Graphics Programs Reference
InDepth Information
second equation. The number of such combinations is infinite. On the other hand,
the equations
2
x
+
y
=
3
4
x
+
2
y
=
0
have no solutionbecause the second equation, being equivalentto2
x
0, con
tradicts the first one. Therefore, any solution thatsatisfies oneequation cannot satisfy
the other one.
+
y
=
IllConditioning
An obviousquestionis:whathappens when the coefficient matrix is almost singular;
i.e., if
is very small? In order to determinewhether the determinant of the coefficient
matrix is “small,”we needareference againstwhich the determinantcanbemeasured.
This reference iscalled the
norm
of the matrix, denotedby

A

A
.Wecan then say that
the determinant issmall if

A

<<
A
Several normsofamatrix have beendefinedinexisting literature, such as
n
n
n
A
i j
A
i j
A
=
A
=
max
1
(2.5a)
≤
i
≤
n
i
=
1
j
=
1
j
=
1
A formal measureofconditioning is the
matrix condition number
, definedas
A
−
1
cond(
A
)
=
A
(2.5b)
If this numberis close to unity, the matrix is wellconditioned. The condition number
increases with the degree of illconditioning, reaching infinity forasingular matrix.
Note that the conditionnumberis not unique, but dependson the choice of thematrix
norm. Unfortunately, the condition numberisexpensivetocompute for large matri
ces. Inmost cases it issufficienttogaugeconditioning by comparing the determinant
with the magnitudes of the elements in the matrix.
If the equations are illconditioned,small changes in the coefficient matrix result
in largechanges in the solution. As an illustration,consider the equations
2
x
+
y
=
3
x
+
1
.
001
y
=
0
thathave the solution
x
002is
much smaller than the coefficients, the equations are illconditioned. The effect of
illconditioning can be verifiedbychanging the second equation to 2
x
=
1501
.
5,
y
=−
3000.Since

A
=
2(1
.
001)
−
2(1)
=
0
.
+
1
.
002
y
=
0
=
.
=−
and resolving the equations. The result is
x
1500. Note that a 0.1%
change in the coefficientof
y
produceda100% change in the solution.
751
5,
y
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