Summary of some useful formulas
Taylor polynomial of degree k (Note typo in (7.7), (7.8) - sums start at j = 0): Pk (x) = 1 (j) f (x0 )(x - x0 ). j! j=0
k
Error in Taylor polynomial Pk (x): If f C k+1 (Int[x0 , x]), then |f (x) - Pk (x)| = |f (k+
CHAPTER 5 EXERCISES
1. Reconsider problem 4 of chapter four.
(a) Sketch the vector eld for this model. Determine the location of each
equilibrium in the state space. Can you tell from the vector eld which of the
equilibria are stable?
The dynamical system
AMATH 352 HW1
1. a) b) f = @(x) x .* cos(x) f= @(x) x .* cos(x) quad(f, 0, 1) ans = 3.817732902243389e-001 2. From script prob2.m :
true = .381773291; f = @(x) x .* cos(x); sprintf('%-10s %-23s %-15s %-8s', 'tol', 'integral', 'error', 'f evals') for
1. An automobile manufacturer makes a profit of $1,500 on the sale of a certain model.
It is estimated that for every $100 of rebate, sales increase by 15%.
(a) What amount of rebate will maximize profit? Use the five-step method, and model as
a one-varia
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb. 28th 2014
Homework 6
Homework 6 will be posted later this afternoon
Due Friday Mar. 7th in class
Floating point arithmetic
Fact
Almost a
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb 24th 2014
Homework 5
The due date of homework 5 is extended to Friday Feb. 28th
Orthogonal bases
Let V be a real linear space with an inn
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb. 19th 2014
Homework 5
Homework 5 will be posted this afternoon.
Due next Wednesday in class.
Midterm
If you have questions about your mi
Name:
AMATH 352
Homework Set 7
Due: Monday, May 22nd, in class.
(1) In this problem, well think about systems that are given to us in factored form.
Let A Rmn , and suppose that we are given a factorization
A = LR,
L Rmk ,
R Rkn .
Assume that k is much le
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb. 10th 2014
Midterm Exam
Friday Feb. 14th in class
No notes, no calculator, no phone, scratchd paper will be
provided
Review all lecture
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb. 7th 2014
Midterm Exam
Friday Feb. 14th in class
No notes, no calculator, no phone, scratchd paper will be
provided
Review all lecture s
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb. 26th 2014
Gram-Schmidt process
Consider the basis vectors a1 , a2 , a3 . We want to build the
sequence of orthonormal vectors q1 , q2 , q3
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Mar. 7th 2014
Homework 7
Homework 7 will be posted this afternoon
Due Friday March 14th in class
GradeBook
Please double check your homework
CHAPTER 2 EXERCISES
1. (a) Step 1: Ask the question.
Variables:
x = Blue whale population (whales)
y = Fin whale population (whales)
B = Blue whale population growth rate (whales / year)
F = Fin whale population growth rate (whales / year)
T = Total popul
CHAPTER 3 EXERCISES
1. (a) The following graph shows price (p) as a function of time (t). Over the first 5 days
price is nearly a linear function of time and falls about one cent per day. As time goes by
the price curve flattens so that the rate of declin
CHAPTER 4 EXERCISES
1. (a) The dynamical system equation is x' = F(x) where x = (x1, x2) and F = (f1, f2) with
f1(x1, x2) = r1*x1 - a1*x1^2 - b1*x1*x2
f2(x1, x2) = r2*x2 - a2*x2^2 - b2*x1*x2
and the state space is the set of all x satisfying x1 >= 0 and x
Applied linear algebra and numerical analysis
AMATH 352 - Spring 2014
Yuqi Wu
Department of Applied Mathematics
March 31st 2014
Instructor
Dr. Yuqi Wu
Email: [email protected]
Oce: Lewis 116
Assistants
Teaching Assistant: Peng Zheng
Email: [email protected]
Applied linear algebra and numerical analysis
AMATH 352 - Spring 2014
Yuqi Wu
Department of Applied Mathematics
April 4th 2014
Homework 1
Homework 1 has been posted on the course webpage.
Written part is due on Friday Apr. 11th in class.
Programming pa
Applied linear algebra and numerical analysis
AMATH 352 - Spring 2014
Yuqi Wu
Department of Applied Mathematics
Apr. 9th 2014
Review of Linear Space
A real linear space consists of a set of objects V along with two
operations + (addition) and (scalar mult
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb 5th 2013
Range of a Matrix
Consider a matrix A Rmn
Denition
The range is the subspace of all vectors produced by the matrix A
R(A) = cfw_Ax
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Feb. 3rd 2014
Homework 4
Homework 4 will be posted later today, due Monday Feb. 10th.
Range of a Matrix
Consider a matrix A Rmn
Denition
The r
Name:
AMATH 352
Homework Set 7
Due: Friday, March 4th, in class.
(1) Symmetric positive definite and positive semidefinite matrices.
(a) Let A Rm,n be any matrix. Show that AT A and AAT are always positive
semidefinite.
(b) Suppose that m > n. Show that A
Name:
AMATH 352
Homework Set 3
Due: Online, by class time, Monday April 17.
(1) On the first day in class, we saw that if you are interested in approximately solving
Ax = b,
it makes sense to find the least-norm solution x by considering the problem
1
min
AMATH 352
Homework Set 1
Due: Monday, April 3rd, in class.
Name:
Student number:
Matlab code can be pasted in a verbatim environment. Below, I paste
the code to compute the 150th term in a Fibonacci recurrence relation.
xk = 1;
xk1 = 1;
for k = 1:150
xk2
Name:
AMATH 352
Homework Set 5
Due: Monday, May 15th, on canvas (by class time).
(1) In this exercise, fill in the gaps to our logistic regression derivation
(guided practice). We are given data (ai , bi ) for i = 1, . . . , m individuals. Each
ai Rn is a
Name:
AMATH 352
Homework Set 5
Due: Monday, May 8th, on canvas (by class time).
(1) Compute the QR decomposition of the following matrix. You can either do it by
hand, showing the steps, or program up a code, and give the code. You can use the
matlab qr c
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
January 6th 2013
Instructor
Dr. Yuqi Wu
Email: [email protected]
Oce: Lewis 116
Research Interests:
Scientic Computing
Linear Algebra
Computati
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
Jan. 15th 2014
Sets
A set is a collection of elements.
Example
R2 is the set of all column vectors with two real components.
Sets
Example
F (R
Applied linear algebra and numerical analysis
AMATH 352 - Winter 2014
Yuqi Wu
Department of Applied Mathematics
January 10th 2014
Homework 1
Homework 1 has been posted on the course webpage.
Written part is due on Friday Jan. 17th in class.
Programming
Name:
AMATH 352
Homework Set 2
Due: Monday, April 10th, in class.
(1) There are two important methods we can use to find the root of f (x) = 0. The first
is called Newtons method. We take an initial x0 somewhere close to the root, and
then we define
f (xk
Name:
AMATH 352
Homework Set 7
Due: Monday May 29th by 12:30pm on Canvas.
(1) Symmetric positive definite and positive semidefinite matrices.
(a) Let A Rm,n be any matrix. Show that AT A and AAT are always positive
semidefinite.
(b) Suppose that m > n. Sh