4.3. The QR Reduction
Reading Trefethen and Bau (1997), Lecture 7
The QR factorization of a matrix A 2 <m n is
A = QR
cfw_ Q 2 <m m is an orthogonal matrix
cfw_ R 2 <m n is upper triangular
cfw_ Assume (for the moment) m n
Motivation:
cfw_ An alternate to
4.2. Rotations and Re ections
Reading: Trefethen and Bau (1997), Lecture 10
Example 1. Consider
c
sin
Q = ;os ;os
sin c
cfw_ Q is orthogonal since QT Q = I
cfw_ y = QT x is a rotation of x through
Let
x = jjxjj2 cos
Multiply
sin ]T
cos ; sin
; sin
cos
cos
4. Orthogonalization and Least
Squares
4.1. Orthogonality and the
Singular Value Decomposition
Reading Trefethen and Bau (1997), Lecture 2
Orthogonal vectors:
cfw_ Orthogonal bases are better conditioned than others
De nition 1: fx1 x2
xng 2 <m is orthogo
3.3. Block Tridiagonal Systems
Example 2.4 produced the block tridiagonal system of the
form
Ax = b
or
2
A1
6 C2
6
6
4
B1
A2 B2
.
Cn An
(1a)
32 3 2 3
x
b
7 6 x1 7 6 b1 7
76 2 7 = 6 2 7
7 6 . 7 6 . 7
54 5 4 5
xn
bn
cfw_ Ak , Ck, Bk, = 1 : , are
cfw_ The bl
3.2. Banded and Pro le Systems
If A 2 <n n has lower bandwidth p and upper bandwidth q
then
cfw_ L has lower bandwidth p
lij = 0 for j > i and i > j + p
cfw_ U has upper bandwidth q
uij = 0 for i > j and j > i + q
Example 1. A 6 6 matrix with p = 2 and q
3. Special Linear Systems
3.1. Symmetric Positive
De nite Systems
Reading: Trefethen and Bau (1997), Lecture 23
Simplify Gaussian elimination when A has special properties
cfw_ Symmetry
cfw_ positive de niteness
cfw_ Banded structure
cfw_ Sparsity
cfw_ Bl
2.5. Accuracy Estimation
Reading: Trefethen and Bau (1997), Lecture 22
Accuracy:
cfw_ With roundo , how accurate is a solution obtained by
Gaussian elimination?
cfw_ How sensitive is a solution to perturbations introduced by
round o error (stability)?
cfw
2.4. Pivoting
Reading: Trefethen and Bau (1997), Lecture 21
The Gaussian factorization and backward substitution fail
when u = 0, i = 1 : n
cfw_ The system need not be singular, e.g.,
01
11
ii
cfw_ The factorization can proceed upon a row interchange
i.e.
2.3. Factorizations
Reading: Trefethen and Bau (1997), Lecture 20
Solve the n n linear system
Ax = b
(1)
by Gaussian elimination
cfw_ Gaussian elimination is a direct method
The solution is found after a nite number of operations
cfw_ Iterative methods nd
2.2. Finite Precision Arithmetic
Reading: Trefethen and Bau (1997), Lectures 12-15
Real numbers are approximated by oating point numbers
Real number:
e
x = d0:d1d2 dt;1dt
(1a)
The base of the number system
di -digits, 0 di <
d0 6= 0 unless x = 0
e Exponen
1.2. Special Matrices
Complex Matrices
cfw_ A 2 C m n implies that aij are complex numbers
aij =
ij +i ij
ij
i = 1 2 : m
ij 2 <
cfw_ Operations are the same except for:
Transposition: if aij =
ij + i ij
j = 1 2 : n
then
C = AH = AT cij = aji =
Inner produ