Circular Interval Arithmetic Applied on LDM
T
for
Linear Interval System
Stephen Ehidiamhen Uwamusi
Department of Mathematical SciencesFaculty of Natural Sciences, Kogi State University Anyigba, Kogi State, Nigeria
Email: [email protected]
Abstract–The paper considers the LDMTFactorization of a general nxn matrix arising from system of interval linear equations. We paid special emphasis on Interval Cholesky Factorization. The basic computational tool used is the square root method of circular interval arithmetic in a sense analogous to Gargantini and Henrici as well as the generalized square root method due to Petkovic which enables the construction of the square root of the resulting diagonal matrix. We also made use of Rump’s method for
multiplying two intervals expressed in the form of midpoint-radius respectively. Numerical example of matrix factorization in this regard is given which forms the basis of discussion. It is shown that LDMT even though is a numerically stable method for any diagonally dominant matrix it also can lead to excess width of the solution set. It is also pointed out that in spite of the above mentioned objection to interval LDMTit has in addition , the advantage that in the presence of several solution sets sharing the same interval matrix the LDMT Factorization requires to be computed only once which helps in saving substantial computational time. This may be found applicable in the development of military hard ware which requires shooting at a single point but produces multiple broadcast at all other points.
Keywords – Circular Interval Arithmetic, Linear System, LDMT, Cholesky Factorization, Multiple Broadcast.
I. I
NTRODUCTIONThe paper presents interval based circular arithmetic for the solution of the interval linear system of equation
Ax = b (1.1)
Given [A]=
[
a
ij]
I
(
R
nxn),
[
b
]
I
(
R
n)
,we aim to characterize and enclose the solution set]}
[
],
[
,
{
x
R
Ax
b
A
A
b
b
S
n
(1.2)and the symmetric solution set
sym
S
={xRn Axb,A AT [A][A]T,b[b]}. (1.3) These sets often occur when measurements in experimental data are made under uncertainty. We usually are able to tackle this problem only if the matrix A is nonsingular by employing interval arithmetic. As a follow up to this discussion, we will expect the readers to be acquainted with the basics of interval arithmetic as described for instance in [1], [2], and [3].Thus in the presentation of our method we follow [4].
II. C
IRCULARI
NTERVALA
RITHMETICIn this paper emphasis will be placed on the applications of circular interval arithmetic due to ici [2] which also applied in [5] , and [4].We define a circular disk as follows:
Definition ([2]:
A circular disk (circular region), denotes a closed point set in the extended complex plane whose projection on the Riemann Sphere is bounded by a circle. Depending on whether the point is infinity, an exterior point, a boundary point or an interior point, a circular region thus is either a closed disk, a closed half-plane or the complement of an open disk. For example, the set of a disk x with centre a and radius r is written as
x
(
a
,
r
)
.In order to work in the realm of circular interval arithmetic we first of all express the coefficients of interval matrix in the form of midpoint-radius matrix. Therefore given coefficients entries of an interval matrix
,
[ , ] [ ] [ , ]
ij ij i j
A a a
a a r
Where
1 2
1 2
( )
( )
ij ij
ij ij
a a a
r a a
the operations for midpoint radius representation [5] given that A=<a, r> and B = <b, r2> are as follows:
A+B =
a
b
,
r
1
r
2
A-B =
a
b
,
r
1
r
2
AB =
a
.
b
,
a
r
2
b
r
1
r
1r
2
B
1
=
D
b
r
B
D
r
D
b
,
,
20
2 2
It follows that
B
B
A
B
A
0
),
1
(
All operations are inclusion isotone. For the sake of completeness we will include in our work the computation of power and square root of a disk (see e.g. [6], and [7]):
}
)
(
,
{
,
,
1
n n n
j j n n
j n n
a
r
a
a
r
a
j
n
a
r
a
Copyright © 2014 IJECCE, All right reserved
)
,
(
)
(
1
)
(
}
,
({
1 1 1
1
r
a
N
k
r
a
k
j
k
a
r
r
a
a
r
a
rad
k j
k
o j
k k
k
(1.7)
The disk inversion adopted is taken to be
2 2 2 2 1
,
,
)
1
(
1
r
a
r
a
r
a
a
ij , see for instance, [8].Our motive in this paper is to use the above mentioned properties of circular interval arithmetic to compute the
T
LDM
Factorization .III. I
NTERVALB
ASEDD
IRECTL
INEARS
OLVERS: LDM
TF
ACTORIZATIONIn this section we discuss some well known direct linear solvers in the form of interval arithmetic computation.
