Homework 5 Numerical Linear Algebra 1 Fall 2011
Due date: beginning of class Wednesday, 11/9/11
Problem 5.1
Let A Rnk have rank k . The pseudoinverse for rectangular full columnrank matrices
behaves much as the inverse for nonsingular matrices. To see th
Solutions for Homework 3 Numerical Linear Algebra 1
Fall 2011
Problem 3.1
Problem 3.1.6 Golub and Van Loan p. 93
Solution: This is a standard incremental algorithm based on a partitioning of the matrices
involved. The algorithm is O (n2 ) over all when we
Solutions for Homework 4 Numerical Linear Algebra 1
Fall 2011
Problem 4.1
Given that we know the SVD exists for any complex matrix A Cmn , assume that A
Rmn has rank k with k n, i.e., A is real and it may be rank decient, and show that the
SVD of A is al
Numerical Linear Algebra Midterm Exam
Takehome Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
No collaboration with anyone
Due beginning of Class Wednesday, October 26, 2011
Question
1. Structured Schur
complements
2. LU
Factoriza
Comments on Program 1 Numerical Linear Algebra 1 Fall 2011
The conversion process from the generic coordinate form to any of the basic
compressed row or column data structures can be done in a straightforward
way using an approach based on sorting. The co
Fall 2011
Numerical Linear Algebra
MAD 593201
Details
Time and Place : MWF 9:05 AM 9:55 AM , 201 Love Building
Instructor: K. A. Gallivan (50306, 318 Love Building, gallivan@math.fsu.edu)
Oce Hours: 8:00 AM  9:00 AM and 11:00 AM 1:00 PM, MWF and mee
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
#11. http:/en.wikipedia.org/wiki/Monte_Carlo_methods_in_finance http:/www.riskglossary.com/link/monte_carlo_method.htm We call [11] the crude Monte Carlo estimator. Formula [12] for its standard error is important for two reasons. First, it tells us that
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
Chapter 5
Stochastic Dierential
Equations
We would like to introduce stochastic ODEs without going rst through the
machinery of stochastic integrals.
5.1
It Integrals and It Dierential Equations
o
o
Let us start with a review of the invariance principle.
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
I
cfw_hon.l(t 10r. l,)uaa,r ^0tr\ 1
L+n*
tvvL.
o
Snra "grtgul
5g,l*tu 4 t >I
Qjff^tH.*+ (\) w, 1c\rtr\'r6
*/
@ ,!
\/
th. c onVu,/o , /t1rt9,^0
' U 1 ) u Jtb)
./
./
ui(o)' uo
< tnt = T , v t'l *. I ncncqMh
t = L 2'4 b ,'a b z "
\ds = .ttl Cf^ )  uJ[\^
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
FSU MATHSCS, MAD 593201, fall 2006
1 of 2
http:/www.scs.fsu.edu/~rtempone/SDE_course_fall06/MAD5932_01_c.
MAD 593201, fall 2006
Teacher
Raul Tempone (rtempone at scs fsu edu)
[OFFICE HOURS:] Tuesday, Thursday, from 10:00 to 11:0 hrs at Dirac 442.
Addit
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
2a. Formulate and motivate a forward Euler method for approximation of the Stratonovich SDE dXt = a(t; Xt )dt + b(t; Xt ) dW: 2b. Consider the dierential equation
dX
= aXt dt;
where Xt 2 R2 and the matrix a has two real eigenvalues 1 = 1 and 2 = 105. Then
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
3a. Assume that St is the price of a single stock. Derive a Monte Carlo and a PDE method R to determine the price of a contingent claim with the contract 0T ht; St dt, for a given function h, instead of the usual contract maxST K; 0 for European call opti
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
The Monte Carlo method and options models
3a. In the risk neutral formulation a stock solves the SDE
dS=S (t) = rdt + dW (t);
with constant interest rate r and volatility . Show that
S (T ) = S (0)exp(rT 2=2T + W (T );
and use that to simulate the price
f
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
Numerical Methods for SDEs, Fall 2006. Course Instructor: Ral Tempone.
u
Homework Set 1, due Thursday Sept 7.
Last revised, Aug 26, 2006.
Exercise 1 Prove the following identities by taking limits of the Forward Euler method:
T
T
tdW (t) = T W (T )
0
W (
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
Numerical Methods for SDEs, Fall 2006. Course Instructor: Ral Tempone. u
Homework Set 2, due Thursday Sept 14.
