Graphics Programs Reference
In-Depth Information
the procedure, let us solve the equations
4
x
1
−
2
x
2
+
x
3
=
11
(a)
−
2
x
1
+
4
x
2
−
2
x
3
=−
16
(b)
x
1
−
2
x
2
+
4
x
3
=
17
(c)
Elimination phase
The eliminationphase utilizes only one of the elementary op-
erationslistedinTable 2.1—multiplying oneequation (say,equation
j
) by a constant
λ
and subtracting itfromanother equation (equation
i
). The symbolic representation
of thisoperationis
Eq. (
i
)
←
Eq. (
i
)
−
λ
×
Eq. (
j
)
(2.6)
The equationbeing subtracted, namelyEq. (
j
), iscalled the
pivot equation
.
Westart the eliminationbytaking Eq. (a)tobe the pivot equation and choosing
the multipliers
λ
so astoeliminate
x
1
fromEqs. (b) and (c):
Eq. (b)
←
Eq. (b)
−
(
−
0.5)
×
Eq. (a)
Eq. (c)
←
Eq. (c)
−
0.25
×
Eq. (a)
After thistransformation, the equations become
4
x
1
−
2
x
2
+
x
3
=
11
(a)
3
x
2
−
1
.
5
x
3
=−
10
.
5
(b)
−
1
.
5
x
2
+
3
.
75
x
3
=
14
.
25
(c)
Thiscompletes the first pass. Nowwe pick (b) as the pivot equation and eliminate
x
2
from (c):
Eq. (c)
←
Eq. (c)
−
(
−
0.5)
×
Eq. (b)
which yields the equations
4
x
1
−
2
x
2
+
x
3
=
11
(a)
3
x
2
−
1
.
5
x
3
=−
10
.
5
(b)
3
x
3
=
9
(c)
The eliminationphase is now complete. The originalequationshave beenreplaced
by equivalentequationsthatcan beeasily solvedbyback substitution.
As pointed out before, the augmented coefficient matrix is amoreconvenient
instrumentforperforming the computations. Thus the originalequations wouldbe
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