The LU decomposition of a square matrix A consists of computing two matrices L and U respectively, where L is the lower triangular matrix and U is the upper triangular matrix such that A=LU. The major steps in the construction of an LU decomposition is the Gauss elimination process which has been extended to interval matrices, [9], [10] for example. In this process an interval LU decomposition can be introduced which may not satisfy A=LU exactly but only the weaker relation
LU
A
that is suitable for the enclosure of inner approximation of the tolerance solution sets. This is due to fact that the space of real intervals is not a linear space, [10] .It may be imperative to find out the necessary and sufficient conditions for the applications of Gauss elimination process assuming that the interval matrix A has no degenerate entries.
As pointed out in [11]), interval Gauss elimination is feasible if [A] is an M- matrix. The feasibility of interval Cholesky Factorization is dependent on the existence of interval Gaussian algorithm.
We present here interval Cholesky Factorization :
Algorithm
Step 1:
“LL
T Decomposition’’: For j= 1 to n do2 1 1
1 2
]
[
]
[
]
[
j
k jk jj
jj
a
l
l
for i = j+1 to n do
jj jk j
k ik ij
ij
l
l
l
a
l
]
[
]
[
]
[
]
[
1
Step 2:
Forward substitution For i= 1 to n do]
[
]
[
]
[
]
[
]
[
ii j ij i
i
l
y
l
b
y
Step 3:
Back substitution For i= n down to 1 doii n
i
c j ji i
c
l
x
l
y
x
1]
[
]
[
]
[
]
[
,ICh ([A],[b])=
[
x]
cHaving obtained
LL
T decomposition of A we will now solve forLy = b
y
x
L
T
where it is assumed that
L
ii
0
, i=1,2, …,n.The LU decomposition has the tendency that L tends to be well conditioned where as the condition number of the matrix A moves in U, [12], and [13]. We will describe the LDMT as a kind of Preconditioner for the solution of the interval linear system of equation (1.1). Here the interval matrix A is decomposed into three parts namely, L ,D, M are both unit lower triangular matrices, D is the diagonal part of the matrix. Usually we distribute the matrix D between L and M such that
2 1
2 1
ˆ
,
ˆ
)
(
,
MD
U
LD
L
D
D
sign
D
D
D
(3.1)
for symmetric positive definite matrix A.
On the other hand, once the LDMTdecomposition of A is obtained, the solution to system (1.1) may be found by solving the following 3 steps :
flop
z
x
M
flops
n
y
Dz
flops
b
Ly
n T
n
2 2
2 2
,
,
,
(3.2)
Following [13], it can be shown that the computed solution x to the system Ax=b obtained via the LDMT is bounded by
b
x
H
A
)
(
)
(
0
]
ˆ
ˆ
5
3
[
A
L
D
M
U
2nU
H
T
(3.3)M
and
D
L
ˆ
,
ˆ
ˆ
are the computed versions of L, D and M respectively.On the other hand, given that L=M, in the case of symmetric positive definite matrix A, the diagonal elements coming from the entries of U will also be positive definite.
nn
U
U
U
D
22 11
2 1
(3.4)
and by letting
T T
L
L
LDL
A
obtain
we
LD
L
1 1
1
,
2 1
(3.5) The factorization given in (2.12) is known as Cholesky Factorization of A.
Below we give our algorithmic structure of
LDM
TFactorization
Algorithm
(1) Input the matrix A and the right hand side vector b. (2) Compute the
inf
lation
for the matrix A. (3) Obtain theLDM
T Factorization for the intervalmatrix A by using the circular interval arithmetic described in section 1.
(4) Obtain the square root of the matrix D where D=
Diag
[
A
11],
[
A
22],
.
.
.
[
A
nn]
using procedure described in section 1.(5) Form
L
L
D
andU
sign
(
D
)
D
.
M
(6) Verify the bound for
N(
L
)
computing2
K
U
L
by inverse power iteration, k<1 (see e.g. [14]).(7) Provide inclusion for the solution to the interval linear system by using equation 3.2 or any other simplified form of it.
