Orthogonality-Preserving Cubic Spline Formulation Using Mamadu-Njoseh Polynomials

Ebimene James Mamadu; Jude Chukwuyem Nwankwo; Omoregbe Charles Osahon; Vincent Odiaka; Ebikonbo-Owei Anthony Mamadu; Henrietta Ify Ojarikre; Jonathan Tsetimi; Nkonyeasua Ignatius Njoseh; Ewoma Justice Aluya

doi:10.25259/JQUS_11_2025

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10.25259/JQUS_11_2025

Orthogonality-Preserving Cubic Spline Formulation Using Mamadu-Njoseh Polynomials

Ebimene James Mamadu^1,*, Jude Chukwuyem Nwankwo², Omoregbe Charles Osahon³, Vincent Odiaka⁴, Ebikonbo-Owei Anthony Mamadu^1,, Henrietta Ify Ojarikre¹, Jonathan Tsetimi¹, Nkonyeasua Ignatius Njoseh¹, Ewoma Justice Aluya¹

1Department of Mathematics, Delta State University, Abraka, 330106, Nigeria

2Department of Mathematics, University of Delta, Agbor, 331105, Nigeria

3Department of Mathematics, University of Port Harcourt, Port Harcourt, 500101, Rivers State, Nigeria

4Department of Computer Science, Nottingham Trent University, 50 Shakespeare Street, Nottingham, NG1 4FQ, United Kingdom

* Corresponding author: Dr. Ebimene James Mamadu Department of Mathematics, Delta State University, Abraka, 330106, Nigeria. mamaduebimene@gmail.com

Received: 2025-08-28, Accepted: 2025-11-12, Epub ahead of print: 2026-02-03,

Licence

This is an open-access article distributed under the terms of the Creative Commons Attribution-Non Commercial-Share Alike 4.0 License, which allows others to remix, transform, and build upon the work non-commercially, as long as the author is credited and the new creations are licensed under the identical terms.

How to cite this article: Mamadu, EJ, Nwankwo JC, Osahon OC, Odiaka V, Mamadu EOA, Ojarikre HI, et al. Orthogonality-preserving cubic spline formulation using Mamadu-Njoseh polynomials. J Qassim Univ Sci. doi: 10.25259/JQUS_11_2025

Abstract

Objectives

This study aims to develop an orthogonality-preserving cubic spline formulation using Mamadu–Njoseh polynomials as basis functions, with the objective of improving numerical stability, computational efficiency, and accuracy in spline-based interpolation.

Material and Methods

The proposed approach reformulates classical cubic spline segments by expressing them in terms of Mamadu–Njoseh polynomial bases. The formulation explicitly incorporates the orthogonality and weighted integration properties of these polynomials while maintaining standard interpolation constraints and boundary conditions. Analytical construction and theoretical considerations are employed to ensure consistency with the fundamental behavior of classical cubic splines.

Results

The resulting spline formulation successfully preserves orthogonality within each spline segment and demonstrates enhanced numerical stability and computational efficiency compared to standard cubic spline methods. The use of Mamadu–Njoseh polynomials improves accuracy while retaining the smoothness and continuity properties characteristic of classical cubic splines.

Conclusion

The orthogonality-preserving cubic spline formulation presented in this paper offers a robust and reliable framework for scientific computing and engineering applications. By combining the smoothness of classical cubic splines with the structured advantages of Mamadu–Njoseh orthogonal polynomial bases, the method provides an effective alternative for accurate and stable interpolation.

Keywords

Cubic spline

Interpolation

Mamadu-Njoseh polynomials

Natural spline

Orthogonality

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Introduction

Spline theory, as we know it mathematically today, originated from the work of Isaac Jacob Schoenberg in the 1940s to 1960s. These foundational works, defining splines as spaces of rigorously defined functions and introducing the concepts of B-splines and cardinal splines, are still the bases for research in this area. The draftsman defines the spline model—its equations, structure, and theoretical properties. These craftsmen would bend long, thin, and flexible strips of wood or metal into smooth curves between a series of points when laying out the surfaces for cars, airplanes, and ships. The strip would only bend smoothly at the anchor point, avoiding sharp angles and strain. This mechanical exercise was the intuitive precursor to what mathematicians would eventually identify as the natural cubic spline, a curve that passes smoothly through certain specified data points and which does not “wiggle” unnecessarily.^[1,2]

