Math 477/577 Computer Assignment 3 Charles Tier
For each question, please submit the Matlab code along with an execution of the code.
1. For this question let m= 50 and n= 12. Define tas an m-vector consisting of a linearly
space grid from 0 to 1 using MATLAB’s linspace command. Using MATLAB’s vander
and fliplr, define an m×nmatrix Afor the least squares problems of fitting the grid by a
polynomial of degree n−1. Let bbe the function cos(4t)evaluated on the grid. Calculate
and print (to 16 digit precision) the least squares solution xby the following 5 methods:
(a) Formation and solution by the normal equations using MATLAB’s \command.
(b) QR factorization computed using a routine mgs that you must write that uses the mod-
ified Gram-Schmidt.
(c) QR factorization using MATLAB’s qr.
(d) x=A\b.
(e) SVD using MATLAB’s svd.
You should produce 5 table of twelve coefficients. In each list, use a red pen to circle the
digits that appear to be incorrect due to rounding error. Explain the differences.
2. Let Aby a 10 ×10 random matrix with entries drawn from the standard normal distribution
minus twice the identity. Write a program to plot ketAk2for t∈[0,2] on a log scale. Com-
pare the result to the straight line etα(A)where α(A) = max <(λj)is the spectral abscissa
of A. Run the program for ten random matrices Aand discuss the results. What property of
a matrix leads to the graph of ketAk2that is oscillatory as t→ ∞?
Hints: The definition of the matrix exponential is
etA =
∞
X
k=0
Aktk
k!.
Use randm and expm to deal with normally distributed random numbers and matrix expo-
nentials, respectively.