Environmental Engineering Reference
In-Depth Information
Sidebar 1.1: Over- and Underdetermined Systems
Both divisions,
/
and
\
, are also possible for non-square matrices. In the latter
case the linear system A
B is over- or underdetermined. In the case of
an overdetermined system the equations usually can not be fulfilled exactly.
Then the solution X is computed in the sense of least square minimization, i.e.
X minimises the expression A
X
¼
B. In the following example the right
side of the system is a column vector b: The solution x is also a column
vector.
X
The solution of the overdetermined system A
x
¼
b is given in the
MATLAB
answer. The second part of the answer gives the deviation in
each of the three elements. The example can be interpreted as linear curve
fitting on three given points (see graphic). For such a task the user may also
use the polynomial curve fitting tool of MATLAB
®
, which is introduced in
Chap. 10. As an exercise, the novice may confirm the obtained result for
x
by
using the graphical user interface tool.
The division operator can also be applied for underdetermined systems
®
A
B, where the number of columns of A exceeds the number of rows. In
that case the solution is usually not unique, and MATLAB
X
¼
delivers just one
®
solution.
Moreover the well-known function sin, cos, exp can be performed on matrices.
The exponential function of a matrix is based on the formula
1
1
i!
A
i
exp
ðAÞ¼
(1.10)
i¼
0
in which the powers of A are performed with matrix multiplication. Try:
