1.
Part 4b:
NUMERICAL LINEAR ALGEBRA
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LU Decomposition
Matrix Inverse
System Condition
Special Matrices
Gauss-Seidel
2.
LU Decomposition:
In Gauss elimination, both coefficieints and constants are
munipulated until an upper-triangular matrix is obtained.
a11 x1 a12 x2
b1
'
'
a23 x3 ... a2 n xn
'
b2
''
'
a33 x3 ... a3' n xn
'
a22 x2
a13 x3 ... a1n xn
b3''
...
(n
ann 1) xn
(
bnn
1)
In some applications, the coefficient matrix [A] stays constant while
the right-hand-side constants vector (b) changes.
[L][U] decomposition does not require repeated eliminations. Once
[L][U] decomposition is applied to matrix [A], it can be repeteadly
used for different values of (b) vector.
3.
Decomposition methodology:
Solution to the linear system
A x
or
b
A x
b
0
The system can also be stated in an upper triangular form:
U x
d
or U x
d
0
Now, suppose there exist a lower triangular matrix (L) such that
L U x
d
A x
b
Then, it follows that
L U
A
and
L d
b
Solution for (x) can be obtained by a two-step strategy (explained
next).
4.
Decomposition strategy:
A x
b
Decomposition
U
L
L (d )
Apply backward
substitution to
calculate (x)
U ( x)
(b)
Apply forward
substitution to
calculate (d)
(d )
The process involves one decomposition, one forward
substitution, and one backward substitution processes.
Once matrices L and U are computed once; manipulated constant
vector (d) is repeatedly calculated from matrix L; hence vector (x).
5.
LU Decomposition and Gauss Elimination:
Gauss elimination processes involves an LU decomposition in itself.
Forward elimination produces an upper triangular matrix:
.. .. ..
U
0 .. ..
0 0 ..
In fact, while U is formed during elimination, an L matrix is formed
such that (for 3x3)
1
0
f 21
1
0
f 31
L
0
f 32
1
where
f 21
a21
a11
a31
a11
f 31
A
f 32
L U
'
a32
a22
…
This decomposition is unique when
the diagonals of L are ones.
6.
EX 10.1 : Apply LU decomposition based on the Gauss elimination for Example
9.5 (using 6 S.D.):
Coefficient matrix:
3
0.2
0.1
7
0.3
0.3
A
0.1
0.2
10
Forward elimination resulted in the following upper triangular form:
3
0.1
0.2
U
0 7.00333
0.293333
0
0
Lower triangular matrix will have
L
1
f 21
f 31
0
1
f 32
0
0
1
1
a21
a11
a31
a11
10.0120
0
0
1
0
'
a32
'
a22
1
1
0.0333333
0.100000
0
0
1
0
0.0271300 1
7.
Check the result:
1
A
0 3
0.0333333
L U
0
1
0 0 7.00333
0.100000
0.0271300 1 0
We obtain:
0
0.2
0.293333
10.0120
compare to:
3
A
0.1
0.1
7
0.0999999
0.3
0.2
0 .3
0.2 9.99996
3
0.2
0.1
7
0.3
0.3
A
0.1
0.2
10
Some round-off error is introduced
To find the solution:
Calculate (d) by applying one forward substitution.
L (d )
(b)
Calculate (x) by applying one back substitution.
U ( x)
(d )
[L] facilitates
obtaining modified
(b) each time (b) has
been changed during
calculations.
8.
EX 10.2: Solve the system in the previous example using LU decomposition:
We have:
A
1
L U
0
0 3
0.0333333
1
0 0 7.00333
0.100000
0.0271300 1 0
0.1
0
0.2
0.293333
10.0120
> Apply the forward substitution:
1
0
0 d1
0.0333333
1
0 d2
0.100000
0.0271300 1 d 3
7.85
19.3
71.4
d1
7.85
d2
19.5617
d3
70.0843
> Apply the backward substitution:
3
0. 1
0.2
x1
0 7.00333
0.293333 x2
0
10.0120
0
x3
7.85
19.3
71.4
x1
3
x2
2.5
x3
7.00003
9.
Total FLOPs with LU decomposition
n3
3
O n2
same as Gauss
elimination
Crout Decomposition:
1
A
L U
(Doolittle decomposition/factorizaton)
..
1
1
A
L U
U
(forming)
(Crout decomposition)
..
1
row operation
Column ro operation
They have comperable performances.
Crout decompositon can be implemented by a
concise series of formulas. (see the book).
Storage can be minimized:
L
> No need to store 1’s in U.
(forming)
> No need to store 0’s in L and U.
> Elements of U can be stored in zeros of L.
A
(remaining)
10.
Matrix Inverse
If A is a square matrix,there exist an A-1,s.t.
A A
1
A
1
A
I
LU decomposition offers an efficient way to find A-1.
A x
b
decomposition
U
L
forward
substitution
Backward
substitution
For constant
vector, enter (I:,i )
(ith column of the
identity matrix.)
L (d )
1
:, j
U ( A ) (d )
( I: , j )
Solution gives ith
column of A-1.
11.
EX 10.3 : Use LU decomposition to determine the inverse of the system in EX 10.1
3
0.2
0.1
7
0.3
0. 3
A
0.1
0.2
10
Corresponding upper and lower triangular matrices are
3
U
0.1
1
0 7.00333
0.293333
0
L
10.0120
0
0
0
0.0333333
0.2
1
0
0.100000
0.0271300 1
To calculate the first column of A-1 :
> Forward substitution:
1
0
0.0333333
0.100000
0 d1
1
d1
0 d2
0
d2
0.03333
0.0271300 1 d 3
0
d3
0.1009
1
1
12.
> Back substitution:
3
0.1
0.2
x1
1
x1
0.33249
0 7.00333
0.293333 x2
0.03333
x2
0.00518
0
10.0120
0.1009
x3
0.01008
0
x3
To calculate the second column
First
column
of A-1
To calculate the third column
b1
0
x1
0.004944
b1
0
x1
0.006798
b2
1
x2
0.142903
b2
0
x2
0.004183
b3
0
x3
0.00271
b3
1
x3
0.09988
We finally get
0.33249
A
1
0.004944 0.006798
0.00518 0.142903 0.004183
0.01008
0.00271
0.09988
13.
Importance of Inverse in Engineering Applications:
Many engineering problems can be represented by a linear
equation
A x
System design
matrix
b
Response
(e.g., deformation)
Stimulus
(e.g., force)
The formal solution to this equation
x
A
1
b
For a 3x3 system we can write explicitly
x1
a111b1 a121b2
a131b3
x2
1
1
a21 b1 a22 b2
1
a23 b3
x3
a311b1 a321b2
a331b3
There is a linear relationship
between stimulus and response.
Proportionality constants are the
coefficients of A-1 .
14.
System Condition
Condition number indicates ill-conditioning of a system.
We will determine condition number using matrix norms.
Matrix norms:
A norm is a measure of the size of a multi-component entity
(e.g., a vector)
x3
n
x1
xi
1-norm
i 1
x
1/ 2
n
x
x
2
2
i
x
e
i 1
n
x
xi
p
i 1
2-norm
(Euclidean norm)
x1
1/ p
p
2
p-norm
x2
15.
We can extend Euclidean norm for matrices:
n
1/ 2
n
Ae
2
a i, j
(Frobenius norm)
i 1 j 1
There are other norms too…, e.g.,
n
A
max
1 i n
aij
(row-sum norm)
aij
(column-sum norm)
j 1
n
A
max
1 j n
i 1
Each of these norms returns a single (positive) value for the
characteristics of the matrix.
16.
Matrix Condition Number:
Matrix condition number can be defined as
Cond A
A
A
1
( Cond A 1 )
If Cond [A] >> 1
ill-conditioned matrix
It can be shown that
i.e., the relative error of the
x
x
Cond A
A
A
For example;
[A]
contains element of t S.F.
(precision of 10-t)
Cond [A] 10c
computed solution cannot
be larger than the relative
error of the coefficients of
[A] multiplied by the
condition number.
(x) will contain
elements of (t-c) S.F.
(precision of 10c-t)
17.
EX 10.4 : Estimate the condition number of the 3x3 Hilbert matrix using row sum
norm
1 1/ 2 1/ 3
Hilbert matrix is
A
inherently illconditioned.
1/ 2 1/ 3 1/ 4
1/ 3 1/ 4 1/ 5
First normalize the matrix by dividing each row by the largest coefficient:
1 1/ 2 1/ 3
A
1 2 / 3 1/ 2
1 3/ 4 3/ 5
Row-sum norm:
1 1/ 2 1/ 3
A
1 1/ 2
1/ 3 1.833
1 2 / 3 1/ 2
1
2 / 3 1/ 2
2.1667
1 3/ 4 3/ 5
1
3/ 4
2.35
3/ 5
A
2.35
18.
Inverse of the scaled matrix:
9
A
1
36
30
18
96
90
this part takes
the longest
time of
computation.
10
60
60
Row-sum norm:
9
A
1
36
18
96
30
90
10
A
60
1
36
96
60 192
60
Condition number:
Cond A
(2.35)(192 )
451 .2
matrix is ill-conditioned.
e.g., for a single precision (7.2 digits) computation;
c log(451.2) 2.65
(7.2-2.65)=4.55 ~ 4 S.F. in the solution!