Schoenberg’s contribution was to provide a mathematical theory for this convenient device. He gave an interpretation of splines as piecewise polynomials that are properly smoothed at the knots, the points where the segments meet. Thereby, splines represent a mathematically elegant, computationally efficient, and well-structured finite-dimensional function space. Perhaps his most significant contribution was the introduction of B-splines, or basis splines, which formed a stable, local method of creating all splines. B-splines only impact the curve in a small area and form a partition of unity, so they combine nicely without inducing instability. The regular, shift-invariant property of splines became even clearer when the knots were placed at regular intervals and resulted in the so-called cardinal splines, a concept that eventually found applications in approximation theory, signal processing, and numerical analysis.^[2]

Splines have also become essential in geometric modeling and computer-aided design. They are perfect for designing curves and surfaces across a variety of applications, from consumer goods to airplane wings and car bodies, thanks to their stability, local control, and smoothness qualities. The flexibility of spline-based modeling was significantly increased with the extension to non-uniform rational B-splines (NURBS), enabling designers to depict precise circles, ellipses, and other conic sections. Almost all contemporary computer-aided design/computer-aided manufacturing (CAD/CAM) systems now include these built-in tools. Splines are used in image processing, computer graphics, and the numerical solution of partial differential equations because they can be extended into multiple dimensions to produce smooth surfaces and volumes outside of design.^[3-5]

The cubic spline is one of the most relevant and popular forms of spline functions, both from a theoretical standpoint and a practical point of view. It dates back to the drafting habits of engineers who used flexible wooden or metallic strips to smoothly connect certain points. This analogy in real life helps us to understand the cubic spline more intuitively: It is the smoothest curve that can be drawn freely passing through given points and does not oscillate too much in the other dimensions while maintaining balance and continuity in the entire interval. It is this property of cubic functions that helps us understand why the spline is formed of cubic polynomials. Cubic polynomials are the simplest polynomials that can satisfy the conditions of continuity and the smoothness of the curve and its first two derivatives.^[4] A cubic spline’s smoothness is its distinguishing characteristic. The cubic spline finds a balance between piecewise linear interpolation, which produces sharp corners at the data points, and higher-degree polynomial interpolation, which can oscillate excessively when many points are involved. Over the whole set of intervals, it preserves the curve’s continuity, including its slope and curvature. This results in a curve that is both mathematically stable and visually appealing. To make the curve appear to gently extend beyond the data range, the natural cubic spline specifically requires that the curve have no artificial bending at the endpoints. For applications where boundary behavior needs to be tightly controlled, this makes it especially appropriate.^[5-7]

Cubic spline interpolations are piecewise cubic polynomials with $C^{2}$ continuity. Once boundary conditions (natural or periodic) are selected, the solution is unique. They interpolate a given set of data points while ensuring continuity of the function and its first and second derivatives across all intervals. Mathematically, each spline on the subinterval $(x_{i}, x_{i + 1})$ belongs to the polynomial space $φ_{3} (x)$ , The vector space of all real polynomials of degree at most 3.^[8] The power of cubic splines lies in their combination of accuracy, stability, and efficiency. They offer an outstanding interpolation tool, typically producing results that are far better than those derived from fitting a single high-degree polynomial to every data point. The systems of equations needed to calculate cubic spline coefficients, which frequently involve basic tridiagonal matrices, are organized and easy to solve. Cubic splines are now a common method in numerical analysis due to their computational ease.^[9] Cubic splines are essential for smoothing noisy and irregular data, in addition to interpolation. Forcing the curve to go through each data point precisely is frequently undesirable in these situations.^[10] In the cubic spline framework, smoothing splines instead achieves a balance between preserving overall smoothness and fitting the data. This concept is widely used in machine learning, signal processing, and statistics to identify patterns while removing chance fluctuations.^[11-14]

Njoseh and Mamadu introduced the Mamadu-Njoseh polynomials as a family of orthogonal polynomials developed to provide a structured framework for solving problems in applied mathematics.^[15-17] Since their introduction, these polynomials have been widely recognized for their usefulness as basis functions in a range of numerical methods that address real-life challenges in the physical sciences and engineering. Their distinctive feature lies in their orthogonality and weighted integration properties, which ensure stability, accuracy, and consistency in computational schemes. At the same time, cubic splines have long been regarded as a classical and versatile tool for interpolation and approximation, particularly valued for their ability to produce smooth curves that preserve continuity, balance, and natural boundary conditions.

