(x)= fs (x) for each i ∈ is injective.e generated by A and monomials of the form f Ag, and (Φ) Thee following Examples 1 and 2eare given to calculatee is an ideal of Stieltjes operators. the evaluatione matrixe E with integral conditions at arbitrary point on the interval [1,2] and a generalization Deﬁnition 6. [3] A set of integro-differential operators for {x0,...,xs−1}⊂ R respectively. Γ = {γ0,...,γs} is called a Tchebycheff system, or simply a T-system for a ﬁnite linearly independent set 1 M = {m0,...,ms}, if the evaluation matrix Γ M is Example 1.Consider the integral conditions f (x) dx = 2, 0 nonsingular for all sets of s + 1 evaluation points c ∈ R. 1.25 1.4 1.6 R γ γ The operator 0,..., i form a complete Tchebycheff f (x) dx = 1, f (x) dx = 1, f (x) dx = 2, system, or simply CT − system, if {γ0,...,γi} is a 0 0 0 T-system for each i = 0,...,s. 1R.85 R2 R f (x) dx = 2 and f (x) dx = 3. In symbolic notations, 0 0 weR have Θ = {E1A,RE1.25A,E1.4A,E1.6A,E1.85A,E2A} and 2 Symbolic formulation of polynomial its associated values Ω = {2,1,1,2,2,3}. Then the interpolation following evaluation matrix is calculated as in (2):

Consider the interpolation problem (Θ,Ω) given in 1.00 0.50 0.33 0.25 0.20 0.17 1.25 0.78 0.65 0.61 0.61 0.64 Section 1.1 where Θ = {θ0,...,θs 1} of the form − 1.40 0.98 0.91 0.96 1.08 1.25  {EcA : c ∈ R}, the monomials of integral conditions. E ≈ . From Proposition 1, the polynomial interpolation problem 1.60 1.28 1.37 1.64 2.10 2.80  1.85 1.71 2.11 2.93 4.33 6.68  is regular with respect to Θ if and only if there exists a   2.00 2.00 2.67 4.00 6.40 10.67 finite linearly independent set M = {m0,...,ms−1} of S   such that the evaluation matrix E defined as in   −8 equation (2) is regular. Indeed, there exists a unique Since det(E ) ≈−1.40 × 10 6= 0, there exists a unique interpolating polynomial with respected to Θ. fs(x) ∈ S satisfying Θ if and only if there exists a set M = {1,...,xs−1}⊂ S such that the evaluation matrix E Example 2.Consider the monomials is regular. e Θ = {E A,E ,...,E A}. Then evaluation matrix is The classification of the interpolation problem x0 x1 xs−1 given by depends on the type of the functionals θ ∈ Θ that have to Θ be matched with the polynomial. If is a finite set of 1 2 1 s E K x0 2 x0 ··· s x0 monomials of the form { c : c ∈ } with associated 1 2 1 s values Ω = {αθ : θ ∈ Θ}, then such type of interpolation x1 2 x1 ··· s x1 E =  . . . . , (5) is called Lagrange or Newton interpolation and the points . . .. . c K are called nodes. If Θ is of the form   ∈ x 1 x2 ··· 1 xs  {E Di : c ∈ K,i ∈ N} with Ω, then it is called Hermite or  s−1 2 s−1 s s−1 c   Birkhoff interpolation depending on i consecutive or not. and the determinant of the evaluation matrix (5) is It is seen that, there is no existing method available for Θ of the form {E A : c ∈ K} with given Ω to find a x x ···x c det(E )= 0 1 s−1 ∏ (x − x ). polynomial f x . Hence, we develop an algorithm to j i s( ) s! 0≤ j

