Exercise · ZW-RUNGE-PHENOMENON-010 · ★★☆ Intermediate

Why does \( f(x)=\frac{1}{1+25x^2} \) produce oscillations with equally spaced nodes even though it is infinitely differentiable on \( [-1,1] \)?

Choose one option.

Show answer
  • A. It has complex poles at \( x=\pm i/5 \) close to the interval

Explanation

The complex poles at \( \pm i/5 \) lie close to the interval and limit the region where interpolation converges, so the error grows near the endpoints. The other statements are false: \( f(0)=1 \), \( f \) is continuous on \( [-1,1] \), and evenness does not cause oscillations.

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Review the concept: Runge phenomenon

The Runge phenomenon is the oscillatory, divergent behavior of polynomial interpolation errors at high degree when the nodes are equally spaced on an interval.

\[f(x)=\frac{1}{1+25x^2},\quad x\in[-1,1]\]

More practice problems: runge phenomenon

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