Math 477/577 — Computer Assignment 4, due Oct.26, 2006
1. Take m= 50, n= 12. Using MATLAB’s linspace, define tto be the m-vector corresponding
to linearly spaced grid points from 0 to 1. using MATLAB’s vander and fliplr, define Ato
be the m×nmatrix associated with least squares fitting on this grid by a polynomial of degree
n−1. Take bto be the function cos(4t) evaluated on the grid. Now, calculate and print (to
sixteen-digit precision) the least squares coefficient vector xby six methods:
(a) Formation and solution of the normal equations, using MATLAB’s \,
(b) QR factorization computed using the mgs routine from Computer Assignment 3,
(c) QR factorization computed via the house routine from Computer Assignment 3,
(d) QR factorization computed by MATLAB’s qr,
(e) x=A\bin MATLAB,
(f) SVD, using MATLAB’s svd.
(g) The calculations above will produce six lists of twelve coefficients. In each list, shade with
red pen the digits that appear to be wrong (affected by rounding error). Comment on what
differences you observe. Do the normal equations exhibit instability? You do not have to
explain your observations.
2. The usual Gaussian elimination algorithm involves a triply nested loop. The version given
in the classnotes (as LU factorization) involves two explicit for-loops, and the third loop is
implicit in the vectors U(j, k :m) and U(k, k :m). Rewrite this algorithm with just one
explicit for-loop indexed by k. Inside this loop, Uwill be updated at each step by a certain
rank-one outer product. This “outer product” form of Gaussian elimination may be a better
starting point than the LU factorization algorithm from the classnotes if one wants to optimize
computer performance. Implement both the LU factorization algorithm from the classnotes and
the “outer product” version discussed here, and test and time them both with the matrices A =
gallery(’lehmer’,N) for N=100*[1:10].