(precision of ~10-4)
19.
Iterative refinement:
This technique especially useful for reducing round-off errors.
Consider a system:
a11 x1 a12 x2
a13 x3
b1
a21 x1 a22 x2
a23 x3
b2
a31 x1 a32 x2
a33 x3
b3
Assume an approximate solution of the form
a11 x1o
o
a12 x2
o
a13 x3
b1o
a21 x1o
o
a22 x2
o
a23 x3
o
b2
a31 x1o
o
a32 x2
o
a33 x3
o
b3
We can write a relationship between exact and approximate
solutions:
x1
x1o
x1
x2
o
2
x
x2
x3
o
x3
x3
20.
Insert these into the original equations:
a11 ( x1
x1 ) a12 ( x2
x2 ) a13 ( x3
x3 )
b1
a21 ( x1
x1 ) a22 ( x2
x2 ) a23 ( x3
x3 )
b2
a31 ( x1
x1 ) a32 ( x2
x2 ) a33 ( x3
x3 )
b3
Now subtract the approximate solution from above to get
a11 x1 a12 x2
a13 x3
b1 b1o
a21 x1 a22 x2
a23 x3
o
b2 b2
e2
a31 x1 a32 x2
a33 x3
o
b3 b3
e3
e1
This a new set of simultaneous linear equation which can be
solved for the correction factors.
Solution can be improved by applying the corrections to the
previous solution (iterative refinement procedure)
It is especially suitable for LU decomposition since constant
vector (b) continuously changes.
21.
Special Matrices
In engineering applications, special matrices are very common.
> Banded matrices
aij
0
BW=3
if
i
j
(BW 1) / 2
tridiagonal system
> Symmetric matrices
aij
a ji
or
A
A
T
> Spare matrices (most elements are zero)
only black areas
are non-zero
BW
22.
Application of elimination methods to spare matrices are not
efficient (e.g., need to deal with many zeros unnecessarily).
We employ special methods in working with these systems.
Cholesky Decomposition:
This method is applicable to symmetric matrices.
A symmetric matrix can be decomposed as
i 1
aki
A
L L
T
or
lki
lij lkj
j 1
for i 1,2,...,k 1
lii
k 1
lkk
2
lkj
akk
j 1
Symmetric matrices are very common in engineering applications. So, this
method has wide applications.
24.
Gauss-Seidel
Iterative methods are strong alternatives to elimination methods
In iterative methods solution is constantly improved so there is no
concern of round-off errors.
As we did in root finding,
> start with an initial guess.
> iterate for refined estimates of the solution.
Gauss-Seidel is one of the most commonly used iterative method.
For the solution of
A ( x) (b)
We write each unknown in the diagonal in terms of the other
unknowns:
25.
In case of a 3x3 system:
x1
x2
x3
b1 a12 x2 a13 x3
a11
Start with initial guesses x2 and x3
b2
Use new x1 and old x3
a21 x1 a23 x3
a11
b3 a31 x1 a32 x2
a33
Use new x1 and x2
calculate new x1
calculate new x2
calculate new x3
iterate...
In Gauss Seidel, new estimates are immediately used in
subsequent calculations.
Alternatively, old values (x1 , x2 , x3) are collectively used to
calculate new values (x1 , x2 , x3) Jacobi iteration (not
commonly used)
27.
For the second iteration, we repeat the process:
x1
7.85 0.1( 2.794524) 0.2(7.005610)
3
x2
19.3 0.1(2.990557) 0.3(7.005610)
7
x3
71.4 0.3(2.990557) 0.2( 2.499625)
10
2.990557
2.499625
7.000291
The solution is rapidly converging to the true solution.
t
0.31%
t
0.015%
t
0.0042%
28.
Convergence in Gauss-Seidel:
Gauss-Seidel is similar to the fixed-point iteration method in root
finding methods. As in the fixed-point iteration, Gauss-Seidel
also is prone to
> divergence
> slow convergence
Convergence of the method can be checked by the following
criteria:
n
aii
aij
j 1
j i
that is, the absolute value of the diagonal
coefficient in each of the row must be
larger than sum of the absolute values of
all other coefficients in the same row.
(diagonally dominant system)
Fortunately, many engineering applications fulfill this
requirement.
29.
Improvement of convergence by relaxation:
After each value of x is computed using Gauss-Seidel equations,
the value is modified by a weighted average of the old and new
values.
xinew
xinew (1
) xiold
0
2
If 0< <1
underrelaxation (to make a system converge)
If 1< <2
overrelaxation (to accelerate the convergence )
The choice of
is empirical and depends on the problem.
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