Reliable Computing 8: 115–122, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
Sharp Bounds on Interval Polynomial Roots
ELDON R. HANSEN
654 Paco Drive, Los Altos, CA 94024, USA
G. WILLIAM WALSTER
Sun Microsystems, Inc., 901 San Antonio Road, UMPK16-304, Palo Alto, CA 94303-4900, USA
(Received: 28 June 1999; accepted: 12 June 2001)
Abstract. An algorithm is developed for computing sharp bounds on the real roots of polynomials
with interval coefﬁcients. The procedure is introduced by studying interval quadratic equations.
A recurring problem in interval computations is bounding the real roots of a qua-
dratic equation with interval coefﬁcients. In this paper, we give a simple procedure
for bounding interval roots, which are deﬁned as follows: Let the interval quadratic
function be expressed as F(x)=[
(x)]. Given a point x for which
(x) ≤ 0 ≤
let R be the largest interval containing values of x such that every point in R also
satisﬁes (1.1). We shall call R an “interval root” of F(x).
We then show that the procedure can be extended to ﬁnd the interval roots of
2. Quadratic Equations
Consider a quadratic equation
+ Bx + C =0
where A =[a
a], B =[b
b], and C =[c
c] are intervals.
For example, with
2], B =[−4
2], and C =[−3
1] the graph
in Figure 1 displays the
situation in which there is only one interval root. In this note we consider how to
2002 Sun Microsystems, Inc. and Eldon R. Hansen. All rights reserved.
It is common practice to write an interval linear equation as, say, Bx + C = 0, even though
substituting the solution interval X = −C
B for x does not produce zero using interval arithmetic.
For convenience, this convention is followed in the quadratic case.
All graphs have been produced using GrafEq. See: www.peda.com/grafeq. GrafEqis based
on interval arithmetic and can directly graph interval functions and implicit relations.