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(a) First of all, read Professor Saito’s Lecture Notes through NS_LECTURE_11.pdf,
since in Problem 3(c) you will need to compute the QR decomposition of a matrix A
using Householder Triangularization by hand. Eldén also has a good description of
Householder Triangularization. However his notation is different than Saito’s. Due to
time constraints we will only cover Saito’s LECTURE_11.pdf in class.
(b) You may also want to review the material on eigenvalues, eigenvectors,
transformation to the eigenbasis and real symmetric matrices in Sections 6.1, 6.2, and 6.4
of Strang. This material is both important and useful.
NOTE: THERE IS A NEW FILE NAMING CONVENTION, WHICH YOU CAN
FIND ON CANVAS. YOU WILL LOSE 25 POINTS IF YOU FAIL TO FOLLOW
IT.
Problem 1 (60 points total) (Reverse Engineering) Let A be a 2 × 2 matrix such that for
some nonzero vectors x, y ∈ R2,
Ax= x,
(1a)
Ay = −y.
(1b)
(a) (5 points) Find the characteristic polynomial p (λ) = d e t (A − λ I).
(b) (5 points) Suppose you know the first column of A is [ 2, 1] T . Find the components b
and d of the second column vector [b, d]T of A so that
(c) (5 points) If (1a) and (1b) hold for some nonzero vectors x, y ∈ R2, show that x and
y are linearly independent.
(d) (5points) Compute A101.(NOTE: Even if you did not solve Parts(a)-(c),you can use
the description of the matrix A in the initial statement of Problem 1 to solve this
problem.) As with all questions, show enough work so we can see your method and give
due credit.
(e)(5points) Given the matrix A you found in Part(a), find an invertible matrix S and a
diagonal matrix Λ such that
−1
A=SΛS
(3)
. (f) (5 points) Given the diagonal matrix Λ you found in Part (e), what are the entries d1
and d2 on the main diagonal of Λ called?
Problem2 (60points) Let
where ε is a small positive number (e.g., 10−8) so that ε2 can be ignored numerically,
assuming that it is in an equation with another number which is O(1). If you are unsure
whether ε2 should be regarded as 0 or as 10−8 compute the expression you are unsure of
in MATLAB and see what result you obtain.
Problem 3
(a) (15 points) Compute the reduced QR factorization
Gram-Schmidt algorithm by hand.
(b) (15points) Compute the reduced QR factorization
Schmidt algorithm by hand.
using the classical
using the modified Gram-
(c) (15points) Compute the full QR factorization A=QR using the Householder
triangularization by hand.
(d) (15points) Check the quality of these results by computing the Frobenius norm of
for the results obtained by the CGS and MGS algorithms and
for the result obtained by the Householder triangualarizartion.
Problem 3 (50 points) Consider an operator E that acts in the following way. First, one is
given an input vector x. Then, one computes the arithmetic mean (average) of the entries
of x, that is
Finally, one creates the output vector y, whose entries are all equal to
. In other words,
a) (10points) Let n=3. Express E as a matrix; i.e. providing the entries of E.
(b) (10 points) What are the dimensions of the column space C(E) and the nullspace N(E)
(c) (10 points) Is E an orthogonal projector, an oblique projector, or not a projector at all?
(d) (10points) Express the matrix E as the sum of outer products.
(e) (10points) Express the matrix E as the sum of outer products of unit vectors.
IN Problem 4 BELOW YOU MUST TURN IN A WELL-DOCUMENTED CODE
THAT PRODUCES ALL OF THE OUTPUT IN PART Problem 4(a)–(f). IF YOUR
CODE FAILS TO OUTPUT FIVE LISTS OF TWELVE COEFFICIENTS AS STATED
IN PART Problem 4(f) YOU WILL RECEIVE NO CREDIT FOR THIS PROB- LEM.
The reason I am emphasizing this issue is because in Problem 04 on HW 02 a significant
number of students’ MATLAB programs only output the solutions to Parts (a)-(c) and did
not include code for the remaining parts of the problem. Please don’t make that mistake
this time. 🙂
Problem 4 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 the m × n matrix associated with least squares fitting
on this grid by a polynomial of order n − 1. Take b to be the function cos(4t) evaluated on
the grid. Now, calculate and print (to 16-digit precision) the least squares coefficient
vector x by the following three methods.
Place your results in a table similar to Table 1 below. I will provide a template for this
table on PIAZZA.
(a) (10points) Solve the normal equation explicitly by computing
using LU decomposition. (There is an example for how to solve an equation of the form
Ax = b in the MATLAB documentation. Type ’lu’ in the search rectangle and then click
on examples and scroll down.)
(b) (10 points) Using the MATLAB implementation CGS.m of the classical GramSchmidt algorithm CGS, which can be downloaded from CANVAS.
(c) (10 points) Using the MATLAB implementation MGS.m of the modified GramSchmidt algorithm MGS, which can be downloaded from CANVAS.
(d) (10points) QR factorization using MATLAB’s qr, which is based on the Householder
triangularization.
(e) (10points) x=Ab in MATLAB, which is also based on QR factorization.
(f) (10points) The calculations above will produce five lists of twelve coefficients.
(i) Create a table with five columns. In each column put one of these lists, in the
or- der in which you created them: (a), (b), (c), (d), and (e). Make sure you
place a heading above each column of the table identifying which method
produced the coefficients in that particular column. I have given you an
example in Table 1 be- low.
(ii) You will need to include the LaTeX package
in your preamble in order to complete Parts (f) and (g) below. If you also create
the following LaTeX macro
in your preamble then you can highlight the trailing digits as required in
Problem (f) with the following LaTeX code:
1.000000000rd{996607} → 1.000000000996607
This example is from Row 01 Column 05 in Table 1.
(iii) In each list that you created in Problem Problem 4 (f) (i) use the the
LaTeX command
to produce, for example,
in order to highlight the digits, which appear to be incorrect; i.e., affected by rounding
error.
(g) (10 points) Comment on the differences you observe.
For example, do the normal equations exhibit instability?
Give an explanation for what you observe vis-à-vis with regards to the accuracy of
each of the five algorithms you used to compute the numerical solution of (4). You
must try to explain your observations. You must try to explain your
observations without help from a classmate or a friend. However, you may
post questions concerning this problem – including this part – on PIAZZA, and I
Problem 5 (35points) Let
(a) (15 points) What are the pseudoinverse A† and the projector P = AA† for this
example? Give exact answers (not by MATLAB).
(b) (15 points) Find the exact SOLUTIONS
problem Ax ≈ b; i.e., not by MATLAB.
to the least squares
(c) (05 points) The condition number of a rectangular shape matrix A is
Compute the condition number of A in this example using
MATLAB.
LECTURE 11
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– 02 –
– 03 –
– 04 –
– 05 –
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