MATERIAL & METHODS

In this paper, we extend these two powerful ideas by re-expressing the cubic spline segments in a form that inherits the structural advantages of the Mamadu-Njoseh polynomials.^[18-20] This reformulation does not modify the essential behavior of the cubic spline when compared with established results in the literature, particularly those developed under similar interpolation conditions and boundary constraints. Instead, it strengthens the spline framework by embedding within it the orthogonality and stability properties of the Mamadu-Njoseh polynomial basis.^[21,22] This integration is significant because it provides a spline formulation that achieves improved conditioning, enhanced numerical stability, and more efficient computation while retaining the simplicity and interpretability of classical cubic splines. The resulting method preserves the traditional smoothness and elegance of cubic splines while delivering higher accuracy and robustness in practical computations. By combining the natural strengths of cubic splines with the robust structure of the Mamadu-Njoseh polynomials, the approach presented here offers a consistent and reliable tool for practical applications and opens a pathway for more advanced developments in scientific computing and engineering analysis.

Preliminaries

Spline function

A function $Y (x)$ of degree $n$ with $x_{i}, i = 0 (1) n,$ is called a spline function if it satisfies the following properties:^{^[1^,3^]}

i.

$Y (x) = y_{i}, i = 0 (1) n .$
ii.

On each subinterval $[x_{i}, x_{i + 1}], 1 \leq i \leq n$ , $Y (x)$ is a polynomial in $x$ of degree at most $n .$
iii.

$Y (x)$ and its $(n - 1)$ derivatives are continuous on $(a, b) .$
iv.

$Y (x)$ is a polynomial of degree one for $x < a$ and $x > b .$

Cubic Spline Interpolation

A cubic spline satisfies the following properties:^{^[13^]}

i.

$Y (x) = y_{i}, i = 0 (1) n .$
ii.

On each subinterval $[x_{i}, x_{i + 1}], 1 \leq i \leq n$ , $Y (x)$ is a third-degree polynomial in $x .$
iii.

$Y (x), Y^{'} (x), and Y^{″} (x)$ are continuous on $(a, b) .$

Mamadu-Njoseh Polynomials

These are orthogonal polynomials with interval of orthogonality $(- 1, 1)$ with weight $w (x) = 1 + x^{2},$ satisfying the following properties:^{^[16^]}

i.

$φ_{q} (x) = \sum_{k = 0}^{n} a_{0} x^{i}, a_{0} \neq 0.$
ii.

$(φ_{p} (x), φ_{q} (x)) = 0, p \neq q .$
iii.

$φ_{q} (1) = 1.$

Cubic Spline Interpolation with Mamadu-Njoseh Basis

We use Mamadu-Njoseh polynomials on the interval $[- 1, 1$ ] with weight function $w (x) = 1 + x^{2},$ and normalization $φ_{n (1)} = 1.$ The first four (degree $0, \dots, 3$ ) are given as,

http://www.w3.org/1998/Math/MathML

(1)

\begin{matrix} φ_{0} (x) = 1, φ_{1} (x) = x, φ_{2} (x) = \frac{5 x^{2} - 2}{3}, \\ and φ_{3} (x) = \frac{14 x^{3} - 9 x}{5} . \end{matrix}

Table 1 below shows the results for the computation of $φ_{k} (\pm 1), φ_{k}^{'} (ξ)$ and evaluation of $φ_{k}^{'} (\pm 1),$ then $φ'_{k}^{'} (ξ)$ and evaluation of $φ'_{k}^{'} (\pm 1) .$

Table 1 Endpoint and derivative values of the Mamadu-Njoseh Cubics at

ξ = \pm 1

$k$	$φ_{k} (ξ)$	$φ_{k} (1)$	$φ_{k} (- 1)$	${φ^{'}}_{k} (ξ)$	${φ^{'}}_{k} (\pm 1)$	${φ^{″}}_{k} (ξ)$	${φ^{″}}_{k} (\pm 1)$
$0$	$1$	$1$	$1$	$0$	$0$	$0$	$0$
$1$	$ξ$	$1$	$- 1$	$1$	$1$	$0$	$0$
$2$	$\frac{1}{3} (5 ξ^{2} - 2)$	$1$	$1$	$\frac{10 ξ}{3}$	$\frac{10}{3}, - \frac{10}{3}$	$\frac{10}{3}$	$\frac{10}{3}, \frac{10}{3}$
$3$	$\frac{1}{5} (14 ξ^{3} - 9 ξ)$	$1$	$- 1$	$\frac{1}{5} (42 ξ^{2} - 9)$	$\frac{33}{5}, \frac{33}{5}$	$\frac{84 ξ}{5}$	$\frac{84}{5}, - \frac{84}{5}$

These simple endpoint values make it convenient to impose interpolation and derivative conditions.

