Solutions for Homework 5 Foundations of Computational
Math 1 Fall 2011
Problem 5.1
Consider the following numbers:
122.9572
457932
0.0014973
5.1.a. Express the numbers as oating point numbers with = 10 and t = 4 using
rounding to even and using choppin

Homework 1 Foundations of Computational Math 1 Fall
2013
Problem 1.1
Consider the following numbers:
122.9572
457932
0.0014973
1.1.a. Express the numbers as oating point numbers with = 10 and t = 4 using
rounding to even and using chopping.
1.1.b. Expr

Homework 2 Foundations of Computational Math 1 Fall
2013
Problem 2.1
Let S1 Rn and S2 Rn be two subspaces of Rn .
2.1.a. Suppose x1 S1 , x1 S1 S2 . x2 S2 , and x2 S1 S2 . Show that x1 and
/
/
x2 are linearly independent.
Solution:
Proof by contradiction.

Homework 3 Foundations of Computational Math 1 Fall
2013
Problem 3.1
Let n = 4 and consider the lower triangular system Lx = f of the form
1
1
1
0
00
21 1
0 0 2 2
31 32 1 0 3 = 3
4
4
41 42 43 1
Recall, that it was shown in class that the column-orie

Homework 4 Foundations of Computational Math 1 Fall
2013
Problem 4.1
Suppose A Rnn is a nonsymmetric nonsingular diagonally dominant matrix with the
following nonzero pattern (shown for n = 6)
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0000
It is known that a di

MAD 5403 Program 5
Morgan Weiss
Written Exercise
Consider the family of linear one-step methods defined by
yn = yn1 + h(fn + (1 )fn1 )
where 0 1.
1) Identify three well-known methods that are in this family and the associated values of .
Proof. The genera

Set 0: Administrivia
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Mathematics 1
Fall 2014
1
Course
Time and Place : MWF 10:10 AM 11:00 AM , 201 Love Building
Instructor: K. A. Gallivan (5-0306, 318 Lo

Set 1: Representation, Conditioning and
Error
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2014
1
Finite Precision
All discussions must make clear the assumptions made about the
numbers use

Homework 5 Foundations of Computational Math 1 Fall
2013
Problem 5.1
Let x and y be two vectors in Rn .
5.1.a. Show that given x and y the value of x y
min =
2
is minimized when
xT y
yT y
5.1.b. Show that x = ymin + z where y T z = 0, i.e., x is easily wr

Program 1 Foundations of Computational Math 1 Fall
2013
1.1
During the semester you will nd that changing the precision of your variables and constants
is useful for many reasons. As a result, making this task easy and being able to determine
the associat

Homework 7 Foundations of Computational Math 1 Fall
2013
Problem 7.1
Let A Rnn be a symmetric positive denite matrix, C Rnn be a symmetric nonsingular
matrix, and b Rn be a vector. The matrix M = C 2 is therefore symmetric positive denite.
Also, let A = C

Homework 6 Foundations of Computational Math 1 Fall
2011
Problem 6.1
Suppose you are attempting to solve Ax = b using a linear stationary iterative method
dened by
xk = Gxk1 + f
that is consistent with Ax = b.
Suppose the eigenvalues of G are real and suc

Solutions for Homework 7 Foundations of Computational
Math 1 Fall 2012
Problem 7.1
7.1.a
Let f (x) = x3 3x + 1. This polynomial has three distinct roots.
(i) Consider using the iteration function
1
1 (x) = (x3 + 1)
3
Which, if any, of the three roots can

Solutions for Homework 8 Foundations of Computational
Math 1 Fall 2012
Problem 8.1
Textbook, page 330, Problem 6
Solution:
We are given G : R3 R3 to dene the iteration used to solve a system of nonlinear
equations:
x(k+1) = G(x(k) )
(k )
(k )
(k )
1
1
1

Homework 9 Foundations of Computational Math 1 Fall
2012
Problem 9.1
Let A Rnn be a symmetric positive denite matrix, C Rnn be a symmetric nonsingular
matrix, and b Rn be a vector. The matrix M = C 2 is therefore symmetric positive denite.
Also, let A = C

Comments on Program 1 Foundations of Computational
Math 1 Fall 2012
This note oers some basic comments on points concerning the solution to Program 1 that
sometimes cause problems.
1
Basic Test Routines
In order to assess the factorization produced by you

Program 2 Foundations of Computational Math 1 Fall
2013
Comments on banded LU factorization program:
Let A Rnn be a nonsymmetric and nonsingular matrix with zero/nonzero element
structure that has nonzero elements on the main diagonal, i.e., i,i = 0, the

Homework 6 Foundations of Computational Math 1 Fall
2013
Problem 6.1
Suppose you are attempting to solve Ax = b using a linear stationary iterative method
dened by
xk = Gxk1 + f
that is consistent with Ax = b.
Suppose the eigenvalues of G are real and suc

Homework 8 Foundations of Computational Math 1 Fall
2013
Problem 8.1
8.1.a
Let f (x) = x3 3x + 1. This polynomial has three distinct roots.
(i) Consider using the iteration function
1
1 (x) = (x3 + 1)
3
Which, if any, of the three roots can you compute wi

Set 2: Finite Arithmetic and Numerical
Stability
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2014
1
Sources
Additional sources for this set:
Matrix Algorithms, Volume 1: Basic Decompositio

Set 5: Linear Least Squares
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2014
1
Simple Example
Arithmetic Mean: =
1
n
n
i=1
i
n
(i )2
= argmin f () = argmin
R
R
i=1
n
2(i ) and f (min ) = 0

$
'
Set 13: Nonlinear Equations Part 3
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2014
&
%
1
$
'
Local Model-based Methods
Recall, we used the idea of replacing a scalar root nding problem

$
'
Set 11: Nonlinear Equations Part 1
Kyle A. Gallivan
Department of Mathematics
Florida State University
Foundations of Computational Math 1
Fall 2014
&
%
1
$
'
Overview
Iteration for nonlinear equations will be discussed:
solving scalar nonlinear equa

Fall 2014
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: 8:00 10:00, 11:00 12:00 on Monday,

Homework 2 Foundations of Computational Math 1 Fall
2014
Problem 2.1
Consider the vector space R4
2.1.a. Specify a subspace of R4 with dimension 2 by giving a basis for the subspace.
2.1.b. Show that the basis for a subspace is not unique by giving anothe

Homework 1 Foundations of Computational Math 1 Fall
2014
Problem 1.1
Suppose the n-bit 2s complement representation is used to encode a range of integers,
2n1 x 2n1 1.
1.1.a. If x 0 then x is represented by bit pattern obtained by complementing all
of the

Homework 3 Foundations of Computational Math 1 Fall
2014
Problem 3.1
Recall that a unit lower triangular matrix L Rnn is a lower triangular matrix with diagonal
elements eT Lei = ii = 1. An elementary unit lower triangular column form matrix, Li , is
i
an

Homework 4 Foundations of Computational Math 1 Fall
2014
Problem 4.1
Let A Rnn and its inverse be partitioned
A=
A1 =
A11 A12
A21 A22
A11 A12
A21 A22
where A11 Rkk and A11 Rkk .
4.1.a. Show that if, S = A22 A21 A1 A12 , the Schur complement of A with resp

Homework 6 Foundations of Computational Math 1 Fall
2014
Problem 6.1
6.1.a. Textbook page 241, Problem 2
6.1.b. Textbook page 241, Problem 4
6.1.c. Textbook page 241, Problem 5
Material in textbook Sections 1.7 and 5.1 is useful for these problems.
Proble