Preventing Runge Phenomenon Through Interpolation Functions with Varying Degrees

In numerical analysis, interpolation serves as a fundamental numerical approximation technique that constructs a function passing exactly through known data points. However, when using high-degree polynomials for interpolation, one may encounter the Runge phenomenon - characterized by severe oscillations near the endpoints of the interpolation interval.

To overcome this issue, several methods can be employed including piecewise low-degree interpolation or specialized node distributions. Common solutions involve:

Piecewise Linear Interpolation: This approach divides the entire interval into smaller subintervals and applies linear functions within each segment. While computationally simple (implementable through linear interpolation algorithms like numpy.interp() in Python or interp1() in MATLAB), the resulting curve lacks global smoothness.

Cubic Spline Interpolation: Utilizes third-degree polynomials within each subinterval while maintaining continuity of function values, first derivatives, and second derivatives at connecting nodes. Implementation typically involves solving tridiagonal systems of equations (using functions like scipy.interpolate.CubicSpline in Python or spline() in MATLAB), balancing smoothness with effective Runge phenomenon prevention.

Chebyshev Node Interpolation: By strategically selecting node positions (Chebyshev points) that minimize interpolation error through cosine distributions, this method effectively suppresses Runge phenomenon. Algorithm implementation often requires coordinate transformation and specialized polynomial evaluation routines.

In practical applications, selecting an appropriate interpolation method requires careful consideration of computational complexity, accuracy requirements, and smoothness constraints. For beginners, understanding the differences and applicable scenarios of these techniques forms the cornerstone of mastering numerical interpolation methods.