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New Recursive Divided-Difference Interpolation Method

This paper introduces and rigorously proves a new recursive divided-difference method that directly determines the coefficients of an interpolation polynomial in the monomial basis by sequentially reconstructing the divided-differences table, thereby avoiding the need for explicit Newton expansion or solving Vandermonde systems.

Original authors: A.N. Samijonov, N.S. Mamatov, B.N. Samijonov, N.A. Niyozmatova

Published 2026-10-20

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CC BY 4.0

Original authors: A.N. Samijonov, N.S. Mamatov, B.N. Samijonov, N.A. Niyozmatova

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of mathematics and engineering, there is a constant need to draw a smooth, continuous line that connects a scattered set of data points. Imagine you have measured the height of a plant at different times, or the temperature at various locations, and you want to know what the value is at a moment or place you didn't measure. To do this, scientists use a tool called polynomial interpolation, which creates a single, flexible curve that passes exactly through every known point. For over two hundred years, the standard way to build this curve has been to arrange the data in a specific triangular table of differences, a method that is efficient for adding new points but leaves the final formula in a form that is difficult to use for many practical calculations. Engineers and computer scientists often need the curve written in a standard format, where the influence of each power of the variable is clearly separated, but getting there usually requires a tedious algebraic expansion or solving a massive, complex system of equations that can become unstable and inaccurate as the curve gets more complicated.

A team of researchers from universities in Japan, Uzbekistan, and South Korea has proposed a new way to solve this specific problem. Instead of building the curve in the traditional way and then trying to translate it into the standard format, or forcing the data into a rigid system of equations, they developed a method that peels away the curve layer by layer, directly revealing the numbers needed for the standard format. Their approach, which they call a new recursive divided-difference interpolation method, treats the process of finding the curve's coefficients not as a single construction job, but as a sequence of reductions. The researchers proved mathematically that by looking at a specific column of their difference table, they can identify the value of the highest power in the equation. Once that value is found, they subtract its contribution from the original data points, effectively lowering the complexity of the problem by one degree. They then repeat this process on the remaining data to find the next highest power, and so on, until every number in the final equation is determined.

The paper demonstrates that this method works by showing exactly how the process unfolds with a concrete example. The researchers started with a known curve and a set of five data points. They first built their table of differences and noticed that a specific column contained identical values. This repetition was the key signal that allowed them to identify the coefficient for the highest power of the variable. They then took that number, multiplied it by the appropriate power of the input values, and subtracted the result from the original measurements. This subtraction left behind a new set of values that corresponded to a simpler curve with one less power. They rebuilt their table with these new values, found the next repeating column, identified the next coefficient, and subtracted again. By the time they reached the end of the sequence, they had successfully reconstructed the entire equation, including the constant term, without ever having to expand the original formula or solve a large system of linear equations.

The authors emphasize that their method is distinct from the traditional approaches that rely on expanding the polynomial or solving a Vandermonde system, which is a specific type of matrix equation known to become numerically unstable for high-degree curves. Their work provides a rigorous mathematical proof that this step-by-step reduction is valid, showing that the constant value found in a column of differences is not just a property of the curve, but is actually the exact coefficient needed for the standard form. The researchers tested their algorithm with a polynomial of degree three and confirmed that the numbers they derived matched the exact coefficients of the original curve. While the paper does not yet provide a full comparison of how fast this method is compared to existing ones, it establishes a new theoretical pathway that allows for the direct determination of polynomial coefficients through recursive data reduction. This offers a fresh computational perspective, turning the divided-difference table from a tool used only for building the curve into a mechanism for deconstructing it directly into its fundamental parts.

Technical Summary: New Recursive Divided-Difference Interpolation Method

Problem Statement
Polynomial interpolation is a fundamental tool in numerical analysis, yet a persistent computational challenge exists in determining the coefficients of an interpolation polynomial in the standard monomial basis ( $P_n(x) = a_n x^n + \dots + a_0$ ). While the Newton form of the interpolation polynomial is computationally efficient for construction using divided differences, converting this form to the monomial basis typically requires one of two inefficient strategies:

Explicit Expansion: Algebraically expanding the Newton basis polynomials and collecting like terms, which increases arithmetic operations and complexity as the degree rises.
Vandermonde Systems: Solving a linear system $Va = y$ , which is known to be numerically unstable for high-degree polynomials due to the exponential growth of the condition number of the Vandermonde matrix.

