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Computer Science/Mathematics 550
Numerical Linear Algebra
Assignment Four
Due 28 February 2012
1. The Lanczos Process. Recall the symmetric Lanczos algorithm. We
choose v1 where v1 2 = 1.
T
a1 = v1 Av1
f1 = Av1 a1 v1 ;, b1 = f1 2 ;
v2 = f1 /b1 ;
for k = 1
Computer Science/Mathematics 550
Numerical Linear Algebra
Assignment Three
Due 14 February 2012
This entire homework will take the full rank least squares problem
yLS = arg minyRn b X y
2
2
and compare it to the perturbed solution
zLS = arg minyRn b + b (
Computer Science/Mathematics 550
Numerical Linear Algebra
Assignment Two
Due 31 January 2012
For problems one and two, let X Rmn have the complete orthogonal
decomposition
X = U CV T
=
k
U1
mk
U2
C11 0
00
= U1 C11 V1T
k
nk
V1T
V2T
where C11 is k k nonsing
Computer Science/Mathematics 550
Numerical Algebra
Assignment One
Due 19 January 2012
1. Let be a norm on Rn such that ei = 1, i = 1, 2, . . . , n where ei
is the ith column of the identity matrix. Show the following:
(a) x x 1 for all x Rn .
(b) Add the
nxFXFxX6 xc FrXxkFvuq
d n r r b y
wp s tq w q yw y q vp q vsw b y b b j y b
rh6|pcXgF4xvbqmsvq$qwmbr$cccrpcmXbrl|qri|pmcfw_r
pb q b r b y w s p y b b u
[email protected]`4c`$
b p j b` Y n b p y y j b y s b y i y bq b
ryqr2ycasWmjbrc`2ljclk
Chapter 3
Matrix Computational Concerns
We give some basics on the analysis and design of numerical algorithms for
linear algebra computations in oating point arithmetic. Basic notions about
computing in oating point arithmetic are given in 3.1. In 3.5, w
4.2 The Givens Orthogonal Factorization Let the upper triangular factor R be thought of as being produced by the recurrance p m-k R11 0
103
R1
=
1 = Q T X1
(4.69)
tk-1 m - tk-1
Sk ~ Xk
= QT QT Xk , tk = kp k-1 1 k t = QT k-1 m - tk-1 Sk ~ Xk
(4.70)
tk-1
Chapter 4
QR Factorization Methods
When a matrix X Rmn has full column rank, it can be factored into
X=Q
R
0
(4.1)
where Q Rmm is orthogonal and R Rnn is upper triangular and nonsingular. This factorization, called the QR decomposition, is a complete
orth
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cfw_z
2g ( g r$jir!
( li k f h gg f
s l z h i k g f z i f i cfw_ h s q p cfw_ z s q p l z fh i z l cfw_ f g
!2jf26!mrje r2g~ 2!eHr2$
vre!Vi e$eege$02ge!$igji
~ ps q p n f z fh i f l z fh i kh f
x qz ! p
y
I v y
#
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g f cF cH H a f S g E Q F
t o v I v y
y
H Qf Q k i
[email protected]
u
y y
F E C H G a g H S g U c H Si F Qf c C S F S C g G F
2R6AhcPwhiPPr96R!p2me2hHm$g xham$g R!k
Q g q F
QUQ C
'
dUc
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Computer Science/Mathematics 550 Numerical Linear Algebra Final Exam Due 27 April 2012, 5pm No late exams, please! 1. Exercise 3.4, p.49, Hansen book. Choose n = 16, 32, 64. 2. Distance from Orthogonality. Let X, U Rmn and let C, V Rnn . Suppose that C is
Computer Science/Mathematics 550
Numerical Linear Algebra
Assignment Five
Due 29 March 2012
1. Consider the n n symmetric tridiagonal matrix
T =
b
a
b
.
.
.
. .
.
.
.
.
.
.
.
.
0
b
a
b
0
b
a
a
b
0
.
.
.
b
a
b
.
.
.
Show that
T vk = k vk
where
vk =
2