Exercise · ZW-RUNGE-PHENOMENON-002 · ★☆☆ Basic
What type of nodes reduces the Runge phenomenon?
Choose one option.
Show answer
- A. Chebyshev nodes clustered toward the endpoints
Explanation
Chebyshev nodes cluster toward the endpoints and decrease the nodal product. Equally spaced nodes are the cause of the problem, and the others are not designed for interpolation on the interval.
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-003 Which classic function is used to illustrate 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…