Solutions for Homework 4 Foundations of Computational
Math 1 Fall 2010
Problem 4.1
Recall that an elementary reector has the form Q = I + xxT Rnn with x
2
= 0.
4.1.a. Show that Q is orthogonal if and only if
=
2
or = 0
xT x
4.1.b. Given v Rn , let = v and

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Set 6: Conditioning, Stability and
Factorizations
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
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Overview
Example of stability analysis: sum of n scalars
Conditioning of so

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Set 5: Basics Part 3
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
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Sources
Additional sources for this set:
Matrix Algorithms, Volume 1: Basic Decompositions, G. W. Stewart

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Set 4: Basics Part 2
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
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Finite Precision
All discussions so far have assumed we can compute with elements
of R and by extension C

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Set 3: Linear Least Squares
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
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Simple Example
Arithmetic Mean: =
1
n
Pn
i=1
i
n
X
( i )2
= argmin f ( ) = argmin
R
f ( ) =
R
n

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Set 2: Solving Linear Systems
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
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The Problem
A Cnn , x Cn and b Cn
Given A and b nd x where
Ax = b
n equations and n unknowns

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Set 7: Iterative Methods for Solving
Equations: Part 1
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
'
$
Overview
The second half of the course deals with the application of the

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Set 8: Iterative Methods for Solving
Equations: Part 2
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
'
Examples for Jacobi and Gauss-Seidel
$
3 7 1
3
04
3 3 6
7 4 1 A1 = 7
A2

Homework 3 Foundations of Computational Math 1 Fall
2010
The solutions will be posted on Wednesday, 9/22/09
Problem 3.1
Suppose A Rnn is a nonsymmetric nonsingular diagonally dominant matrix with the
following nonzero pattern (shown for n = 6)
0 0 0 0
0

Solutions for Homework 2 Foundations of Computational
Math 1 Fall 2010
Problem 2.1
Let n = 4 and consider the lower triangular system Lx = f of the form
1
0
00
1
1
21 1
0 0 2 2
31 32 1 0 3 = 3
41 42 43 1
4
4
Recall, that it was shown in class that t

Solutions for Homework 1 Foundations of Computational
Math 1 Fall 2010
Problem 1.1
This problem considers three basic vector norms: . 1 , . 2 , .
1.1.a Prove that .
1
1.1.b Prove that .
.
is a vector norm.
is a vector norm.
1.1.c Consider . 2 .
i Show tha

Foundations of Computational Math I Exam 2
Take-home Exam
Open Notes, Textbook, Homework Solutions Only
Due beginning of Class Wednesday, December 1, 2010
Question
1. Iterative Methods
for Ax = b
2. Iterative Methods
for Ax = b
3. Nonlinear Equations
4. N

Foundations of Computational Math I Exam 1
Take-home Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Tuesday 19 October, 2010
Question
Points
Points
Possible Awarded
1. Basics
15
2. Bases and Orthogonality
20
3. Factorization
30
Com

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Set 13: Line Search and Quasi-Newton for
General Nonlinear Systems
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
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Other Methods for Nonlinear Systems
General line search and

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Set 12: Unconstrained Optimization
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
'
$
Unconstrained Smooth Optimization
Problem 12.1. Given f (x) : Rn R, solve
min f (x)
xRn
to n

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Set 11: Optimization-based Methods for
Systems of Equations
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
'
$
Optimization-based System Solving
Want to solve F (x) = 0, f : Rn

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Set 10: Nonlinear Equations Part 2
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
'
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Systems of Nonlinear Equations
Let F (x) : Rn Rn be a vector-valued function of n variables.

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Set 9: Nonlinear Equations Part 1
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
'
$
Overview
Iteration for nonlinear equations will be discussed:
solving scalar nonlinear equat

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Set 1: Basics
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2010
&
1
%
'
$
Scalars, Vectors and Matrices
Scalars and their operations are assumed to be from
the eld of real numbers (R)
t

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Set 0: Administrivia
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Mathematics 1
Fall 2010
&
1
%
'
Course
$
Time and Place : MWF 10:10 AM 11:00 AM , 201 Love Building
Instructor: K. A. Gallivan (5-0

Homework 1 Foundations of Computational Math 1 Fall
2010
The solutions will be posted on Wednesday, 9/8/09
Problem 1.1
This problem considers three basic vector norms: . 1 , . 2 , .
1.1.a. Prove that .
1
1.1.b. Prove that .
.
is a vector norm.
is a vector

Foundations of Computational Math I Exam 2
Take-home Exam
Open Notes, Textbook, Homework Solutions Only
Due beginning of Class Wednesday, December 1, 2010
Question
1. Iterative Methods
for Ax = b
2. Iterative Methods
for Ax = b
3. Nonlinear Equations
4. N

Foundations of Computational Math I Exam 1
Take-home Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Tuesday 19 October, 2010
Question
Points
Points
Possible Awarded
1. Basics
15
2. Bases and Orthogonality
20
3. Factorization
30
Com

Fall 2010
Foundations of Computational Mathematics 1
MAD 5403
Details
Time and Place : MWF 10:10 AM 11:00 AM , 201 Love Building
Instructor: K. A. Gallivan (5-0306, 318 Love Building, [email protected])
Oce Hours: 9:00 AM 10:00 AM and 11:00 AM 1:00

Solutions to Program 3 Foundations of Computational
Math 1 Fall 2010
1
General Task
In assignment you will implement solvers for nonlinear systems and apply them to a specic
system of 2 nonlinear equations. You will also analyze some of your iterations to