Exercise · ZW-RUNGE-PHENOMENON-003 · ★☆☆ Basic
Which classic function is used to illustrate the Runge phenomenon?
Choose one option.
Show answer
- D. \( f(x)=\frac{1}{1+25x^2} \)
Explanation
The function \( \frac{1}{1+25x^2} \) on \( [-1,1] \) is the standard example, since its derivatives grow rapidly. The other functions are smooth and do not show the problem noticeably.
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
- ZW-RUNGE-PHENOMENON-001 What does the Runge phenomenon describe?
- ZW-RUNGE-PHENOMENON-002 What type of nodes reduces the Runge phenomenon?
- ZW-RUNGE-PHENOMENON-004 Which factor of the interpolation error grows sharply near the endpoints with equally spaced nodes?
- ZW-RUNGE-PHENOMENON-005 What happens to the error when the degree increases with equally spaced nodes for f(x)=(1)/(1+25x^2…
- ZW-RUNGE-PHENOMENON-006 You interpolate f(x)=(1)/(1+25x^2) on [-1,1] with equally spaced nodes. Which statement best descri…
- ZW-RUNGE-PHENOMENON-007 If the interpolation error is written as (f^(n+1)(xi))/((n+1)!)prod(x-x_i) , what happens when swit…
- ZW-RUNGE-PHENOMENON-008 What is the main reason piecewise interpolation avoids the Runge phenomenon?
- ZW-RUNGE-PHENOMENON-009 For f(x)=(1)/(1+25x^2) on [-1,1] , compare the maximum of |prod(x-x_i)| for n=10 with equally space…