at given arbitrary nodes {Exi : xi ∈ R} with associated Newton’s divided difference formula as follows a αθ αθ ,...,αθ i αθ i αθ αθ values { 0 s−1 } , where ai = i+1 and i = x , 0 0 i f s(x)= + (x − x0)∑ for i = 0,1,...,s − 1. Following Newton’s divided x0 x0(x1 − x0) difference interpolations formula, we have αθ e e e e e 0 e + (x − x0)(x − x1)∑ x0(x0 − x1)(x0 − x1) xr[Er+1 f (x)] − xr+1[Er f (x)] f [xr,xr+1]= + ··· (8) xr+1xr[xr+1 − xr] + (x − x0)(x − x1)···(x − xs−2) x αθ − x αθ r r+1 r+1 r αθ = 0 xr+1xr[xr+1 − xr] × ∑ x0(x0 − x1)(x0 − x1)···(x − xs−1) αθ αθ r r+1 s−1 = + = a0 + a1x + ··· + as−1x . xr(xr+1 − xr) xr+1(xr − xr+1) αθ s−1 r Hence, we have f (x)= a + a x + ··· + a x , where = ∑ , e es 0 e 1 s−1 xr(xr+1 − xr) coefﬁcients ai, for i = 0,...,s − 1, are calculated as ai = (i + 1)ai. e similarly, we have the (s − 1)th divided differences The following theorem gives generalization of above formulation.e αθ Θ θ θ 0 Theorem 2Let = { 0,..., s−1} be a ﬁnite set of the f [x0,x1,...,xs−1]= ∑ . (7) S ∗ x0(x0 − x1)···(x0 − xs−1) form {EcA : c ∈ R} ⊂ with associated values Ω α θ Θ = { θi : i ∈ } ⊂ R. Then there exists a unique s polynomial fs(x)= a0 + a1x + ··· + asx satisfying Θ if and only if the evaluation matrix E is regular, and fs(x) is Theorem 1The (s − 1)th divided differences can be given by, e expressed as the quotient of two determinants of order s s−1 e each. fs(x)= a0 + (2a1)x + ··· + (sas−1)x , (9)

ai where ai = . e i+1 Proof.From equation (7), we can write To verify the algorithme in Theoreme 2, we provide a test examplee to construct a polynomial interpolation with α ∑ θ ∏1 j

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2 s 1 x0 x0 ··· x0 and hence, we have ...... 2 s 1 xl−1 xl−1 ··· xl−1 2 s−1 0 xl 2xl ··· (s − 1)xl 93 2 3 2 s−1 f4(x)= 15 − 53x + x − 12x c0 0 x 2x ··· (s − 1)x 2 l+1 l+1 l+1 ...... e 160 2 3 D2 = + − + 224x − 224x + 64x c1 2 s−1 3 0 xl+m−1 2xl+m−1 ··· (s − 1)xl+m−1 2 s−1 xl+m xl+m xl+m 243 1053 2187 2 324 3 1 2 3 ··· s + − x + x − x c2 2 xs−1 5 5 10 5 xl+m+1 xl+m+1 l+m+1 1 2 3 ··· s 15 72 2 3 ...... + − + 34x − x + 12x c3...... 2 2 2 s−1 xs−1 xs−1 xs−1 1 2 3 ··· s αθ j and αθ = xiαθ , αθ = . i i j x j Now, we present another algorithm to find a The denominator D2 is non-zero, for the evaluation matrix polynomial that satisfies the given Stieltjes conditions, i.e. is regular. e not only on the function values and its derivatives, but In the following example, we find a polynomial with also for arbitrary integral conditions, as a quotient of two Stieltjes conditions using the algorithm presented in determinants in the following theorem. This quotient Theorem 3. represents a particular case of general remainder theorem [14, p. 75]. Example 4.Consider the conditions Θ = {E0,E 1 ,E1D,E 4 A,E2A} with Ω = {1,2,0,3,−1}. 2 3 By Theorem 3, we have Theorem 3Let Θ = {E ,··· ,E , E D, x0 xl−1 xl − 136 + 136 + 28 x + 167 x2 + 1373 x3 − 29 x4 E D E D E A E A E A 3645 3645 1215 1944 14580 243 xl+1 ,··· , xl+m−1 , xl+m , xl+m+1 ,..., xs−1 : f5(x)= 1− , ∗ 136 xi ∈ R;0 < l < l + m < s − 1;l,m ∈ N} ⊂ S with − 3645 associated values Σ αθ : θ Θ R. Then there = { i i ∈ } ⊂ e after simplification, we have exists a unique polynomial fs(x) satisfying Θ if and only if the evaluation matrix E is regular, and fs(x) is given by, 21 2505 2 1373 3 435 4 e f5(x)= 1 + x + x + x − x . 34 1088 544 136 e D1 Ife we choose Ω = {c0,c1,c2,c3,c4}, then we get fs(x)= 1 − , (10) D2 39 9 181 15 e f (x)= 1 − x + x2 + x3 − x4 c 5 17 272 136 34 0 where e 384 1104 2 928 3 240 4 + x − x + x − x c1 D 17 17 17 17 1 = 1 1 x x2 xs−1 ··· 62 849 2 413 3 60 4 αθ 2 s + x − x + x − x c2 0 1 x0 x0 ··· x0 17 68 34 17 ...... ...... 243 48843 2 20169 3 1215 4 + − x + x − x + x c3 αθ 2 s l−1 1 xl−1 xl−1 ··· xl−1 17 1088 544 136 s−1 αθ 2 l 0 xl 2xl ··· (s − 1)xl 21 687 441 45 s−1 2 3 4 αθ 0 x 2x2 s 1 x + − x + x − x + x c4. l+1 l+1 l+1 ··· ( − ) l+1 34 272 136 34 ...... ...... , s−1 αθ 2 l+m−1 0 xl+m−1 2xl+m−1 ··· (s − 1)xl+m−1 s 1 x x2 x − 3 Error estimation αθ l+m l+m l+m l+m 1 2 3 ··· s s 1 x x2 x − αθ l+m+1 l+m+1 l+m+1 l+m+1 1 2 3 ··· s The error involved due to the approximation of a function e ...... f x by a polynomial interpolant f x is calculated as ...... ( ) s( ) s 1 follows e x x2 x − αθ s−1 s−1 s−1 s−1 1 2 3 ··· s Es( f ,Θ)= f (x) − fes(x). (11)

