For \( f(x)=\frac{1}{1+25x^2} \) on \( [-1,1] \), compare the maximum of \( |\prod(x-x_i)| \) for \( n=10 \) with equally spaced nodes versus Chebyshev nodes. Which statement is correct?
Choose one option.
Show answer
B. The maximum with equally spaced nodes is larger, especially near the endpoints
With equally spaced nodes the nodal product reaches much larger values near the endpoints than with Chebyshev nodes. The options claiming equality or that Chebyshev is larger reverse the comparison, and the one claiming equally spaced is smaller is also false.
The Runge phenomenon is the oscillatory, divergent behavior of polynomial interpolation errors at high degree when the nodes are equally spaced on an interval.