Now, given the nodes $x_{i} < x_{i + 1}$ with function values $y_{i}, y_{i + 1},$ we define the map

http://www.w3.org/1998/Math/MathML

(2)

ξ = \frac{2 (x - x_{i})}{h_{i}} - 1, h_{i} = x_{i + 1} - x_{i},

so that $x = x_{i} \leftrightarrow ξ = - 1, x_{i + 1} \leftrightarrow ξ = 1.$

Representing the local cubic on $(x_{i}, x_{i + 1})$ as:

http://www.w3.org/1998/Math/MathML

(3)

s_{i} (x) = \sum_{k = 0}^{3} c_{i, k} φ_{k} (ξ),

with the Mamadu-Njoseh basis given by (1), and $c_{i, k}^{'} s$ are constant parameters to be estimated. Thus, the interpolation constraints at $x = x_{i} and x = x_{i + 1}$ are

http://www.w3.org/1998/Math/MathML

(4)

s_{i} (x_{i}) = c_{0} - c_{1} + c_{2} - c_{3} = y_{0} .

http://www.w3.org/1998/Math/MathML

(5)

s_{i} (x_{i + 1}) = c_{0} + c_{1} + c_{2} + c_{3} = y_{1} .

Now, the derivative transform is,

http://www.w3.org/1998/Math/MathML

(6)

\frac{d}{d x} = \frac{d ξ}{d x} \frac{d}{d ξ} = \frac{2}{h_{i}} \frac{d}{d ξ},

so that at $ξ = \pm 1$ ,

http://www.w3.org/1998/Math/MathML

(7)

s_{i} (x_{i}) = \frac{2}{h_{i}} (c_{1} - \frac{10}{3} c_{2} + \frac{33}{5} c_{3}) = d_{i},

http://www.w3.org/1998/Math/MathML

(8)

s_{i} (x_{i + 1}) = \frac{2}{h_{i}} (c_{1} + \frac{10}{3} c_{2} + \frac{33}{5} c_{3}) = d_{i + 1},

where $d_{i}$ denotes the first derivative at the node $x_{i} .$ The $d_{i}$ are known for a natural $C^{2}$ spline.

Solving the linear system $(4, 5, 7, 8)$ for $c_{i}, i = 0 (1) 3,$ we have,

http://www.w3.org/1998/Math/MathML

(9)

(\begin{matrix} c_{0} = \frac{3}{40} d_{i} h_{i} - \frac{3}{40} d_{i + 1} h_{i} - \frac{1}{2} (y_{i} + y_{i + 1}), \\ c_{1} = - \frac{5}{112} d_{i} h_{i} - \frac{5}{112} d_{i + 1} h_{i} - \frac{33}{56} y_{i} + \frac{33}{56} y_{i + 1}, \\ c_{2} = - \frac{3}{40} d_{i} h_{i} + \frac{3}{40} d_{i + 1} h_{i}, \\ c_{3} = \frac{5}{112} d_{i} h_{i} + \frac{5}{112} d_{i + 1} h_{i} + \frac{5}{56} y_{i} - \frac{5}{56} y_{i + 1} . \end{matrix})

If $d_{i}, d_{i + 1}$ are known, the local coefficients are immediate. Hence, the local cubic is

http://www.w3.org/1998/Math/MathML

(10)

s_{i} (x) = y_{i} φ_{0} (ξ) + y_{i + 1} φ_{1} (ξ) + y_{i + 2} φ_{2} (ξ) + y_{i + 3} φ_{3} (ξ),

where,

http://www.w3.org/1998/Math/MathML

\begin{matrix} φ_{0} (ξ) = 1, φ_{1} (ξ) = ξ, φ_{2} (ξ) = \frac{5 ξ^{2} - 2}{3} and \\ φ_{3} (ξ) = \frac{14 ξ^{3} - 9 ξ}{5} \end{matrix}

To enforce global $C^{2},$ we compute the second derivative at the right end of the interval (i.e, at $x_{i + 1}),$ using

http://www.w3.org/1998/Math/MathML

(11)

{s^{″}}_{i} (x) = {(\frac{2}{h_{i}})}^{2} \sum_{k = 0}^{3} c_{i, k} {φ^{″}}_{k} (ξ) .

