Math 609D
Quiz 8
due by 9am November 12, 2012
1. (5 points) Solve the following differential equation by hand (no computers or calculators).
2
y + 2t y = 2tet ,
y(0) = 2.
Solution:
The equation is already written in standard form y + p(t)y = g(t) so p(t)
Let f (x) = ln(x). The derivatives of f (x) are f (x) = f (x) = 1 x
1 x2 2 f (x) = 3 x 6 f (x) = 4 x 24 f (x) = 5 x and, in general, (n 1)! . xn Hence, the nth degree Taylor polynomial about x0 = 1 is f (n) (x) = (1)n1
n
Pn (x) =
i=1
(1)i1
(x 1)i i
and th
MATH 609-600, Homework #3, Solutions
(1) (20 pts) Problem 14, page 325 from the book.
Solution: Let P (x) be the interpolation polynomial and x0 = 0, x1 , . . . , xn1 be the nodes
of interpolation. Then we have
f (x) P (x) =
f (n) ()
(x 0)(x x1 ) (x xn1 )
MATH609-601
Homework #2
September 27, 2012
1
1. Problems
This contains a set of possible solutions to all problems of HW-2. Be vigilant since
typos are possible (and inevitable).
(1) Problem 1 (20 pts) For a matrix A Rnn we dene a norm by
A =
max
xRn ,x=0
MATH609-600
Homework #1
September 18, 2012
Bojan Popov
1
1. Problems
This contains a set of possible solutions to all problems of HW-1 and an additional practice problems. Be vigilant since typos are possible (and inevitable).
(1) Problem 1 (20 pts) Prove
MATH 609-600, Exam #2 Solutions
November 18, 2014
(1) (20 pts) Determine the nodes and weights for the formula with highest degree of accuracy
1
x2 f (x) dx A0 f (x0 ) + A1 f (x1 ).
1
What is the degree of accuracy of this rule?
Solution: This is the two
MATH 609-600
Exam #1, Numerical Linear Algebra
Solutions
(1) (10 pts) Let A Rnn be nonsingular. Prove that AT A is positive dente.
Solution: For real matrices, A is called positive denite if
xT Ax > 0, x Rn , x = 0.
Here, A Rnn and nonsingular. Then, if x
MATH 609-600
Homework #2
Vector and matrix norms
This homework is designed so you get familiarized with the concept of matrix norm and
the relationships between varuious norms. Solve any set of problems for 100 points.
(1) (20 pts) For a matrix A Rnn we d
Math 609D
Quiz 6
due by 9 am October 15, 2012
Here are some values of a function f (x);
f (1.300) = 0.341074
f (1.400) = 0.471061
f (1.450) = 0.538767
f (1.475) = 0.573271
f (1.500) = 0.608198
f (1.525) = 0.643541
f (1.550) = 0.679295
f (1.600) = 0.752006
Math 609D
Quiz 5
due 9 am, Oct 8, 2012
You are going to approximate the function f (x) = x + 1 using the nodes x0 = 1, x1 = 1.5,
x2 = 2.0, and x3 = 3. The values of f that you need are f (1) = 1.41421, f (1.5) = 1.58114, f (2.0) =
1.73205, and f (3) = 2.0
quiz3solution.nb
1
Quiz 3 Solution
After some preliminary analysis it is evident that the only values of x of interest are near 4.0 and 0.3.
[email protected]@xD, 8x, 3, 5<, PlotRange 8-0.1, 0.1<D;
0.1 0.075 0.05 0.025 3.5 -0.025 -0.05 -0.075 -0.1 4 4.5 5
[email protected]_D := -
Math 609D
Quiz 11
due by 9 am December 3, 2012
This Quiz is concerned with the matrix
1
2
A=
1
2
1
2
1
3
2
5
7
6
1
4
1
4
1. (10 points) Show that A does not have a Doolittle LU decomposition. Do this by assuming that there
exist matrices L and U
1
l
L = 2
Math 609D
Quiz 10
due by 9 am November 26, 2012
This Quiz is concerned with the system of equations
x1 x2 + x3 2x4 = 5
2x1 0x2 + 4x3 6x4 = 12
3x1 5x2 + 0x3 3x4 = 11
x1 + 5x2 + 0x3 + 4x4 = 10.
1. (10 points) Solve this system by hand using partial pivoting
Math 609D
Quiz 9
due by 9 am Nov 19, 2012
1. (10 points) Do Exercise 11a on p. 315. Do all computations by hand and be sure to derive the
local truncation error from the error formula for polynomial interpolation. Give all the details of your
computations
Math 609D
Quiz 4
September 21, 2012
Justify all of your answers to the problems below.
1. (10 points) Let cfw_pn be the sequence dened by
pn+1 =
2
5 1
,
pn +
3
3 p2
n
p0 = 1.
a) Compute p1 , p2 , p3 , p4 , and p5 .
b) From a) it appears that the sequence
Math 609D
Quiz 7
due by 9 am October 22, 2012
1. (5 points) Find the degree of precision of the quadrature formula
1
f (x) d x =
1
Solution:
1
5
8
5
f ( 0.6) + f (0) + f ( 0.6).
9
9
9
5 8 5
+ + = 2.
9 9 9
1
1
5
5
0.6 +
0.6 = 0.
If f (x) = x then
f (x) d
Math 609D
Quiz 1
August 31, 2012
Instructions: Your answers must be sent to [email protected] by 9 am CDT on Monday, September 3.
Your answers must include your calculations and reasoning. State which theorems you are using to justify
your answers. Jus
Solution for Quiz 2 If b2 4ac > 0, the quadratic equation ax2 + bx + c = 0 has two real solutions x1 , x2 given by x1 = and b + b2 4ac 2a (1)
b2 4ac x2 = 2a By rationalizing the numerator we have the algebraically equivalent formulas b x1 = and x2 = 2c b
MATH 609-602 Homework #1 Matrix algebra and direct methods for linear systems Solve any set of problems for 100 points. 1. (20 pts) (problem 11, p. 157 of your textbook) Prove that the inveres of a nonsingular upper triangular matrix is also upper triangu