Reliable Computing (2006) 12: 239–243
Solving Overdetermined Systems of
Interval Linear Equations
654 Paco Drive, Los Altos, CA 94024, USA, e-mail: firstname.lastname@example.org
G. WILLIAM WALSTER
Sun Microsystems, Mailstop UMPK16–160, 16 Network Circle, Menlo Park, CA 94025, USA,
(Received: 16 September 2002; accepted: 8 July 2005)
Abstract. An algorithm is developed to compute interval bounds on the set of all solutions to an
overdetermined system of interval linear equations.
Given the real (n
n) matrix A and the (n
1) column vector b, the linear system
Ax = b (1.1)
is consistent if there is a unique (n
1) vector x for which the system in (1.1) is
satisﬁed. If the number of rows in A and elements in b is m = n, then the system is
said to be either under- or overdetermined depending on whether m<nor n<m.
In the overdetermined case, if m − n equations are not linearly dependent, there is
no solution vector x that satisﬁes the system. Solutions in the underdetermined case
are not unique.
In the point (non-interval) case, there is no generally reliable way to decide if
an overdetermined system is consistent or not. Instead a least squares approximate
solution is generally sought. In the interval case, it is possible to delete inconsistent
cases and bound the set of solutions to the remaining consistent equations. In this
note, we consider the problem of solving overdetermined systems of equations in
which the coefﬁcients are intervals. That is, we consider a system of the form
x = b
is an interval matrix of m rows and n columns with m>n. The interval
has m components. Such a system might arise directly or by linearizing
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