IV. N
UMERICALE
XAMPLESConsider the following two real matrices taken from [15] as problems 1 and 2:
Example 1:
Consider the linear interval system Ax = b
where
A =
0
.
1
0
.
1
0
.
8
0
.
3
5
.
1
0
.
4
8
.
0
2
.
4
6
.
1
,
1
1
1
b
We will give an nxn matrix in form of LDMT from the given system using the procedure described in sections 1 and 2.
Using
-inflation approach where
=
,
(see, e.g., [9]), we are able to transform the given real point matrix to the form of Rump’s midpoint radius matrix wherein we take
=1% .We simply allowed the coefficients of the matrix A in the given problem to be subjected to an uncertainty of 1% tolerance while the right hand side is left unperturbed. We thus have the following decomposition of A as LDMTwith results given as shown below.[1,1] [0, 0 ] [0, 0]
[0.50000078, 0.001876564] [1,1] [0, 0]
[0.2000000312, 0.001501845] [ 0.500005907, 0.005927785] [1,1]
L
[8.0, 0.01] [0, 0] [0, 0]
[0, 0] [ 3.999999969, 0.013516863] [0, 0]
[0, 0] [0, 0] [1.999993298, 0.029621757]
D
[1,1] [ 0.125000195, 0.00140781445] [0.125000195, 0.00140781445]
[0, 0] [1,1] [0.250002875, 0.004235496]
[0, 0] [0, 0] [[1,1]
U
Results for problem 1 are displayed in table 1.
Table 1: showing results obtained by using interval
T
LDM
for problem 1 RESULTSMidpoint Vector x, Rad Vector x
[ 0.027342962, 0.004562175]
[-0.25625207, 0.00657419]
[0.525003024, 0.018342856]
Example
2”
Given a matrix A defined by
A=
14
3
2
3
2
2
2
2
4
, we take
1% where
is an inflation parameter for the coefficients of the matrix A.Copyright © 2014 IJECCE, All right reserved
[2.000000000, 0.002501564] [0, 0] [0, 0]
1.000025000, 0.010025249] [0.999974999, 0.015191520] [0, 0]
[1.000025000, 0.0101002525] [ 2.000361656, 0.010102525] [2.999750530, 0.054927066]
L
T
L
=
]
054927066
.
0
,
999750530
.
2
[
]
000000000
.
0
,
000000000
.
0
[
]
000000000
.
0
,
000000000
.
0
[
]
010102525
.
0
,
000361656
.
2
[
]
015191520
.
0
,
999974999
.
0
[
]
000000000
.
0
,
000000000
.
0
[
]
0101002525
.
0
,
000025000
.
1
[
]
010025249
.
0
,
000025000
.
1
[
]
002501564
.
0
,
000000000
.
2
[
Traditional Cholesky Factorization of problem 2 gives
L =
L
D
=
3
2
1
0
1
1
0
0
2
,
T
L
=
3
0
0
2
1
0
1
1
2
The solution to interval system of problem 2 is obtained in the form
b
L
L
x
T1 1 and is presented in table 2 below.Problem 3:
Consider the linear system Ad = b,
where
A=
9
3
2
1
1
6
5
.
0
4
1
2
11
4
3
2
1
10
, d=
4 3 2 1
d
d
d
d
, b=
1
2
1
2
1
In this note we will give an nxn matrix in form of LDMT from the given system using the procedure described in sections 1 and 2 .
