Numerical Methods in Trigonometry
•
Section 6.2
137
point
of the function cos
x
. No matter where you start, you end up getting “drawn” to it.
Figure 6.2.2 shows what happens when starting at
x
=
0: taking the cosine of 0 takes you to
1, and then successive cosines (indicated by the intersections of the vertical lines with the
cosine curve) eventually “spiral” in a rectangular fashion to the fixed point (i.e. the solution),
which is the intersection of
y
=
cos
x
and
y
=
x
.
Recall in Example 5.10 in Section 5.2 that we claimed that the maximum and minimum of
the function
y
=
cos 6
x
+
sin 4
x
were
±
1
.
90596111871578, respectively. We can show this by
using the opensource program Octave.
2
Octave uses a
successive quadratic programming
method to find the minimum of a function
f
(
x
). Finding the maximum of
f
(
x
) is the same as
finding the minimum of
−
f
(
x
) then multiplying by
−
1 (why?). Below we show the commands
to run at the Octave command prompt (
octave:n>
) to find the minimum of
f
(
x
)
=
cos 6
x
+
sin 4
x
. The command
sqp(3,’f’)
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 Fall '11
 Dr.Cheun
 Calculus, Trigonometry, Computer program, command sqp

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