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3.1 Error term as a quotient of determinants 1 and T ♦T = 1 − P; and P ∈ S [D,A] is the projector onto Ker(T ) along Θ ⊥, computed as The error term in equation (11) can be represented by a quotient of determinants as given in equation (10) for s−1 i integral conditions, i.e. P = ∑ x θi, (15) i=0 e f (x) 1 x ··· xs−1 T −1 T here (θ0,θ1,...,θs−1) = E (θ0,θ1,...,θs−1) . x xs αθ 0 0 0 x0 2 ··· s Simplifying the equation (14) using the fact that 1 − P x xs αθ 1 1 is projector,e e we gete 1 x1 2 ··· s ...... Es( f ,Θ) = (1 − P) f , x xs αθ x s−1 ··· s−1 Θ s−1 s−1 2 s Es( f , )= s . as an operator to calculate the error, we have Es = 1 − P ∈ x0 x0 x0 ··· S [D,A], P is a projector computed as in equation (15). 2 ss x1 x1 The following theorem gives the generalization of above x1 2 ··· s . . . . observation...... x xs Theorem 4Given any finite set Θ of Stieltjes conditions x s−1 ··· s−1 s−1 2 s such that the evaluation matrix is regular, the error In other words, the quotient is a linear combination of induced due to the approximation of a function f (x) by a { f (x),1,...,xs−1} in which the coefficient of f (x) is 1. polynomial interpolation is computed as In the following section, we provide a symbolic Θ formulation of the error Es( f ,Θ) over integro-differential Es( f , ) = (1 − P) f (x), algebras. where P is the projector operator as given in equation (15). The following two examples 5 and 6 show the 3.2 Symbolic formulation of error estimation computation of the error evaluate operator E = 1 − P and errors at various points on the interval [1,2] with integral To formulate the error term Es( f ,Θ) as in equation (11) and Stieltjes conditions respectively. in symbolic form, we first choose a differential operator T , whose fundamental system is {1,x,...,xs−1}. Such Example 5.Consider the data of integral conditions of the differential operator T is of the form form Θ = {E1A,E1.2A,E1.5A,E2A} at arbitrary points from the interval [1,2]. Then by equation (15) the ds+1 T = . (12) projector operator is given by dxs+1 2 3 The polynomial interpolant satisfies the differential P =(36 − 144x + 141x − 40x )E1A 2 3 equation + (−52.08 + 225.69x − 234.38x + 69.44x )E1.2A 2 3 T f = 0, + (21.33 − 99.56x + 112x − 35.56x )E1.5A + (−2.25 + 11.25x − 13.88x2 + 5x3)E A. Θ fs = Θ f . 2 e Therefore, we have semi-inhomogeneous boundary value For simplicity, if f (x)= ex, then using the algorithm in e problem (inhomogeneous differential equation and Theorem 4, the error involved due to the approximation of homogeneous boundary conditions) as follows f (x) by the polynomial interpolation is Θ TEs( f , )= T f , (13) E4( f ,Θ) = (1 − P) f (x) Θ Θ Es( f , )= 0. = ex − 0.919 − 1.46x + 0.197x2 − 0.540x3. Using the symbolic analysis for boundary value Error at x 1.4 is calculated as E f 1.4 ,Θ 6.152 problems developed by Rosenkranz et al. [8], we have = 4( ( ) )= − × 10−4. Similarly E f 1.9 ,Θ 6.209 10−3. following symbolic solution for the boundary value 4( ( ) )= × If f x sin 0.2x e0.6x, then problem given in equation (13) over integro-differential ( )= ( ) algebras ♦ E4( f ,Θ) = (1 − P) f (x) Es( f ,Θ) = (1 − P)T T f , (14) = sin(0.2x)e0.6x + 0.00856 where T ♦ ∈ S [D,A] is the fundamental right inverse of T , 2 3 computed from the given fundamental system, i.e. T T ♦ = − 0.249x − 0.042x − 0.0789x .