Using
-inflation approach where
=
,
in the sense of [9], we are able to transform the given real pointmatrix to the form of Rump’s midpoint radius matrix
wherein we take
=10-2 .We simply allowed the coefficients of the matrix A in the given problem to be subjected to an uncertainty of 1% tolerance while the right hand side is left unperturbed. We thus have the following decomposition of A as LDMTwith results given as shown below.[10, 0.01] [ 1, 0.01] [ 2, 0.01] [ 3, 0.01]
[0, 0] [11.16, 0.01565719] [ 2.32, 0.01127421] [ 0.52, 0.02369]
[0, 0] [0, 0] [5.609039225, 0.023512823] [ 1.495905001, 0.025212022]
[0, 0] [0, 0] [0, 0] [8.267557638, 0.029027503]
U
[1, 0] [0, 0] [0, 0] [0, 0]
[0.16, 0.00401702] [1, 0] [0, 0] [0, 0]
[0.16, 0.00401702] [ 0.030586541, 0.001498657] [1, 0] [0, 0]
[0.04, 0.00049922] [0.18351925, 0.001500692] [ 0.473206769, 0.007018406] [1, 0]
L
1 2
[3.16227766, 0.001581534] [0, 0] [0, 0] [0, 0]
[0, 0] [3.340658618, 0.002344251] [0, 0] [0, 0]
[0.0] [0, 0] [2.368341028, 0.004969199] [0, 0]
[0, 0] [0, 0] [0, 0] [2.875336091, 0.005052109]
D
[1, 0] [ 0.1, 0.001022] [ 0.2, 0.001402] [ 0.3, 0.001602]
[0, 0] [1, 0] [0.207885302, 0.001596379] [0.046594981, 0.002259456]
[0, 0] [0, 0] [1, 0] [0.266695406, 0.006768603]
[0, 0] [0, 0] [0, 0] [1, 0]
T
M
We may as well distribute the diagonal elements between the Lower triangular matrix and the Upper
L
=L
D
We display results for problem 3 as table3
Table 2: showing
LDM
T Results for problem 3X Results:
Midpoint x, radius x 1
x
[ 0.05124,0.0022232
x
[-0.1955, 0.002038]3
x
[0.1023, 0.002923]4
x
[0.2011, 0.001205]V.
CONCLUSIONWe have been able to use our algorithms to factorize the given problems via circular interval arithmetic. It was shown that existence of Interval Cholesky Factorization is dependent on the existence of interval Gaussian algorithm. It is shown that
LDM
T even though is a numerically stable method for any diagonally dominant matrix it also can lead to excess width of the solution set. It is also pointed out that in spite of the above mentioned objection to intervalLDM
T it has in addition , the advantage that in the presence of several solution sets sharing the same interval matrix theLDM
T Factorization requires to be computed only once which helps in saving substantial computational time. This may be found applicable in the development of military hard- ware which requires shooting at a single point but produces multiple broadcast at all other points.R
EFERENCES[1] G.Alefeld, and G. Mayer. Interval analysis: theory and applications. Journal of Computational Mathematics 121, (2000), 421-464
[2] I Gargantini, and P. Henrici. Circular Arithmetic and the determination of polynomial zeros. Numer. Math.18, (1972), 305 -320
[3] S. Ning and RB Kearfott. A comparison of some methods for solving linear interval equations SIAM J. Numer. Anal. 34(1), (1997), 1289–1305
[4] S.Markov. On the Algebraic properties of Intervals and some Applications.Reliable computing 7(2), (2001), 113-127. [5] S.M Rump Fast and Parallel interval arithmetic. BIT 39 (3),
(1999),539-560.
[6] M.S Petkovic. On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic. Computing 27, (1981), 37-55
[7] M.S Petkovic. and L.D Petkovic. Complex interval arithmetic and its applications. John Wiley, ISBN 3-527 –40134– 2 , 1998.
[8] M.S Petkovic, and D.V Vranic. The convergence of Euler-like method for the simultaneous inclusion of polynomial zeros. Computers and Mathematics with Applications. 39 (7-8), (2000), 95-105.
[9] S.E Uwamusi. Towards acceleration of Rump’s Fast and Parallel
Circular interval arithmetic for enclosing solution of non linear system of equations.Scientific Research and Essays 2(11) (2007), 476-481.
[10] A.Neumaier. Interval methods for systems of equations.Cambridge University Press Cambridge, 1990.
[11] G. Alefeld, and G. Mayer. The Cholesky Method for Interval Data. Linear Algebra and Its Applications, 194:, (1993), 161-182.
[12] J. Stoer, and R. Bulirsch. Introduction to Numerical analysis. Springer- Verlag, 1980.
[13] G.H Golub, and C.F Vanloan. Matrix computations. The Johns Hopkins University Press, Baltimore, Maryland, 1983.
[14] S.M Rump. Validated Solution of Large Linear Systems, (in R.Albrecht et al , edts), Validation numerics: theory and applications, vol. 9 of Computing Supplementum, (1993), 191-212.
