Formula for Hermite multivariate interpolation and partial fraction decomposition

We present a new formula for the Hermite multivariate interpolation problem in the framework of the Chung–Yao approach. By using the respective univariate interpolation formula, we obtain a direct and explicit solution to the classical partial fraction decomposition problem for rational functions, including the real case.

💡 Research Summary

The paper introduces a novel explicit formula for multivariate Hermite interpolation based on the Chung–Yao framework and demonstrates its powerful application to the classical partial‑fraction decomposition of rational functions, including cases with multiple and complex conjugate roots.

The authors begin by recalling the Chung–Yao Lagrange interpolation scheme: given n + k hyperplanes L₁,…,L_{n+k} in ℝᵏ that are in general position, the N = C_{n+k}^k intersection points x_α (α∈I_{n+k}^k) are distinct. For any data {c_α} there exists a unique polynomial p of total degree ≤ n satisfying p(x_α)=c_α, and it can be written in the Lagrange form p(x)=∑_α c_α p*_α(x) with p*α(x)=A_α∏{i∈α}L_i(x).

When the hyperplanes are only admissible (any k of them intersect in a point, but more than k never do), each intersection point x(i) acquires a multiplicity m_i defined as the number of hyperplanes passing through it minus k + 1. The Hermite multivariate interpolation problem then asks for a polynomial p of degree ≤ n whose partial derivatives D^α p(x(i)) match prescribed values for all multi‑indices |α|≤m_i−1. The total number of conditions remains N, guaranteeing a unique solution (Theorem 1.3).

The central contribution is Proposition 1.4, which provides a Lagrange‑Taylor representation of the Hermite interpolant. Define the global vanishing polynomial ϕ(x)=∏{j=1}^{n+k}L_j(x) and, for each point x(i), the local vanishing factor ϕ_i(x)=∏{j∉α(i)}L_j(x). For a sufficiently smooth function f, let T_{f/ϕ_i, x(i), m_i−1}(x) be the multivariate Taylor polynomial of order m_i−1 of the quotient f/ϕ_i at x(i). Then the unique Hermite interpolant p_f is given by

 p_f(x)=∑{i=1}^s ϕ_i(x)·T{f/ϕ_i, x(i), m_i−1}(x).

The proof checks two properties: (1) each term ϕ_i(x) contains at least m_r linear factors that vanish at any other point x(r), so all derivatives up to order m_r−1 vanish there; (2) using the multivariate Leibniz rule, the derivatives at x(i) reproduce the prescribed data because the Taylor polynomial reproduces the derivatives of f/ϕ_i up to the required order. Consequently, the Hermite interpolant can be assembled without solving a large linear system—one simply computes local Taylor polynomials and multiplies by the appropriate vanishing factors.

The authors then apply this formula to the univariate case, obtaining a direct method for partial‑fraction decomposition. For a rational function R(x)=p(x)/q(x) with deg p=m and deg q=n+1, they first separate the polynomial part via Euclidean division, yielding a remainder r of degree ≤ n. When the roots of q are distinct, the classical Lagrange formula reproduces the well‑known decomposition r(x)/q(x)=∑_{i=0}^n p(x_i)/q’(x_i)·1/(x−x_i).

When q has multiple roots, they group the roots as {d₁,…,d_s} with multiplicities {m₁,…,m_s}. Defining ψ_i(x)=∏_{j≠i}(x−t_j) (where the t_j are all roots) and using the grouped Lagrange formula (2.6) together with the Lagrange‑Taylor representation, they derive

 r(x)=∑{i=1}^s q_i(x)·∑{j=0}^{m_i−1} (1/j!)·(r/q_i)^{(j)}(d_i)·(x−d_i)^j,

where q_i(x)=q(x)/(x−d_i)^{m_i}. Dividing by q(x) yields the explicit partial‑fraction expansion

 p(x)/q(x)=s(x)+∑{i=1}^s∑{j=0}^{m_i−1} c_{ij}/(x−d_i)^{j+1},

with coefficients c_{ij}= (1/j!)·(r/q_i)^{(j)}(d_i). No linear system solving is required; the coefficients are obtained by evaluating derivatives of the known remainder r.

For real rational functions, complex roots appear in conjugate pairs. The authors treat this by grouping each pair into a quadratic factor (x²+u_νx+v_ν) and defining η_ν(x)=q(x)/(x²+u_νx+v_ν)^{μ_ν}. Using divided differences, they construct real coefficients E_{νk}, M_{νk}, N_{νk} that appear in the final real partial‑fraction form (2.15):

 p(x)/q(x)=s(x)+∑{ν=1}^s∑{k=0}^{m_ν−1} E_{νk}(x−a_ν)^{m_ν−k-1}
     +∑{ν=1}^σ∑{k=0}^{μ_ν−1} (M_{νk}x+N_{νk})(x²+u_νx+v_ν)^{μ_ν−k-1}.

All coefficients are expressed solely through divided differences of p and the vanishing polynomials ψ_ν, η_ν, guaranteeing they are real.

In summary, the paper delivers a compact, constructive Lagrange‑Taylor formula for multivariate Hermite interpolation and shows how this formula yields a straightforward, explicit algorithm for partial‑fraction decomposition of rational functions with arbitrary root multiplicities, including the fully real case with complex conjugate pairs. The approach eliminates the need for solving large linear systems and provides closed‑form expressions for all coefficients, representing a significant advance in both theoretical interpolation theory and practical symbolic computation.