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Example 6.Consider a set of Stieltjes conditions of the [9] S. Thota and S. D. Kumar, Symbolic method for polynomial form Θ = {E1,E1.2D,E1.5A,E2} at arbitrary points of interpolation with Stieltjes conditions, Proceedings of [1,2]. Then error evaluate operator, computed similar to International Conference on Frontiers in Mathematics, pp. Example 5, is given by 225–228 (2015). [10] S. Thota and S. D. Kumar, Solving system of higher- 2 3 E4 = 1 − (−3.244 + 8.169x − 4.577x + 0.651x )E1 order linear differential equations on the level of operators, International Journal of Pure and Applied Mathematics, Vol. 2 3 E D − (0.489 − 3.339x + 4.153x − 1.303x ) 1.2 106, No. 1, pp. 11–21 (2016). 2 3 − (3.058 − 6.671x + 4.656x − 1.042x )E1.5A [11] S. Thota, On a new symbolic method for initial value 2 3 problems for systems of higher-order linear differential − (−0.342 + 1.837x − 2.407x + 0.912x )E2. equations, International Journal of Mathematical Models and Methods in Applied Sciences, Vol. 12, pp. 194–202 (2018). If f (x)= cos(0.4x), then error evaluate function is [12] S. Thota and S. D. Kumar, Symbolic algorithm for a system E ( f ,Θ)= cos(0.4x) − 1.108 + 2.023x − 2.594x2 + 0.706x3. of differential-algebraic equations, Kyungpook Mathematical 4 Journal, Vol. 56, No. 4, pp. 1141–1160 (2016). Using this error evaluate function, we can compute [13] S. Thota and S. D. Kumar, A new method for general errors at various points of the interval [1,2]. solution of system of higher-order linear differential equations, International Conference on Inter Disciplinary Research in Engineering and Technology, Vol. 1, pp. 240–243 (2015). 4 Conclusion [14] P. J. Davis, Interpolation and Approximation, Blaisdell, Waltham, MA. (1963). In this paper, we developed an algorithm to construct a polynomial interpolation with a finite set of integral conditions, also discussed an algorithm to compute the polynomial interpolation with Stieltjes conditions as a Srinivasarao Thota quotient of two determinants using integro-differential completed his M.Sc. operators. The symbolic formation of the error involved in Mathematics from Indian due to the approximation of a given function by the Institute of Technology proposed interpolation is discussed. Several examples are Madras, India and Ph.D. presented to illustrate the proposed algorithms. in Mathematics from Motilal Nehru National Institute of Technology Allahabad, India. References Srinivasarao Thota’s area of research interest is Computer [1] A. Chakrabarti and Hamsapriye, Derivation of a general Algebra, precisely symbolic methods for differential mixed interpolation formula, J. Comput. Appl. Math., Vol. equations. Presently Working as Assistant Professor at 70, pp. 161–172 (1996). Department of Applied Mathematics, Adama Science and [2] H. de Meyer, J. Vanthournout and G. Vanden Berghe, On a Technology University, Ethiopia. new type of mixed interpolation, J. Comput. Appl. Math. Vol. 30 pp. 55–69 (1990). [3] J. P. Coleman, Mixed interpolation methods with arbitrary nodes, Journal of Computational and Applied Mathematics, Vol. 92, pp. 69–83 (1998). [4] R. C. Brown and A. M. Krall, Ordinary differential operators under Stieltjes boundary conditions, Trans. American Mathematical Society, Vol. 198, pp. 73–92 (1974). [5] R. A. Lorentz, Multivariate Hermite interpolation by algebraic polynomials: A survey, Journal of Computational and Applied Mathematics, Vol. 122, pp. 167–201 (2000). [6] S. Thota and S. D. Kumar, On a mixed interpolation with integral conditions at arbitrary nodes, Cogent Mathematics, 3:115161 (2016). [7] T. Sauer, Polynomial interpolation, ideals and approximation order of refinable functions, Proc. Amer. Math. Soc., (1997). [8] M. Rosenkranz, G. Regensburger, L. Tec, and B. Buchberger, Symbolic analysis for boundary problems: From rewriting to parametrized Grbner bases. In: U. Langer and P. Paule (eds.), Numerical and Symbolic Scientific Computing: Progress and Prospects, Texts and Monographs in Symbolic Computation. Springer, Wien Heidelberg London New-York (2011).

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