Curve Fitting and Interpolation

polyfit(x, y, n) fits an n-th degree polynomial to the data (x, y) using least-squares regression, returning a coefficient vector p ordered from the highest to lowest power, so p(1) is the coefficient of x^n and p(end) is the constant term. Once you have coefficients, polyval(p, xq) evaluates the fitted polynomial at new query points xq, which is essential for both plotting a smooth fitted curve over the original data and for extrapolating predictions beyond the sampled range (with the caveat that extrapolation accuracy degrades quickly for higher-degree polynomials). Choosing too high a degree n relative to the number of data points causes overfitting, where the curve wiggles to pass through noise rather than capturing the underlying trend — a classic bias-variance tradeoff that MATLAB will warn about via a poorly-conditioned polynomial warning when n approaches the number of unique x values.

For nonlinear or non-polynomial models — exponential decay, logistic growth, custom equations — the Curve Fitting Toolbox's fit(x, y, 'exp1') function fits a named model type (like 'exp1' for a*exp(b*x), or 'poly3' for a cubic polynomial) using nonlinear least squares, returning both a fit object and a gof (goodness-of-fit) struct containing statistics like rsquare and rmse. Custom model equations are supported via fittype('a*sin(b*x) + c'), which lets you fit arbitrary parametric forms beyond the built-in library, with optional 'StartPoint' values to help the nonlinear solver converge since a poor initial guess can cause it to settle on a local rather than global optimum. Unlike polyfit, which always finds the unique least-squares solution for a linear-in-parameters model, fit()'s nonlinear solvers are iterative and can fail to converge or converge to different local minima depending on the starting point.

interp1(x, y, xq, method) estimates y-values at new query points xq that lie between existing (x, y) samples, unlike curve fitting which finds an approximate model — interpolation passes exactly through every original data point. The method argument controls the interpolation scheme: 'linear' (the default) connects points with straight segments, 'spline' fits a smooth piecewise cubic that also matches first and second derivatives at each knot for a visually smoother curve, and 'nearest' simply returns the value of the closest known sample, useful for categorical or step-like data. Querying xq outside the range of the original x data returns NaN by default unless you specify an 'extrap' flag or a fixed extrapolation value, e.g. interp1(x, y, xq, 'linear', 'extrap'), and it's important to remember that spline extrapolation can behave wildly since cubic polynomials diverge quickly outside their fitted range.