Substituting the above c-expressions (9) into (11) and simplifying further to obtain the following relations,

http://www.w3.org/1998/Math/MathML

(12)

{s^{″}}_{i} (x_{i + 1}^{-}) = \frac{2}{h_{i}} d_{i} + \frac{4}{h_{i}} d_{i + 1} + \frac{6}{h_{i}^{2}} (y_{i} + y_{i + 1}),

http://www.w3.org/1998/Math/MathML

(13)

{s^{″}}_{i} (x_{i + 1}^{+}) = - \frac{4}{h_{i + 1}} d_{i + 1} - \frac{2}{h_{i + 1}} d_{i + 2} - \frac{6}{h_{i + 1}^{2}} (y_{i + 1} - y_{i + 2}) .

Setting (12) and (13) equal (continuity at $x_{i + 1})$ and rearranging gives for interior nodes $j$ (putting $j = i + 1),$ we obtain the tridiagonal equation:

http://www.w3.org/1998/Math/MathML

(14)

\begin{matrix} \frac{2}{h_{j - 1}} d_{j - 1} + 4 (\frac{1}{h_{j - 1}} + \frac{1}{h_{j}} d_{j}) d_{j} + \frac{2}{h_{j}} d_{j + 1} \\ = - 6 (\frac{y_{j - 1} - y_{j}}{h_{j - 1}^{2}} + \frac{y_{j} - y_{j + 1}}{h_{j}^{2}}) . \end{matrix}

This holds for $j = 1, \dots, n - 1.$ Now, we set the endpoint second derivatives to zero, i.e.,

At the left endpoint $x_{0} : {s^{″}}_{0} (x_{0}) = 0,$

http://www.w3.org/1998/Math/MathML

(15)

\frac{4}{h_{0}} d_{0} + \frac{2}{h_{0}} d_{1} = - \frac{6}{h_{0}^{2}} (y_{0} - y_{1}) .

At the right endpoint $x_{n} : {s^{″}}_{n - 1} (x_{n}) = 0,$

http://www.w3.org/1998/Math/MathML

(16)

\frac{2}{h_{n - 1}} d_{n - 1} + \frac{4}{h_{n - 1}} d_{n} = - \frac{6}{h_{n - 1}^{2}} (y_{n - 1} + y_{n}) .

Resolving the interior tridiagonal equation (14) together with (15) and (16) yields a $(n + 1) \times (n + 1)$ tridiagonal linear system for the derivative vector $d = {(d_{0}, \dots, d_{n})}^{T} .$ In other words, the resulting $(n + 1) \times (n + 1)$ tridiagonal linear system can be written in the form,

http://www.w3.org/1998/Math/MathML

(17)

A d = B

where,

http://www.w3.org/1998/Math/MathML

\begin{matrix} A = (\begin{matrix} \frac{4}{h_{0}} & \frac{2}{h_{0}} & 0 & \dots & 0 & 0 \\ \frac{2}{h_{0}} & \frac{4}{h_{0}} + \frac{4}{h_{1}} & \frac{2}{h_{1}} & ⋱ & ⋮ \\ 0 & \frac{2}{h_{1}} & \frac{4}{h_{1}} + \frac{4}{h_{2}} & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & ⋱ & ⋱ & \frac{2}{h_{n - 2}} & 0 \\ 0 & ⋱ & \frac{2}{h_{n - 2}} & \frac{4}{h_{n - 2}} + \frac{4}{h_{n - 1}} & \frac{2}{h_{n - 1}} \\ 0 & \dots & 0 & 0 & \frac{2}{h_{n - 2}} & \frac{4}{h_{n - 1}} \end{matrix}), \\ d = {(d_{0}, \dots, d_{n})}^{T}, \end{matrix}

http://www.w3.org/1998/Math/MathML

B = (\begin{matrix} - \frac{6}{h_{0}^{2}} (y_{0} - y_{1}) \\ - 6 (\frac{y_{0} - y_{1}}{h_{0}^{2}} + \frac{y_{1} - y_{2}}{h_{1}^{2}}) \\ ⋮ \\ - 6 (\frac{y_{j - 1} - y_{j}}{h_{j - 1}^{2}} + \frac{y_{j} - y_{j + 1}}{h_{j}^{2}}) \\ ⋮ \\ - \frac{6}{h_{n - 1}^{2}} (y_{n - 1} - y_{n}) \end{matrix}), j = 1, \dots, n - 1.

Solving (17) using the Thomas algorithm^[14] to get all $d_{j} .$ Then compute the local coefficients $c_{i, 0 \dots 3},$ for each interval. Finally, evaluate (3) with $ξ = \frac{2 (x - x_{i})}{h_{i}} - 1,$ for the Mamadu-Njoseh cubic spline interpolating polynomial.

