Non standard linear algebra with error analysis
Júlia Justino
(joint work with Imme van den Berg)
Logic Colloquium 2013
Introduction
In this presentation we will …nd the conditions that guarantee the existence of an admissible solution, in terms of inclusion, on systems of linear equations which have coe¢cients having uncertainties of type o(.) or O(.). We will not use the functional form of neglecting knowned as o(.) orO(.)but an alternative formulation within nonstandard analysis using sets of in…nitesimals knowned as neutrices, introduced by the program of Van der Corput in[2]. This kind of systems will be called ‡exible systems.
Introduction
OVERVIEW:
1 Brief recall of external numbers, in which we distinguish neutrices, a sort of generalized zeros.
2 De…ning what is a ‡exible system of linear equations.
3 Presentation of a general theorem that guarantees the existence of an admissible solution and also a maximal solution produced by Cramer’s rule.
External numbers
The setting of this study it is the axiomatic nonstandard analysis IST
as presented by Nelson in [5].
External numbers are were introduced in 1995 by Koudjeti and Van den Berg in Koudjeti’s thesis and a chapter of "Nonstandard Analysis in Practice" [4] (Springer, F. and M. Diener, eds.), to serve as mathematical models of orders of magnitude within nonstandard analysis.
External numbers
We will use the notions ofin…nitesimal,in…nitely large,limited(real numbers which are not in…nitely large) and appreciable(limited numbers which are not in…nitesimals) numbers.
De…nition
A neutrixis an additive convex subgroup ofR, which is symmetric with
respect to 0.
Examples
Except for f0gand Ritself all neutrices are external sets (with internal
External numbers
Fact
1 Neutrices are totally ordered by inclusion
f0g ε2 ε ε£ £ ω R,
withεan in…nitesimal andω an in…nitely large number.
2 Neutrices are invariant under multiplication by appreciable numbers.
External numbers
De…nition
An external number α is the algebraic sum of a real numberawith a neutrix A:
α=a+A.
1 αis calledzerolessif it is not a neutrix, so 02/α.
2 αis called an absorber of B ifαB B, where B is a neutrix.
Examples
Let ε be a positive in…nitesimal. Then 1+ε£, 1ε + ,εand £ are external numbers. Only £ is not zeroless and ε is an absorber of since ε .
Flexible systems
Not all equations with external numbers can be solved in terms of equalities. For instance, no external number, or even set of external numbers, satis…es the equation x= £ since one should have x £ and £ = £. So will study inclusions instead of equalities. De…nition
Let standard n2N andαij =aij +Aij, β
j =bj +Bj,ξj =xj+Xj for all
i,j 2 f1, ...,ng. We call the quadratic system 8
> <
> :
α11ξ1+ ... +α1jξj+ ... +α1nξn β 1 ..
. ... ... ...
αn1ξ1+ ... +αnjξj+ ... +αnnξn β
n
Flexible systems
ExampleLet ε be a positive in…nitesimal. Then
(FS1) (3+ε )x+ ( 1+ )y =1+ε£ (2+ε£)x+ (1+ε )y = ε£ is a non-singular non-homogenous ‡exible system. Indeed ∆ detA= 3+ε 1+
2+ε£ 1+ε =5+ and β 1max6j6n βj =1+ε£ are zeroless.
De…nitions
If external numbers x,y can actually be found to satisfy (FS1),(x,y)
T is called an admissible solutionofAX B. This solution is maximalif no other (ξ,η)
T
(x,y)
T
satis…es (FS1). If x,y satisfy (FS1)with equalities, (x,y)
Cramer’s rule
The real part of system (FS1)is given by 3x y =1
2x+y =0 which has the exact solution x =
1 5
y = 25 produced by Cramer’s rule. When we apply Cramer’s rule, we get
8 > > > > > < > > > > > :
x = detM1
∆ =
1+ε£ 1+ ε£ 1+ε
∆ =
1+ε£
5+ = 15+
y = detM2
∆ =
3+ε 1+ε£ 2+ε£ ε£
∆ = 2
+ε£
5+ = 25+ .
Cramer’s rule
Not all non-singular non-homogeneous ‡exible systems of linear equations can be resolved by Cramer’s rule.
