Write out the first and second divided differences explicitly and give their geometric meaning in terms of secant slopes.
Observe numerically the divergence of interpolation at equally spaced points on [−5, 5] for =
!
!!!! as → ∞ for 3.63 |x| 3.63. How do its coefficients in the estimate grow? Explain the observed behavior of interpolating coefficients.
Determine (analytically) the spacing ℎ in a table of evenly spaced values of the function = between 1 and 2, so that interpolation with
1st degree polynomial
2nd degree polynomial
in this table will yield a desired accuracy of at least places after zero. Choose your own .
Develop a program that magnifies (monochrome) image by a factor of 2! using bilinear interpolation. Implement this up-scaling in 2 different ways: 1) directly, i.e. calling interp2 once; and 2) times iteratively doubling the size. Compare the outputs and discuss the variation in the output (if any).
