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

Search WWH ::

Custom Search