Looking at the uncertainties of the system (FS1)we notice that the maximum of the uncertainties on matrix Ais while the minimum of the uncertainties on matrixB is ε£, with "ε£.
Cramer’s rule
Theorem
Let n2N be standard. LetA= [αij]be a non-singular matrix, with
αij =aij +Aij and ∆=detA=d+D, and let B= [βi] be a zeroless
vector, with βi =bi +Bi for16i,j 6n. Consider the ‡exible system
AX B whereX = [ξi],with ξi =xi+Xi 2 Efor all i 2 f1,. . .,ng.
If R(A) P(B), ∆is not an absorber of B and B =B, then
X =
2
6 4
detM1
∆
.. .
detMn
∆ 3
7 5
Cramer’s rule
Corollary
In the previous conditions, if R(A) P(B) and∆ is not an absorber of
B, then X =
2
6 4
detM1(b)
∆
.. .
detMn(b)
∆ 3
7
5 is an admissible solution ofAX B.
Corollary
In the previous conditions, if only R(A) P(B), then
X =
2
6 4
detM1(b) d
.. .
detMn(b) d
3
7
Cramer’s rule
De…nition
Consider a non-singular non-homogeneous ‡exible system with matrix representation given by AX B where we denote Aas the maximum of the uncertainties and αas the maximum coe¢cient on matrixA andB as the minimum of the uncertainties on matrix B.
1 R(A) =Aαn 1 ∆ as therelative uncertaintyofA;
2 P(B) =B βas therelative precisionofB.
Example
In the system (FS1)one has R(A) =Aα ∆= (3+ε )
5+ = and
Cramer’s rule
Example
Let ε be a positive in…nitesimal. Consider the non-singular non-homogeneous ‡exible system
(FS2) (3+ε )x+ ( 1+ε )y =1+ε£ (2+ε£)x+ (1+ε )y =ε£.
This system has the same real part as the system (FS1). Also ∆=detA=5+ε£ is not an absorber of B =ε£=B and
R(A) =ε£=P(B). By applying Cramer’s rule we get
x = 15+ε£
y = 2
5 +ε£.
Gauss-Jordan elimination
When solving systems of linear equations, Gauss-Jordan elimination is the main method applied.
By applying Gauss-Jordan elimination to a non-singular
Gauss-Jordan elimination
Example
Consider the ‡exible system (FS1). Gauss-Jordan elimination yields
AjB = 3+ε 1+ j 1+ε£ 2+ε£ 1+ε j ε£
1 3L1
! 1+ε 13+ j 1
3+ε£ 2+ε£ 1+ε j ε£
!
L2 2L1
1+ε 13+ j 13 +ε£
ε£ 53 + j 2
3 +ε£
! 3 5L2 1+ε 13+ j 13 +ε£
ε£ 1+ j 25 +ε£
L1+13L2 !
1+ε£ j 1
5+ε£
Gauss-Jordan elimination
The solution produced by Gauss-Jordan elimination
x= 15 +ε£
y = 2
5 +ε£
is not the same produced by Cramer’s rule, x= 1
5 +
y = 25 + .
Theorem
Consider a non-singular non-homogenous ‡exible system of linear
equations with matrix representation given by AX B.
If R(A) P(B), ∆is not an absorber of B and B =B,
Gauss-Jordan elimination
During the condensation of matrixA, the 3 conditions of the
Theorem guarantee that the real part of ∆=detAand the real part of the maximum of all the minors of Aare limited numbers greater, in absolute value, than ∆.
References
[1] I.P. van den Berg.Nonstandard asymptotic analysis, in Lecture Notes in Mathematics 1249.Springer-Verlag, 1987.
[2] J.G. van der Corput. Neutrix calculus, neutrices and distributions.
MRC Tecnical Summary Report. University of Wisconsin, 1960.
[3] J. Justino and I.P. van den Berg.Cramer’s rule applied to ‡exible systems of linear equations, in Electronic Journal of Linear Algebra, Volume 24, p. 126-152, 2012.
[4] F. Koudjeti and I.P. van den Berg.Neutrices, external numbers and external calculus, in Nonstandard Analysis in Practice, p. 145-170. F. and M. Diener eds., Springer Universitext, 1995.