Properties of the Mamadu-Njoseh cubic spline interpolation

i.

The local cubic spline splits into even/odd parts in $ξ,$ which is convenient for analysis and conditioning.
ii.

The $(φ_{0}, \dots, φ_{3})$ span all cubics in $ξ,$ the Mamadu-Njoseh spline reproduces any cubic polynomial in $x$ exactly.
iii.

If $y \mapsto α y + β,$ then the coefficients transform linearly, i.e, $c_{i},_{k} \mapsto α c_{i},_{k} + β {()}_{k}_{= 0}$ . Thus, the construction is stable under data scaling and shifting.
iv.

Each $s_{i}$ depends only on $(y_{i}, y_{i + 1}, d_{i}, d_{i + 1})$ and $h_{i} .$ No other intervals appear in the local coefficient formula.
v.

The standard tridiagonal system for node derivatives $d_{j},$ the assembled spline is $C^{2} -$ continuous in $(x_{0}, x_{n}) .$
vi.

Boundary conditions: $s {(x}_{0})= s^{″} (x_{n}) = 0$ .

General algorithm

1.

Compute $h_{j} = x_{j + 1} - x_{j}$ for all intervals.
2.

Formulate $A$ and $r$ as in (17).
3.

Solve $A d = r$ for all $d_{j}$ .
4.

For each interval compute $c_{i},_{0 \dots 3}$ using (9).
5.

Evaluate $s_{i} (x) = \sum_{k = 0}^{3} c_{i, k} φ_{k} (ξ)$ with $ξ = \frac{2 (x - x_{i})}{h_{i}} - 1$ .

Numerical Illustrations

In this section, we present numerical examples of comparing the proposed method with the standard natural cubic spline to test its accuracy and effectiveness.

Example 1. Consider the following data:^[14]

$x$	0	1	2	3
$y$	2	-6	-8	2

Using the proposed methodology, cubic splines in different intervals are tabulated as shown in Table 2.

Table 2 Mamadu-Njoseh versus standard natural cubic splines.

Interval	(c₀, c₁, c₂, c₃)	Mamadu-Njoseh cubic spline	Standard natural cubic spline [14]
$(0, 1)$	$(- \frac{109}{50}, - \frac{11}{28}, \frac{9}{50}, \frac{1}{28})$	$\frac{4}{5} x^{3} - \frac{44}{5} x + 2$	$0.8 x^{3} - 8.8 x + 2$
$(1, 2)$	$(- \frac{781}{100}, - \frac{61}{56}, \frac{81}{100}, \frac{5}{56})$	$2 x^{3} - \frac{18}{5} x^{2} - \frac{26}{5} x + \frac{4}{5}$	$2 x^{3} - 3.6 x^{2} - 5.2 x + 0.8$
$(2, 3)$	$(- \frac{363}{100}, \frac{41}{8}, \frac{63}{100}, - \frac{1}{8})$	$- \frac{14}{5} x^{3} + \frac{126}{5} x^{2} - \frac{314}{5} x + \frac{196}{5}$	$- 2.8 x^{3} + 25.2 x^{2} - 62.8 x + 39.2$

Example 2. Consider the cubic spline for every subinterval from the given data:^[14]

$x$	0	1	2	3
$y$	1	2	33	244

Based on the proposed methodology, the results are presented as shown in Table 3.

Table 3: Mamadu-Njoseh versus standard natural cubic splines.

Interval	(c₀, c₁, c₂, c₃)	Mamadu-Njoseh cubic spline	Standard natural cubic spline [14]
$(0, 1)$	$(\frac{12}{5}, \frac{19}{28}, - \frac{9}{10}, - \frac{5}{28})$	$- 4 x^{3} + 5 x + 1$	$- 4 x^{3} + 5 x + 1$
$(1, 2)$	$(\frac{161}{20}, \frac{743}{56}, \frac{189}{20}, \frac{125}{56})$	$50 x^{3} - 162 x^{2} + 167 x - 53$	$50 x^{3} - 162 x^{2} + 167 x - 53$
$(2, 3)$	$(\frac{2563}{20}, \frac{6023}{56}, \frac{207}{20}, - \frac{115}{56})$	$- 46 x^{3} + 414 x^{2} - 985 x + 715$	$- 46 x^{3} + 414 x^{2} - 985 x + 715$

RESULTS AND DISCUSSION

The results and comparisons presented in Tables 2 and 3 clearly show the relationship between the proposed Mamadu-Njoseh spline and the reported standard natural cubic spline.^[14] For each subinterval, both approaches generate spline functions that interpolate the same data points under identical boundary conditions. As seen in Tables 2 and 3, the functional forms of the spline pieces produced by the proposed method match those obtained using the standard natural cubic spline in,^[14] confirming that the interpolation accuracy is fully preserved.

