February , 2012
MATH 2T03
Assignment 3
This assignment is due in the MATH 2T03 locker in the basement of Hamilton
Hall by 3:00pm on Thursday, March 1st 2012.
1. i) Let us consider the function f (x) = x3 2x2 3x + 2 which has roots
on [1.5, 1], [0.5, 1], a
EE103 (Fall 2011-12)
7. LU factorization
factor-solve method
LU factorization
solving Ax = b with A nonsingular
the inverse of a nonsingular matrix
LU factorization algorithm
eect of rounding error
sparse LU factorization
7-1
Factor-solve approach
January 3rd, 2012
MATH 2T03
Assignment 1
This assignment is due in the MATH 2T03 locker in the basement of Hamilton
Hall by 3:00pm on Thursday Jan19th, 2012.
1. i) Let us dene the truncation error function
n
(1)k 2k+1
x
|.
(2k + 1)!
En (x) = | sin x
k=0
Department of Mathematics and Statistics
McMaster University
Math 2T03
Test 1: February 16th, 2012
Name
50 MINUTES TIME LIMIT. Open books and notes.
You may use the back of a page for additional space; please indicate
when you do so. Partial credit will b
Solutions to Math 2T Assignment 4
1. Denoting f (x) = x cos(x), g (x) = cos(x), you can see that f changes sign on the interval [0, 1], since
f (0) = 0 cos(0) = 1 < 0 and f (1) = 1 cos(1) > 0. Therefore, f must have a root in this interval.
Observe furthe
Spring, 2015
Assignment 4
MATH 2T03
1. i) For the following matrix,
1 4
3
A = 2
2
2
calculate by hand the reduced QR factorization by applying Gram-Schmidt
orthogonalization to the columns of A.
ii) Use the QRfactor function to show that for the following
Sample Final Exam, Math 2T03, McMaster University
1. Consider the problem of evaluating the function y = f (x), where f (x) =
there values of x for which this problem is ill-conditioned?
1
.
cos x
Are
[20 marks]
2. i) Consider the system Ax = b, where
2 1
March, 2015
Assignment 5
MATH 2T03
1. Consider the matrix
4i
2
i
2
A = 1 2i
1 1 5
Draw the Gershgorin discs.
2. Briey describe how to compute the following:
i) The dominant eigenvalue and associated eigenvector.
ii) The least dominant eigenvalue and assoc
Math2T03: Mid-Term Test 2
Instructor: Dr. D. Pelinovsky
Date: March 20 2009, 11:30-12:20
NAME:
STUDENT ID NUMBER:
Instructions:
This test paper is printed on both sides of the paper. It includes four questions on six
pages. The last page can be used for r
Math2T03: Mid-Term Test 1
Instructor: Dr. D. Pelinovsky
Date: February 6 2009, 11:30-12:20
NAME:
STUDENT ID NUMBER:
Instructions:
This test paper is printed on both sides of the paper. It includes four questions on six
pages. The last page can be used for
Assignment 2
Questions
1(b), 5(c), 8, 10, 14(b,c), 17
Chapter 2 - Solutions
Q1(b) To check for linear dependence, it is enough to consider a matrix whose columns are
the given vectors and then verify if its null space contains any nonzero vector. With
res
Assignment 3
Questions
1,5(a,b),7,8(b),11,16 (Chap.5)
Chapter 3 - Solutions
Q1 The angle between two vectors x and y is determined by the formula
cos =
x, y
.
x y
For the basis vectors cfw_e1 , e2 , e3 and the inner product given in the exercise, we nd
>
Assignment 1
Questions
3,7,9(e,f),10(c),12,14
Solutions to Assignment 1
Q3 The following function calculates the angle between two vectors in R3 :
function theta=angle_vectors(x,y)
% x and y are column vectors in R3
r=x*y/(sqrt(x*x)*sqrt(y*y);
theta=acos(
January, 2015
Assignment 2
MATH 2T03
This assignment is due in the MATH 2T03 locker in the basement of Hamilton
Hall by 3:00pm on Thursday, February 12th 2015.
1. i) Compute the LU factorization of the matrix
1
1+
1
1
ii) Does the matrix
0 1
1 0
have a LU
January 8th, 2015
Assignment 1
MATH 2T03
This assignment is due in the MATH 2T03 locker in the basement of Hamilton
Hall by 3:00pm on Thursday Jan 22nd, 2015.
1. Use a Casio calculator to nd a fraction p , p < 106 that approximates
q
2 with a truncation e
February , 2015
Assignment 3
MATH 2T03
1. i) If the secant method is used for solving the equation x5 + x3 + 3 = 0
with x0 = 1 and x1 = 1, what is x3 ?
ii) If
(2 exn )
,
xn+1 = xn + xn
e exn1
with x0 = 0 and x1 = 1 what is limn xn
2. Consider the nonlinea
Assignment 1
MATH 2T
7 January 2016
Due in the Math 2T locker in the basement of Hamilton Hall by 15:00 on January 21.
Exercises not to be handed in:
1. Go to the computer lab (BSB 241) or use your own computer to review the matlab tutorials
available on
Gershgorins Theorem for Estimating Eigenvalues
Sean Brakken-Thal
[email protected]
1
Copyright (c) 2007 Sean Brakken-Thal.
Permission is granted to copy, distribute and/or modify this document under the
terms of the GNU Free Documentation License, Version 1.2
February , 2012
MATH 2T03
Assignment 5
This assignment is due in the MATH 2T03 locker in the basement of Hamilton
Hall by 3:00pm on Monday April 2nd 2012.
1. For the following matrix,
1
2
1 2
2
1
1 1
A=
1
1 3
0
2 1
0
4
i) Find the eigenvalues of the follo
February , 2012
MATH 2T03
Assignment 4
This assignment is due in the MATH 2T03 locker in the basement of Hamilton
Hall by 3:00pm on Thursday, March 15th 2012.
1. For the following matrix,
1
1
A=
0
1
2
1
1
1
3
1
0 1
calculate the reduced QR factorization a
January , 2012
MATH 2T03
Assignment 2
This assignment is due in the MATH 2T03 locker in the basement of Hamilton
Hall by 3:00pm on Thursday, February 9th 2012.
1. i) Calculate by hand the LU factorization of the matrix
1 1 4
2 2 0
332
and verify your resu
Chapter 3 of Calculus+ : The Symmetric Eigenvalue Problem
by
Eric A Carlen
Professor of Mathematics
Georgia Tech
c 2003 by the author, all rights reserved
1-1
Table of Contents
Section 1: Diagonalizing 2 2 symmetric matrices
1.1 An explicit formula . . .
Chapter 3 - Solutions
Q1 The angle between two vectors x and y is determined by the formula
cos =
x, y
.
xy
For the basis vectors cfw_e1 , e2 , e3 and the inner product given in the exercise, we nd
> A=[1 2 2;4 4 2;2 6 4];
> e1=[1 0 0];e2=[0 1 0];e3=[0 0
Chapter 2 - Solutions
Q1(b) To check for linear dependence, it is enough to consider a matrix whose columns are
the given vectors and then verify if its null space contains any nonzero vector. With
respect to the canonical basis for M33 , the three given