Re: Explicit Solution of cos(x)=x

the solution of cos(x)= x
x = cos(cos(cos(cos(cos(.... cos(n))))))
When Cos tends to infinite and "n" can be any number.

Hi. It was just an idea for discussion. I understand Tan(x)=x has a
solution...if you know the solution ahead of time.
When "n" can be any number (Op statement), then you can start with 0.5 say,
and hit the Tan button 100,000 times, and you won't arrive at a solution to
Tan(x)=x.
Tan doesn't have a Fixed Point per say because the Tan value of x tends to
get Larger...until it recycles back to a low number again..
Sin values do tend towards 0, but as the limit of x gets smaller, x and
Sin[x] tend to equal.
Therefore they converge very very slowly towards zero.

The Newton equation for Cos has a local minimum, therefore, you can hit Cos
button a reasonable amount of times, and arrive at a solution. No Local
Minimum per say with Sin & Tan.

equ = z - (Cos[z] - z)/(-Sin[z] - 1)

Which does have a minimum of .739 when z=.739

FindMinimum[equ, {z, -1, 1}]

{0.739085..., {z -> 0.739085...}}

FixedPoint[Cos, 0.4]
0.739085...

Tan doesn't converge after 10 iterations. (Staring with any number, not just
the solution of 0.). As you see, it kind of cycles low to high.
FixedPointList[Tan, 0.5, 10]
{0.5, 0.546, 0.608, 0.696, 0.835, 1.105, 1.992, -2.234, 1.28, 3.346, 0.208}

Hitting the Sin button 1,000,000 times doesn't tell us the solution to
Sin(x)=x is 0.

FixedPoint[Sin, 0.5, 1000000]

0.0017320345110059929

Again, just an idea using Local minimum.( & non complex numbers)
--
Dana

"David W. Cantrell" <DWCantrell@xxxxxxxxxxx> wrote in message
news:20070206153326.192$RI@xxxxxxxxxxxxxxxxx
Ronald Bruck <bruck@xxxxxxxxxxxx> wrote:
In article <_35yh.43850$Ts.35626@xxxxxxxxxxxxxxxxxxxxxx>, Dana
<ddelouis@xxxxxxxxxxxxx> wrote:

Another term one will sometimes hear is that the function "Cos" has a
"Fixed Point."
Functions "Sin" and "Tan" do not.
Some math programs have a "FixedPoint" routine that keeps applying the
function over and over till the output doesn't change.

For Example, no matter where you start, the value converges to 0.739...
for Cos.
FixedPoint[Cos, 10.]
0.7390851332151607

FixedPoint[Cos, 2.34]
0.7390851332151607

I thought it was the Fixed Point Theorem, but I can't make much sense
out of it.
http://mathworld.wolfram.com/FixedPointTheorem.html

Both sin and tan have fixed-points; sin(0) = 0, and there are
infinitely many solutions of tan(x) = x. Just look at the graphs
of y = x and y = tan x and you'll understand the latter better.

Not to mention the complex plane, where sine also has infinitely many
fixed-points.

David


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