The key distinction appears in the coefficient sets ( $c_{0}, c_{1}, c_{2}, c_{3}$ ). In the standard natural spline,^[14] the spline is expressed in the monomial basis, which is well known to exhibit numerical sensitivity due to basis non-orthogonality. In contrast, the proposed method represents each cubic segment in the Mamadu-Njoseh orthogonal polynomial basis defined on $(- 1, 1)$ with the weight $w (x) = 1 + x^{2}$ . This orthogonality significantly improves the conditioning of the linear system that determines the coefficients. Consequently, the coefficients shown in Tables 2 and 3 are more stable and less prone to numerical oscillation than those generated using the monomial basis.^[14]

This difference highlights the superiority of the proposed method: although both approaches produce identical interpolants, the Mamadu-Njoseh formulation achieves this through a numerically stable and better-conditioned computational process. This ensures improved robustness, particularly as data size increases or the function exhibits steep gradients.

Figures 1 and 2 further illustrated this comparison. In both examples, the curves generated by the proposed method coincide precisely with those from the standard natural spline.^[14] This confirms that the classical method’s interpolation accuracy is retained. The significance of these figures lies not in visual differences, which do not appear, but in demonstrating that the new method’s enhanced numerical stability does not compromise interpolation quality. Thus, while the graphical outputs remain identical, the underlying construction of the proposed method is superior in stability and coefficient behavior.

Mamadu-Njoseh cubic spline versus standard natural cubic spline for example 1.

Mamadu-Njoseh Cubic Spline Versus Standard Natural Cubic Spline for Example 2.

Hence, the comparisons show that the proposed Mamadu-Njoseh cubic spline matches the standard spline in accuracy, agrees with the benchmark results,^[14] but offers important advantages in stability, conditioning, and numerical reliability.

Conclusion

This paper has shown that the choice of basis functions, whether the standard monomial basis or the Mamadu-Njoseh polynomial basis, does not change the fundamental cubic spline interpolant when identical boundary conditions and interpolation constraints are applied. The core spline curve remains the same. The distinction lies in the mathematical formulation and numerical construction of the spline pieces. The Mamadu-Njoseh approach provides a more stable and better-conditioned computational framework due to its underlying orthogonality and weighted structure, while the standard method relies on monomials, which may be less stable in certain numerical environments.

In practical applications, the choice between the standard natural cubic spline and the Mamadu-Njoseh spline formulation should therefore be guided by the intended use case. For situations requiring enhanced numerical stability, improved conditioning, and reliable performance under large or sensitive datasets, the Mamadu-Njoseh formulation offers clear advantages. The standard natural spline remains adequate for routine interpolation tasks where numerical sensitivity is minimal.

Financial support and sponsorship

Nil

Conflicts of interest

There are no conflicts of interest.

Use of artificial intelligence (AI)-assisted technology for manuscript preparation

The authors confirm that there was no use of Artificial Intelligence (AI)-Assisted Technology for assisting in the writing or editing of the manuscript, and no images were manipulated using AI.

References

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Introduction

MATERIAL & METHODS

Preliminaries

RESULTS AND DISCUSSION

Conclusion

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[1] Schoenberg IJ. Contributions to the problem of approximation of equidistant data by analytic functions. In: I.J. Schoenberg selected papers I.J. Schoenberg selected papers. Birkhäuser Boston; 1988. p. :464-93.
[Google Scholar]

[2] Schoenberg IJ. Spline interpolation and the higher derivatives. Proc Natl Acad Sci USA. 1964;51:24-8.
[CrossRef] [PubMed] [Google Scholar]

[3] Schoenberg IJ. Cardinal spline interpolation, regional conference series in applied mathematics. Philadelphia, US: SIAM; 1973.

[4] de Boor C. A practical guide to splines. New York: Springer-Verlag; 1978.

[5] Schumaker LL. Spline functions: Basic theory (3rd ed.). Cambridge, UK: Cambridge Univ. Press; 2007.

[6] Farin G. Curves and surfaces for CAGD: A practical guide (5th ed.). San Diego, CA: Morgan Kaufmann; 2002.

[7] Powell MJD. Approximation theory and methods. Cambridge, UK: Cambridge Univ. Press; 1981.

