Page 29 - Compact Numerical Methods For Computers

Chapter 2

FORMAL PROBLEMS IN LINEAR ALGEBRA

2.1. INTRODUCTION
A great many practical problems in the scientific and engineering world give rise
to models or descriptions of reality which involve matrices. In consequence, a very
large proportion of the literature of numerical mathematics is devoted to the
solution of various matrix equations. In the following sections, the major formal
problems in numerical linear algebra will be introduced. Some examples are
included to show how these problems may arise directly in practice. However, the
formal problems will in most cases occur as steps in larger, more difficult
computations. In fact, the algorithms of numerical linear algebra are the key-
stones of numerical methods for solving real problems.
Matrix computations have become a large area for mathematical and compu-
tational research. Textbooks on this subject, such as Stewart (1973) and Strang
(1976), offer a foundation useful for understanding the uses and manipulations of
matrices and vectors. More advanced works detail the theorems and algorithms for
particular situations. An important collection of well-referenced material is Golub
and Van Loan (1983). Kahaner, Moler and Nash (1989) contains a very readable
treatment of numerical linear algebra.

2.2. SIMULTANEOUS LINEAR EQUATIONS
If there are n known relationships

A x + A x +. . .+ A x = b i i = 1, 2, . . . , n (2.1)
in n
i2 2
il 1
between the n quantities x with the coefficients A and right-hand side elements
ij
j
b , i = 1, 2, . . . , n, then (2.1) is a set of n simultaneous linear equations in n
i
unknowns x , j = 1, 2, . . . , n. It is simpler to write this problem in matrix form
j
Ax = b (2.2)
where the coefficients have been collected into the matrix A, the right-hand side is
now the vector b and the unknowns have been collected as the vector x. A further
way to write the problem is to collect each column of A (say the jth) into a column
vector (i.e. a ). Then we obtain
j
(2.3)

Numerous textbooks on linear algebra, for instance Mostow and Sampson
(1969) or Finkbeiner (1966), will provide suitable reading for anyone wishing to
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