Polynomial Detection System

Overview

I developed a mathematical interpolation system that reconstructs the unique polynomial of degree at most n − 1 through n points with distinct x-values, then determines the lowest degree consistent with those observations. Exact rational arithmetic preserves coefficient precision without floating-point rounding.

What It Delivers

• Exact interpolation for distinct x-values, with degree detection rather than assuming degree n − 1.
• Rational coefficients and symbolic expressions suitable for further mathematical analysis.
• Explicit reporting when no lower-degree polynomial is established by the data.

What I Did

• Implemented Newton divided differences for incremental polynomial construction and degree identification.
• Used higher-order zero differences to detect when a lower-degree polynomial explains all supplied points.
• Represented decimal inputs as rational fractions, performing coefficient calculations exactly.
• Converted Newton-form polynomials into conventional coefficient form using nested multiplication.
• Applied Lagrange interpolation and barycentric evaluation for an alternative mathematical representation and efficient plotting.
• Avoided constructing ill-conditioned Vandermonde systems, using O(n²) construction and O(n) barycentric evaluation after weight preparation.
• Added symbolic analysis covering factorisation, roots, derivatives and integrals.
• Accounted for interpolation limitations, including high-degree oscillation and node placement, with Chebyshev sampling guidance.

Outcomes

• N points in, exact coefficients out, with O(n²) construction.

Proven Impact

• Exact rational coefficients with no floating-point rounding.
• Lowest consistent degree detected from the data rather than assumed.

Why It Matters

• Combined numerical analysis, exact algebra and algorithm design into an interpretable mathematical reconstruction system.