急求一篇关于矩阵乘法的英语论文,字数大约400左右.

急求一篇关于矩阵乘法的英语论文,字数大约400左右.
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1. Introduction
In this paper, we are concerned with interval matrix multiplication
C = A ⋅ B (1.1)
where A and B are general interval m × p and p × n matrices, respectively. From
the standpoint of interval arithmetic, we can classify the matrix multiplication into
the following four cases:
(P×P) both A and B are point matrices,
(P×I) A is a point matrix and B is an interval matrix,
(I×P) A is an interval matrix and B is a point matrix, and
(I×I) both A and B are interval matrices.
Throughout this paper, the case (P×P) is called point matrix multiplication and
three cases (P×I), (I×P) and (I×I) are generically called interval matrix multiplica-
tion.Recently, utilizing instructions which control the rounding mode defined by
IEEE 754 floating point standard, fast methods [5], [6], [8], [9] have been developed
by one of the authors (S. Oishi) and S. M. Rump to calculate a rigorous inclusion
of the matrix multiplication. The method is so-called rounding mode controlled
computation.Throughout the paper, the inclusion methods proposed by Oishi and
Rump are called Oishi-Rump algorithms.Since the inclusion of matrix multipli-
cation plays an important role in self-validating algorithms, it is useful to reduce
its computational cost. We shall briefly review some examples of application in
Appendix.
Themain point of this paper is to develop fast inclusion methods for interval
matrix multiplication with rounding mode controlled computation. This is accom-
plished using fast algorithms to calculate an upper bound on the product of two
nonnegative n × n matrices in O(n2)flops. We shall also present a fast inclusion
method for complex matrix multiplication. The main result of the article is to show
that the computational cost for enclosing each three cases of the interval matrix
multiplication is almost the same as that for enclosing the point matrix multiplica-
tion.
We shall present numerical results, which elucidate that the proposed algo-
rithms are much faster than the conventional algorithms and that the guaranteed
accuracies obtained by the new algorithms are comparable to those obtained by the
conventional algorithms.
2. RoundingMode Controlled Computation
In this section, we shall briefly review the Oishi-Rump algorithm, which becomes
the base of new inclusion algorithms for the interval matrix multiplication. The
Oishi-Rump algorithm is based on rounding mode controlled computation and
midpoint-radius arithmetic.
Let X =(xij)and Y =(yij )bereal m × n matrices, then the notation X ≤ Y is
defined by
X ≤ Y ⇐⇒xij ≤ yij for all (i, j),
and the notation X ≥ O means that all elements of X are nonnegative. Moreover, a
real interval m × n matrix [A]isdefined by [A]:=[A, A]= {X ∈ Rn× n
| A ≤ X ≤ A},
nd a complex interval m × n matrix [C]isdefinedby
[C]:=[C, C]=[A + i ⋅ B, A + i ⋅ B]=[A, A]+ i ⋅ [B, B]
= {Z = X + i ⋅ Y ∈ Cn× n
| X ∈ [A, A], Y ∈ [B, B]}.
Throughout this paper, we shall express algorithms MATLAB-style. In
ATLAB, for example, the product of a point m × p matrix X =(xij)anda
oint p × n matrix Y =(yij )whose elements are double precision floating point
umbers can be calculated by
S = X ∗ Y.

1年前他留下的回答

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