Last revised, Aug 26, 2006. Exercise 1
a Consider the ordinary differential equation dXt = A Xt dt where Xt R2 and the matrix A has two real eig
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
Numerical Methods for SDEs, Fall 2006. Course Instructor: Ral Tempone. u
Homework Set 3, due Thursday Sept 21.
Last revised, Aug 27, 2006. Exercise 1 Formulate and motivate a forward Euler method for approximation of the Stratonovich SDE dX(t) = a(t, X(t)
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
Numerical Methods for SDEs, Fall 2006. Course Instructor: Ral Tempone. u
Homework Set 4, due Thursday Sept 28.
Last revised, Aug 30, 2006. Exercise 1 Assume that S (t) is the price of a single stock. Derive a Monte Carlo and a PDE method to determine the
Numerical Linear Algebra for Signals, Systems and Control
MAD 5932

Fall 2006
Minicourse Numerical Methods for SDEs, May 2006. Course Instructors: Ernesto Mordecki and Ral Tempone. u
Homework exercises for the course. Mtodo de aprobacin del Curso e o
Se espera que un 25 por ciento de los ejercicios (a elegir por los estudiantes) s
Solutions for Homework 2 Numerical Linear Algebra 1
Fall 2011
Problem 2.1
Let x Cn and y Cn be two arbitrary vectors. Consider determining a circulant matrix
C Cnn such that
y = Cx
2.1.a. Assume that C exists for a given pair (x, y ), show how to construc
Solutions for Homework 1 Numerical Linear Algebra 1
Fall 2011
Problem 1.1
A matrix A Cnn is nilpotent if Ak = 0 for some integer k > 0. Prove that the only
eigenvalue of a nilpotent matrix is 0.
Solution:
There are multiple ways to prove this. The simples
'
$
Set 11: Orthogonal Bases, Least Squares,
Basic Projections
Kyle A. Gallivan
Department of Mathematics
Florida State University
Numerical Linear Algebra 1
Fall 2011
&
1
%
'
$
Representing a Subspace
There are many ways to represent a subspace S of Rn
ORNL/TM12853
Engineering Physics and Mathematics Division
f!
fi I
:
J
../
B OUNDS FOR DEPARTURE FROM NORMALITY A ND
T HE FROBENIUS NORM O F MATRIX EIGENVALUES
S teven L. Lee
M athematical Sciences Section
Oak Ridge National Laboratory
P.O. Box 2008, Bld
Numerical Linear Algebra Midterm Exam
Takehome Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
No collaboration with anyone
Due beginning of Class Wednesday, October 26, 2011
Question
1. Structured Schur
complements
2. LU
Factoriza
Program 1 Numerical Linear Algebra 1 Fall 2011
Due date: via email by 11:59PM on Friday, 9/30/11
1. Implement a conversion routine that takes a symmetric sparse matrix in coordinate
form and converts it to each of the data structures below.
2. Implement a
Program 2 Numerical Linear Algebra 1 Fall 2011
Due date: via email by 11:59PM on Monday, 10/10/11
1. Implement the CooleyTukey FFT. Compare your results to the DFT as well as library
routines that can be found on the internet and in most computational en
Program 3 Numerical Linear Algebra 1 Fall 2011
Due date: via email by 11:59PM on Monday, 10/31/11
General Task
Implement a code that transforms a symmetric matrix to a symmetric tridiagonal matrix
using the appropriate similarity transformations based on
Program 4 Numerical Linear Algebra 1 Fall 2011
Due date: via email by 11:59PM on Wednesday, 11/16/11
General Task
Dene the random matrix A R5010 via
Udiag (1, 101, . . . , 109 )V T
where U R5050 and V R1010 are random orthogonal matrices. The singular val
'
$
Set 0: Review of Basics
Kyle A. Gallivan
Department of Mathematics
Florida State University
Numerical Linear Algebra
Fall 2011
&
1
%
'
$
Scalars, Vectors and Matrices
Scalars and their operations are assumed to be from
the eld of real numbers (R)
th
'
$
Set 1: Problems and Decompositions of
Interest
Kyle A. Gallivan
Department of Mathematics
Florida State University
Numerical Linear Algebra
Fall 2011
&
1
%
'
Problems of Interest
$
Solving systems
square nonsingular and singular
linear least square
'
$
Set 2: Computational Primitives
Kyle A. Gallivan
Department of Mathematics
Florida State University
Numerical Linear Algebra
Fall 2011
&
1
%
'
$
Overview
Dense matrix primitives BLAS 1, 2, and 3
Sparse matrix primitives
Structured dense matrix prim