1. Iwona Skalna Department of Applied Informatics Cracow, Poland
A method for outer interval solution of parametrized systems of linear equations
Iwona Skalna
Department of Applied Informatics
Cracow, Poland
NFS workshop on Reliable Engineering Computing 2004

2. NFS workshop on Reliable Engineering Computing 2004
Parametrized systems
Consider a family of linear algeraic systems
of the following type
where
and
NFS workshop on Reliable Engineering Computing 2004

3. NFS workshop on Reliable Engineering Computing 2004
Solution set
This family of linear algebraic systems is usually
written in a compact form
and is called parametrized system of linear interval
equations. The solution set of such system is defined as
NFS workshop on Reliable Engineering Computing 2004

4. NFS workshop on Reliable Engineering Computing 2004
New method
Presented here method is based on the following
inclusion
where  denotes an interval hull,
NFS workshop on Reliable Engineering Computing 2004

5. NFS workshop on Reliable Engineering Computing 2004
Algorithm
NFS workshop on Reliable Engineering Computing 2004

6. NFS workshop on Reliable Engineering Computing 2004
Results
In this presentation there is only one example
shown, however in the paper some other
examples may be found.
The results produced by the described method are
compared with the results of the method called
Random Sampling of Parameter Intervals which
gives an inner enclosure of the solution set.
NFS workshop on Reliable Engineering Computing 2004

7. NFS workshop on Reliable Engineering Computing 2004
Example – plane truss y
30kN
30kN
30kN
(0, 9)
(4, 9)
(8, 9)
(12, 9)
(16, 9) 1 4 2 3 5 8 6 7 9
(16, 6)
(4, 6)
(8, 6)
(12, 6) 10 11
(16, 3)
(12, 3) 12
(16, 0) x
NFS workshop on Reliable Engineering Computing 2004

8. NFS workshop on Reliable Engineering Computing 2004
Example - description
The exemplary plane truss structure is subjected
to downward forces of 30 [kN] at nodes 2, 3, 4.
All the bar elements have the same Young’s
modulus E=7e10 [Pa], and the same cross-section
area A=0.003 [m2]. The lengths (in meters) of the
bars are shown in the figure.
Now the stiffness of some of the bars (denoted
in the figure with thick red lines) is assumed to
be uncertain by 5%.
NFS workshop on Reliable Engineering Computing 2004

9. Results – x coordinates
RSPI
rel. err.[%]
Method
dx2
-152.4
[-160.4, ] 5
[-160.4, ]
5.3
dx3
-228.6
[-236.5, ]
3.3
[-236.6, ]
3.5
dx4
[-163.7, ]
7.4
[-165.3, ]
8.5
dx5
-76.2
[-87.5, 65.2]
14.7
[-89.1, -63.3]
16.9
dx6
427.4
[419, 435.6]
1.9
[416.5, 438.3]
2.5
dx7
[417.5, 437.2]
2.3
dx8
351.2
[342.8, 359.4]
2.4
[341.3, 361.1]
2.8
dx9
dx10
267.9
[262.2, 273.6]
2.1
[261, 274.7]
dx11
115.5
[109.8, 121.2]
4.9
[108.6, 122.3]
5.9
NFS workshop on Reliable Engineering Computing 2004

10. Results – y coordinates
[x10-4]
RSPI
rel. err.[%]
Method
dy1
-308.2
[-312.5, ]
1.3
[-315.1, ]
2.2
dy2
-251.2
[-255.5, ]
1.6
[-258.1, ]
2.7
dy3
-149.8
[-152.8, ]
1.9
[-153.4, ]
2.4
dy4
-37.1
[-37.9, -36.4]
[-38.2, -36.1]
2.8
dx5
4.3
dx6
1.7
dx7
-154.1
[-157.2, ]
[-158.3, ]
dx8
-32.9
[-33.5, -32.2]
[-33.6, -32.1]
2.3
dx9 -
dx10
-4.3
dx11
-24.3
[-25, -23.5]
3.1
[-25.2, -23.4]
3.8
NFS workshop on Reliable Engineering Computing 2004

11. NFS workshop on Reliable Engineering Computing 2004
Conclusions
The problem of solving parametrized systems
of linear interval equations is very important
in practical applications. Well known classical
methods (e.g. Gauss Elimination) fail since they compute enclosure for the solution set of nonparametrized systems which is generally
much larger then solution set of the parametrized ones. A direct method for solving paraterized systems of linear interval equations was proposed and checked to be usefull in structure mechanics. The method produced tight enclosure for the solutions set of parametrized systems for exemplary truss structures.
NFS workshop on Reliable Engineering Computing 2004