Jökull - 01.12.1973, Page 40

Fig. 1. Well posed problem in two dimensions
where m = 2.
ems to the case when there is one and only
one set of unknowns which solve the equations.
The problem is then referred to as being well
posed. The cases where (a) there are too few
equations to define a unique solution, or (b)
where there are too many equations to yield any
solution at all, are termed as improperly posed.
The former case represents the underdeter-
mined systems and the latter case the over-
determined systems.
Lanczos (1961) has emphasized that it is quite
simple to define meaningful generalized solu-
tions for any finite system of linear equations
regardless of whether the system is proper or
improper in the above sense. The underlying
ideas are readily explained in the case of only
two unknowns. Let and x2 be two unknown
quantities which are to be derived on the basis
of m given linear equations
ailxl + ai2x2 = bi’ i = 1, 2,. .., m (1)
where the constants a(1, ai2, and b( are known.
Degenerate cases where any two of the two-
component vectors (aJ;1, ai2) are linearly de-
pendent are excluded. We can now distinguish
the three main cases referred to above, viz.
Character of the system:
(a) m = 2 well posed
(b) m = 1 underdetermined
(c) m > 2 overdetermined
It is convenient to consider the geometrical
illustrations of these cases in Figs. 1, 2, 3, and
4, where each of the equations (1) above de-
fines a line in the (x1( x2) plane.
In the well posed case, m = 2, illustrated in
Fig. 1, the two lines 1 and 2 intersect at one
and only one point S which defines the unique
solution of the system.
In the underdetermined case, m = 1, illustrat-
ed in Fig. 2, there is oniy one line and all
points on this line satisfy the given single
linear equation. This line, therefore, represents
a one-dimensional solution space where there
is a continuum of solutions. Each point P on
the line defines a vector x = (x^, x2) repres-
ented by the line segment OP and which can
be expressed
X = x0 + s (2)
where xQ is the vector OS perpendicular to the
line 1 and s is an arbitrary vector SP along 1.
Using the standard notation for the U-norm or
the length, |x| = \/ x2i + x22 we see that the
orthogonality of x0 and s implies that
|x|2=|x0|2+ |s|2 (3)
Fig. 2. Underdetermined problem in two dim-
ensions where m = 1.
38 JÖKULL 23. ÁR