Reliable Computing 10: 241–243, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
Fast Multiplication of Interval Matrices
(Interval Version of Strassen’s Algorithm)
MARTINE CEBERIO and VLADIK KREINOVICH
Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA,
(Received: 16 October 2003; accepted: 15 November 2003)
Abstract. Strassen’s algorithm multiplies two numerical matrices fast, but when applied to interval
matrices, leads to excess width. We use Rump’s interval arithmetic to propose an interval version of
Strassen’s algorithm whose only excess width is in second order terms.
Formulation of the problem. Many numerical algorithms—ranging from mathe-
matical physics to ranking webpages—include matrix multiplication, and multipli-
cation of large matrices often takes a signiﬁcant portion of the algorithm’s running
time. It is therefore desirable to multiply matrices fast.
The product C = AB =(c
of the matrices A =(a
and B =(b
A straightforward algorithm for computing the product of two n
therefore requires O(n) arithmetic operations to compute each of n
—to the total of O(n
The ﬁrst faster algorithm was proposed by Strassen in his 1969 paper ; this
algorithm enables us to multiply two matrices in time O(n
) (i.e., in O(n
steps). Since then, even faster algorithms have been proposed; the fastest one 
requires only O(n
For a detailed description of Strassen’s algorithm, its advantages and disad-
vantages, and more recent matrix multiplication algorithms see . In particular,
according to this exposition, one of the main disadvantages of Strassen’s and similar
algorithms is that they are much more numerically unstable than the straightforward
) matrix multiplication.
For example, if we only know the values a
with interval uncertainty—
in other words, if, instead of the actual (unknown) matrices A and B, we only
know the interval matrices A =(a
) that contain A and B—then the
straightforward algorithm leads to the exact range c
, while Strassen’s and
other algorithms lead to excess width; see, e.g., .
The reason for this excess width is simple. In the formula (1), each variable
occurs only once hence, as is all other single-use expressions (SUE), straightforward