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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