Graphics Programs Reference
In-Depth Information
EXAMPLE 4.11
Aroot of the equation P 3 ( x )
x 3
0 x 2
.
Findamore accurate valueofthis root by one application of Laguerre's iterative
formula.
=
4
.
4
.
48 x
+
26
.
1is approximately x
=
3
i
Solution Use the given estimate of the root as the starting value. Thus
x 2
x 3
=
=
=
x
3
i
8
6 i
18
26 i
Substituting these values in P 3 ( x ) and its derivatives, we get
x 3
0 x 2
P 3 ( x )
=
4
.
4
.
48 x
+
26
.
1
=
(18
26 i )
4
.
0(8
6 i )
4
.
48(3
i )
+
26
.
1
=−
1
.
34
+
2
.
48 i
P 3 ( x )
0 x 2
=
3
.
8
.
0 x
4
.
48
=
3
.
0(8
6 i )
8
.
0(3
i )
4
.
48
=−
4
.
48
10
.
0 i
P 3 ( x )
=
6
.
0 x
8
.
0
=
6
.
0(3
i )
8
.
0
=
10
.
0
6
.
0 i
Equations(4.14) thenyield
P 3 ( x )
P 3 ( x )
4
.
48
10
.
0 i
G ( x )
=
=
=−
2
.
36557
+
3
.
08462 i
1
.
34
+
2
.
48 i
P 3 ( x )
P 3 ( x )
10
.
0
6
.
0 i
G 2 ( x )
08462 i ) 2
H ( x )
=
=
(
2
.
36557
+
3
.
1
.
34
+
2
.
48 i
=
0
.
35995
12
.
48452 i
The term under the square root sign of the denominatorinEq. (4.16) becomes
( n
1) nH ( x )
G 2 ( x )
F ( x )
=
2 3(0
08462 i ) 2
=
.
35995
12
.
48452 i )
(
2
.
36557
+
3
.
5
=
.
67822
45
.
71946 i
=
5
.
08670
4
.
49402 i
Nowwe must find which sign in Eq. (4.16) produces the larger magnitude of the
denominator:
|
G ( x )
+
F ( x )
| = |
(
2
.
36557
+
3
.
08462 i )
+
(5
.
08670
4
.
49402 i )
|
= |
2
.
72113
1
.
40940 i
| =
3
.
06448
|
G ( x )
F ( x )
| = |
(
2
.
36557
+
3
.
08462 i )
(5
.
08670
4
.
49402 i )
|
= |−
7
.
45227
+
7
.
57864 i
| =
10
.
62884
Using the minussign, weobtain fromEq. (4.16) the following improved approxi-
mation for the root
n
3
r
=
x
=
(3
i )
G ( x )
F ( x )
7
.
45227
+
7
.
57864 i
=
3
.
19790
0
.
79875 i
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