
222 Introduction to Linear Optimization and Extensions with MATLAB
R
Go to Step 2.
Step 2: x
(2)
= x
(1)
+ (1)d
(1)
=
3/2
1/2
1
+ (1)
−1/4
1/4
0
=
5/4
3/4
1
.
Table 6.1 shows the first 6 of the first 20 iterates of the Newton-Raphson
method. It should be noted that the error between successive iterates, i.e.,
x
(k+1)
− x
(k)
is reduced by half, each iteration, which results in rapid con-
vergence toward the root x
∗
=
1 1 1
.
Table 6.1 First 20 Newton-Raphson iterates for Example 6.1
k x
(k)
1
x
(k)
2
x
(k)
3
d
(k)
1
d
(k)
2
d
(k)
3
x
(k+1)
− x
(k)
0 1 0 1 0.5 0.5 0 0.707107
1 1.5 0.5 1 −0.25 0.25 0 0.353553
2 1.25 0.75 1 −0.125 0.125 0 0.176777
3 1.125 0.875 1 −0.0625 0.0625 0