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In-Depth Information
F
x
2
x
1
x
0
f
(
x
)
Fig. 4.6
The figure shows a function f
D
f.x/ and its linear interpolant F between x
0
and x
1
Then we can define the linear function
f.x
1
/
f.x
0
/
x
1
x
0
F
0
.x/
D
f.x
1
/
C
.x
x
1
/;
(4.65)
which satisfies F
0
.x
0
/
D
f.x
0
/ and F
0
.x
1
/
D
f.x
1
/. We have plotted a function
f and its linear interpolant F in Fig.
4.6
.
Since we have
F
0
.x/
f.x/;
(4.66)
we can compute an approximation x
2
of x
by solving
F
0
.x
2
/
D
0:
(4.67)
The solution of
f.x
1
/
f.x
0
/
x
1
x
0
f.x
1
/
C
.x
2
x
1
/
D
0
with respect to x
2
is given by
f.x
1
/.x
1
x
0
/
f.x
1
/
f.x
0
/
:
x
2
D
x
1
Generally, we have the following algorithm.
