According to Figure 5.9, the interpolating polynomials for...

According to Figure 5.9, the interpolating polynomials for the function $1 /\left(1+x^2\right)$ on the interval $[-3,3]$ based on equally spaced mesh points are very inaccurate near the ends of the interval. Does the natural spline interpolant based on the same 3,5 , and 11 data points exhibit the same inaccuracy?

Key Concepts

Natural Splines

Natural splines are a type of spline interpolation where the second derivative is set to zero at the endpoints, ensuring that the spline behaves linearly outside the interpolation range. This boundary condition helps in avoiding the extreme oscillations that can occur in polynomial interpolation, making natural splines less prone to inaccuracies near the edges.

Spline Interpolation

Spline interpolation uses piecewise polynomials, typically of low degree, that are joined together at the data points. These piecewise functions are constructed to ensure smoothness at the junctions, which helps in reducing oscillations and providing a more stable approximation than high-degree polynomials over the entire interval.

Runge Phenomenon

The Runge phenomenon describes the problem of oscillation at the edges of an interval that occurs when using high-degree polynomial interpolation with equally spaced nodes. This phenomenon is especially pronounced when the function being approximated is not well-behaved near the boundaries, leading to large deviations from the true function.

Polynomial Interpolation

Polynomial interpolation involves approximating a function by a single polynomial that passes through a given set of points. While it can provide a good approximation over a certain range, when high-degree polynomials are used, particularly with equally spaced nodes, they can exhibit oscillatory behavior near the endpoints of the interval.

Transcript

Please give Ace some feedback