Graphics Programs Reference
In-Depth 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
=
Ill-Conditioning
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 well-conditioned. The condition number
increases with the degree of ill-conditioning, 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 ill-conditioned,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 ill-conditioned. The effect of
ill-conditioning 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 re-solving 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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