• Regression (Curve Fitting): Used when the data exhibits significant scatter or error (e.g., experimental measurements with noise). The goal is to derive a single curve that represents the general trend of the data. Crucially, the regression curve does not necessarily pass through any of the specific data points.
• Interpolation: Used when the data is known to be highly precise or exact (e.g., tabulated thermodynamic properties or computer-generated coordinates). The goal is to derive a curve that passes exactly through every single data point.

The coefficients are derived by taking the partial derivatives of the sum of squared residuals with respect to $a_0$ and $a_1$ and setting them to zero. This yields the "Normal Equations":

1. $n a_0 + (\sum x_i) a_1 = \sum y_i$
2. $(\sum x_i) a_0 + (\sum x_i^2) a_1 = \sum x_i y_i$

Solving this $2 \times 2$ linear system directly yields the formulas for $a_0$ and $a_1$.

• Exponential Equation: $y = \alpha e^{\beta x}$ becomes $\ln(y) = \ln(\alpha) + \beta x$. Let $y^* = \ln(y)$, $a_0 = \ln(\alpha)$, $a_1 = \beta$, $x^* = x$.
• Power Equation: $y = \alpha x^{\beta}$ becomes $\log(y) = \log(\alpha) + \beta \log(x)$. Let $y^* = \log(y)$, $a_0 = \log(\alpha)$, $a_1 = \beta$, $x^* = \log(x)$.
• Saturation-Growth-Rate: $y = \frac{\alpha x}{\beta + x}$ becomes $\frac{1}{y} = \frac{\beta}{\alpha} \frac{1}{x} + \frac{1}{\alpha}$. Let $y^* = \frac{1}{y}$, $a_0 = \frac{1}{\alpha}$, $a_1 = \frac{\beta}{\alpha}$, $x^* = \frac{1}{x}$.

While a higher-degree polynomial will technically yield a lower sum of squared residuals, arbitrarily increasing the degree $m$ simply to chase a higher $R^2$ value is a severe error known as overfitting. An overfitted polynomial will weave wildly to hit noise points rather than capturing the true underlying physics. This resulting function becomes highly unstable and nearly useless for making predictions between or beyond the data points. A general rule is to use the lowest degree polynomial that adequately captures the physical trend, validating it against an independent test set.

Instead of fitting polynomials, Fourier approximation uses a series of sine and cosine functions to model periodic phenomena (e.g., tides or vibrations). It's a form of least-squares fitting using orthogonal trigonometric functions.

A polynomial of order $n$ that passes through $n+1$ points. The coefficients $b_i$ are evaluated using finite divided differences.

• Swapping variables: The simplest approach is to swap $x$ and $y$ (making $x$ the dependent variable) and use a standard method like Newton's or Lagrange. However, this only works well if the function is monotonic.
• Root-finding: A more robust approach is to form a new function $g(x) = f(x) - y_{\text{target}}$ and use root-finding methods (like the Secant Method) on the interpolating polynomial.

To mitigate Runge's Phenomenon, we can choose non-equally spaced points called Chebyshev nodes. These nodes are clustered near the ends of the interval and sparse in the center, which minimizes the maximum interpolation error across the domain. For an interval $[a, b]$, these points are linearly scaled from $[-1, 1]$. Using Chebyshev nodes with Newton's or Lagrange polynomials dramatically improves stability.

To interpolate $n$ points where both $f(x_i)$ and $f'(x_i)$ are specified, a polynomial of degree $2n-1$ is required. This ensures the resulting interpolating curve has smooth tangents at the data points, which is particularly useful in computer graphics and path planning.

The most common type of spline. It fits a third-degree polynomial to each interval between data points. To ensure a smooth continuous curve across the entire data set, the cubic spline enforces matching conditions at the internal knots:

• The function values must match at the knots (continuity).
• The first derivatives must match at the knots (smooth slope).
• The second derivatives must match at the knots (smooth curvature).

This results in a tridiagonal system of linear equations that can be solved very efficiently.

Because there are $n$ intervals but conditions only specify the internal connections, two extra boundary conditions at the first and last knot are required to close the system:

• Natural Spline: The second derivative is set to zero at the endpoints ($f''(x_0) = f''(x_n) = 0$). This makes the curve "straight" at the ends.
• Clamped Spline: The first derivatives (slopes) at the endpoints are specified by the user to match a known tangent.
• Not-a-knot: Forces the third derivative to be continuous across the first and last internal knots, effectively treating the first two and last two intervals as single cubic functions.