[8] HO K, Ahlberg JH, Nilson EN, Walsh JL. The theory of splines and their applications. New York, US: Academic press Inc.; 1967.

[9] Zhang WJ, Zhang F, Zhao JH. Research on modification algorithm of cubic B-spline curve interpolation technology. AMM. 2014;687-691:1596-9.
[CrossRef] [Google Scholar]

[10] Xu W. Elements of Bi-cubic polynomial natural spline interpolation for scattered data: Boundary conditions meet partition of unity technique. Stat, Optim Inf Comput. 2020;8:994-1010.
[CrossRef] [Google Scholar]

[11] Gupta S, Sharma DK, Ranta S. A new hybrid image enlargement method using singular value decomposition and cubic spline interpolation. Multimed Tools Appl. 2022;81:4241-54.
[CrossRef] [Google Scholar]

[12] Zhou Y, Mu F, Hu J. Adaptive state updating particle filter tracking algorithm based on cubic spline interpolation. Proc. IEEE EIECS; 2021. Available from: https://doi.org/10.1109/eiecs53707.2021.9588085 [last accessed on 2025 August 20].

[13] Viswanathan S, Pravin C, Arasamudi RB, Pavithran P. Influence of interface trap distributions over the device characteristics of AlGaN/GaN/AlInN MOS‐HEMT using cubic spline Interpolation technique. Int J Numerical Modelling. 2022;35 Available from: https://onlinelibrary.wiley.com/doi/epdf/10.1002/jnm.2936 [last accessed on 2025 August 20]
[Google Scholar]

[14] Cai ZW, Zheng W, Bao YR, Chu HL, Wu M. Error controlled cubic spline interpolation in CNC technology, arXiv preprint, arXiv:2411.03935, 2024. Available from: https://arxiv.org/abs/2411.03935 [last accessed on 2025 August 20].

[15] Njoseh IN, Mamadu EJ. Numerical solutions of fifth order boundary value problems using Mamadu-Njoseh polynomials. Sci World J. 2016;11:21-24.
[Google Scholar]

[16] Mamadu EJ, Ojarikre HI. Gauss-Mamadu-Njoseh quadrature formula for numerical integral interpolation. JAMCS. 2023;38:128-34.
[CrossRef] [Google Scholar]

[17] Mamadu EJ, Tsetimi J. Perturbation by decomposition: A new approach to singular initial value problems with Mamadu-Njoseh polynomials as basis functions. JMSS. 2020;10:15-8.
[Google Scholar]

[18] Mamadu EJ, Njoseh IN, Ojarikre HI. Space discretization of time-fractional telegraph equation with Mamadu-Njoseh basis functions. AM. 2022;13:760-73.
[CrossRef] [Google Scholar]

[19] Onyeoghane JN, Njoseh IN. Enhanced iterative methods using Mamadu-Njoseh polynomials for solving the heston stochastic partial differential equation. J Adv Math Com Sci. 2024;39:1-9.
[CrossRef] [Google Scholar]

[20] Ojada DO, Njoseh IN. Mamadu-Njoseh spectral collocation method for fractional Klein–Gordon equation, Niger. J Sci Environ. 2023;21:75-93.
[Google Scholar]

[21] Onyemarin N, Ojarikre H, Igabari JN. An ARMA modelling approach on infant mortality rate in Nigeria, Niger. J Sci Environ. 2023;21:23-36.
[Google Scholar]

[22] Mamadu EJ, Njoseh IN, Okposo NI, Ojarikre HI, Igabari JN, Ezimadu PE, et al. Numerical approximation of the SEIR epidemic model using the variational iteration orthogonal collocation method and Mamadu-Njoseh polynomials. Preprints.org. 2020. Available from: https://doi.org/10.20944/preprints202009.0196.v1 [last accessed on 2025 August 20].

Orthogonality-Preserving Cubic Spline Formulation Using Mamadu-Njoseh Polynomials

Abstract

Objectives

Material and Methods

Results

Conclusion

Keywords

Cubic spline

Interpolation

Mamadu-Njoseh polynomials

Natural spline

Orthogonality

Introduction

MATERIAL & METHODS

Preliminaries

Spline function

Cubic Spline Interpolation

Mamadu-Njoseh Polynomials

Cubic Spline Interpolation with Mamadu-Njoseh Basis

Properties of the Mamadu-Njoseh cubic spline interpolation

General algorithm

Numerical Illustrations

RESULTS AND DISCUSSION

Conclusion

Financial support and sponsorship

Conflicts of interest

Use of artificial intelligence (AI)-assisted technology for manuscript preparation

References

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