To store your matlab sessions, you can run diary command in matlab.
For the details, type >> help diary at your matlab session.
Once you store the diary file, you can edit this diary file to remove unnecessary
commands (such as help diary, etc., which are not directly related to the
projects).
Please submit the diary file via email to me.
[1] Reduced Row Echelon Forms, Nullspaces, Column spaces, and
Solutions of Ax=b
[1.1] To kick off the project, do Exercise 2.2.23
(page 78).
[1.2] Determine the bases of column space and nullspace
of the matrices in Exercise 3.5.1,
and compare the results with those of HW #1.
[1.3] Implement mrrf.m in Appendix B.
This program also requires to program mc.m.
Check the program by running the examples of page 473.
[1.4] Implement msoln.m in Appendix B.
Then solve Exercise 3.6.7 with msoln.m and compare
results with those of HW#2. Note that msoln.m requres both
mrrf.m
and mc.m.
[2] Inner products and Norms
[2.1] Do Exercise 4.2.25, 4.2.26.
[2.2] Do Exercise 4.3.18.
[3] Gram-Schmidt Method and QR Factorization
[3.1] Do Exercise 4.5.14.
[3.2] Do Exercise 4.6.10.
[4] Orthogonal Projections and Least Squares
[4.1] Do Exercise 4.9.20.
[4.2] Take m=50, n=12. Using
Matlab's linspace, define t to be the m-vector corresponding
to linearly spaced grid points from 0 to 1. Using Matlab's
vander
and fliplr,define A to be m x n matrix associated
with least squares fitting on this grid by a polynomial of degree n-1.
Take b to be the function cos(4t) evaluated on the grid.
Now, calculate and print (to sixteen-digit precision) the least
squares coeffcient vector x by the following methods.
(a) Formation and solutions of the normal equation, using Matlab's
\
operator.
(b) QR factorization computed by Matlab's qr.
(c) x = A\b in Matlab (also based on QR factorization).
(d) The calculations above will produce three lists of 12 coefficients.
In each list, mark with ^^^ 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.
Project Assignment #2 Due Monday June 7, 1999
Please submit the diary file via email to me in the same as way Project #1.
[5] Roots of Polynomials
Do Exercise 5.2.13
[6] The Characteristic Equation
Do Exercise 5.3.36, 5.3.38
[7] Eigenvectors
Implement meig in Appendix B, and compute
the eigenvalues, their multiplicities, and the corresponding eigenvectors
of Exercise 5.3.11 and 5.3.36.
[8] Matrix Polynomials
Do Exercise 5.6.10
[9] The hermitian norm, inner product
Do Exercise 6.1.14, 6.1.15
[10] The Schur decomposition
Similarly to Examples 6.3.3 and 6.3.4, do
Exercise 6.3.11