Use n+1 equally spaced data points to interpolate f(t)=1 /(1+t^2) on an interval -a ≤t ≤a for a=

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Use n+1 equally spaced data points to interpolate f(t)=1...

Use $n+1$ equally spaced data points to interpolate $f(t)=1 /\left(1+t^2\right)$ on an interval $-a \leq t \leq a$ for $a=1,1.5,2,2.5,3$ and $n=2,4,10,20$. Do all intervals exhibit the pathology illustrated in Figure 5.9? If not, how large can $a$ be before the interpolants have poor approximation properties? What happens when the number of interpolation points is taken to be $n=50$ ?

Key Concepts

Polynomial Interpolation

Polynomial interpolation involves constructing a polynomial that exactly passes through a given set of data points. This method is used to estimate the value of a function at intermediate points, and its effectiveness depends on both the distribution of the data points and the degree of the resulting polynomial. Higher degree polynomials can interpolate more points but also may exhibit unwanted oscillatory behavior, especially in regions near the boundaries of the interpolation interval.

Equally Spaced Nodes

Using equally spaced nodes means that the data points are uniformly distributed over the interval. While this is often convenient, it can lead to poor approximation properties for certain functions, particularly when the interval is large. Equally spaced nodes are a common choice in practice, but they are not always optimal for controlling interpolation error, especially at the ends of the interval.

Runge Phenomenon

The Runge phenomenon is the occurrence of large oscillations at the edges of an interval when using high-degree polynomial interpolation with equally spaced nodes. This problematic behavior becomes more pronounced as the degree of the polynomial increases or when the interpolation interval is extended too far. It illustrates that simply increasing the number of interpolation points (or the polynomial degree) does not guarantee better accuracy and can, in fact, lead to divergence near the boundaries.

Effect of Increasing the Number of Interpolation Points

Increasing the number of interpolation points, particularly with equally spaced nodes, can exacerbate the Runge phenomenon because the interpolation polynomial’s degree increases. For moderate values of n, and small interval sizes, the interpolant may approximate the function well, but for a larger n (such as n=50), especially over larger intervals, the oscillations become severe. This highlights the delicate balance between the number of nodes, their placement, and the overall interval size in achieving accurate polynomial interpolation.

Impact of Interval Length on Approximation

The length of the interpolation interval plays a crucial role in the performance of polynomial interpolants. For functions like 1/(1+t^2), as the interval [-a, a] increases, the interpolation error tends to become more significant due to the increased potential for oscillations. There exists a threshold for the interval size, beyond which the interpolants exhibit the pathological behavior associated with the Runge phenomenon, leading to poor approximation properties.

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