Complete MATLAB Polynomial Guide: Polyfit, Polyval & Polyvalm

In this episode, we explore polynomial curve fitting in MATLAB using polyfit, polyval, and polyvalm, with MATLAB Help documentation as the primary reference throughout. We begin with scalar curve fitting, move into fit accuracy and extrapolation failure, and finish with matrix polynomial evaluation and a computational demonstration of the Cayley–Hamilton Theorem. 🔍 Topics covered in this video: 1️⃣ Polynomial Curve Fitting with polyfit What polynomial curve fitting actually means How polyfit(x, y, n) determines polynomial coefficients Why a degree-n polynomial returns n+1 coefficients MATLAB’s automatic centering and scaling for numerical stability (as documented in MATLAB Help) 2️⃣ Evaluating Fits with polyval How polyval evaluates a polynomial at new data points Visualizing fitted curves versus original data Why polyfit and polyval are inseparable (“brothers for life”) 3️⃣ Fitting Over a Limited Interval Why polynomial fits behave well inside the training range Why they diverge badly when extrapolated A clear demonstration of the extrapolation error trap 4️⃣ Fitting the Error Function (erf) Approximating a nonlinear function using a 6th-degree polynomial Computing fit error numerically Using MATLAB tables to compare: True values Polynomial fit Fit error Plotting fit accuracy versus extrapolation failure 5️⃣ Matrix Polynomial Evaluation with polyvalm How polyvalm extends polynomial evaluation to matrices Constructing a Pascal matrix using pascal Generating a characteristic polynomial using poly Evaluating the polynomial at the matrix itself 6️⃣ Cayley–Hamilton Theorem (Computational Proof) Why polyvalm(p, X) ≈ 0 How MATLAB numerically confirms that every square matrix satisfies its own characteristic equation