Existing recursive algorithms, such as the classical Newton method, determine coefficients only for the Newton basis and terminate once those coefficients are found. They do not utilize the recursive structure of divided differences to directly yield monomial coefficients without intermediate algebraic expansion or linear system solving.

Methodology
The authors propose a new recursive algorithm that determines monomial coefficients directly from interpolation data by iteratively reconstructing the divided-difference table. The method operates on the following principles:

Recursive Divided-Difference Table: The authors define a recursive structure where all elements of a column in the divided-difference table are retained, rather than just the first element used in classical Newton interpolation.
Coefficient Identification via Invariance: The core theoretical insight is that when a column of recursively computed divided differences yields a constant value across all its elements (i.e., $\Delta^{(k)}_1 = \Delta^{(k)}_2 = \dots = \Delta^{(k)}_{n-k+1}$ ), this constant value corresponds exactly to the coefficient of the highest-degree term ( $x^k$ ) in the monomial basis.
Iterative Reduction: Once the highest-degree coefficient ( $a_k$ ) is identified, the algorithm subtracts the contribution of the term $a_k x^k$ from the original interpolation values ( $y_i$ ). This generates a new set of zeroth-order divided differences corresponding to a polynomial of degree $k-1$ .
Reconstruction: A new divided-difference table is constructed from these updated values, and the process repeats to identify the next highest coefficient ( $a_{k-1}$ ). This continues sequentially down to the constant term ( $a_0$ ).

Key Contributions

Direct Monomial Determination: The paper introduces a method to compute monomial coefficients without explicitly expanding the Newton polynomial or solving a Vandermonde system.
Recursive Reduction Strategy: The authors develop a strategy where the interpolation problem is transformed into a sequence of lower-degree interpolation problems by subtracting the contribution of the determined highest-degree term at each step.
Theoretical Justification: The method is supported by rigorous mathematical proofs, including:
- Lemma 3.1: Establishes the representation of the interpolation polynomial in terms of the proposed recursive divided differences.
- Theorem 3.2: Proves the necessary and sufficient condition that a constant column in the recursive divided-difference table corresponds to the specific monomial coefficient.
Algorithmic Implementation: A step-by-step iterative algorithm (Algorithm 3.1) is provided, detailing the reconstruction of the table and the update of zeroth-order differences to isolate coefficients from $a_n$ down to $a_0$ .

Results
The paper validates the proposed method through numerical experiments:

Correctness: A numerical example using a cubic polynomial ( $y = 2x^3 + 3x^2 + 5x + 7$ ) demonstrates the step-by-step recursive reconstruction. The algorithm successfully identified the coefficients $a_3=2$ , $a_2=3$ , $a_1=5$ , and $a_0=7$ by observing the constant values in the divided-difference columns after each reduction step.
Accuracy: The reconstructed coefficients matched the exact coefficients of the polynomial, confirming the theoretical validity of the approach.
Efficiency: The method demonstrated that coefficients can be determined directly from interpolation data, avoiding the intermediate algebraic expansion required by traditional conversion methods.

Significance and Claims
The authors position this work as a novel computational interpretation of divided differences. They claim that the method establishes a direct recursive link between divided differences and monomial coefficients, fundamentally altering the role of divided differences from merely constructing Newton polynomials to serving as intermediate quantities for reconstructing monomial coefficients.

The paper explicitly states that it does not provide a comparative analysis of computational complexity or numerical stability against existing methods (such as Vandermonde solvers or basis-conversion algorithms) within this study. Instead, it presents the method as a new alternative recursive computational scheme. The authors note that future research will address the computational complexity, numerical stability, and efficiency of the algorithm across various node distributions and compare it with existing methods. The primary contribution is the theoretical framework and the demonstration that monomial coefficients can be derived recursively through data reduction without auxiliary linear algebra or polynomial expansion.

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