z:=1/4*(4+4*I*3^(1/2))^(1/3)+1/((4+4*I*3^(1/2))^(1/3));
RI:=[Re(z), Im(z)];
map(shake, RI);

  Error, (in evalr/shake) Not implemented, evalr/Re

Note that z = cos(1/9*Pi) is real, using 
convert(z, trig); expand(%); simplify(%); 

It may be, that shake and evalr are needed both ?

If I understand your suggestion, then one can consider a product
of intervals (but one would have to make sure, that functions are
defined there - excluding the mentioned branch cuts).

As soon as functions are applied those may be mapped to something
being no longer of that form

Ok, at least now I have an impression, why it comes not handy as
just a command to be applied.

PS: there is another issue - at least for me the help for shake
does not actually tell me, what it does and it suggestes me, that
it gives absolute error in an interval.

However shake(0) gives INTERVAL(0. .. 0.), shake on 1e-20 is
still positive, shake(1)= INTERVAL(.9999999990 .. 1.0000000010)
using Digits = 10.

So it seems to give a relative error. In that case shake(1) shows,
that the interval is 2 * Float(10, -ampl), despite the help.

z:=1/4*(4+4*I*3^(1/2))^(1/3)+1/((4+4*I*3^(1/2))^(1/3));
RI:=[Re(z), Im(z)];
map(shake, RI);

  Error, (in evalr/shake) Not implemented, evalr/Re

Note that z = cos(1/9*Pi) is real, using 
convert(z, trig); expand(%); simplify(%); 

It may be, that shake and evalr are needed both ?

If I understand your suggestion, then one can consider a product
of intervals (but one would have to make sure, that functions are
defined there - excluding the mentioned branch cuts).

As soon as functions are applied those may be mapped to something
being no longer of that form

Ok, at least now I have an impression, why it comes not handy as
just a command to be applied.

PS: there is another issue - at least for me the help for shake
does not actually tell me, what it does and it suggestes me, that
it gives absolute error in an interval.

However shake(0) gives INTERVAL(0. .. 0.), shake on 1e-20 is
still positive, shake(1)= INTERVAL(.9999999990 .. 1.0000000010)
using Digits = 10.

So it seems to give a relative error. In that case shake(1) shows,
that the interval is 2 * Float(10, -ampl), despite the help.

The following would fail

x := I*x+3-I:
map(shake, [Re(I*x+3-I), Im(I*x+3-I)]);
  
  Error, recursive assignment
  Error, (in shake) Not implemented evalr/Im

I guess, that it is not trivial to expand it to Complex
and be aware, that it should work as input for functions.

Say 'lousy' is along the complex axis or the Negatives
and one wants a logarithm of it will even stumble into
branch cuts.

May be a reason, why it is avoided up to now?

The following would fail

x := I*x+3-I:
map(shake, [Re(I*x+3-I), Im(I*x+3-I)]);
  
  Error, recursive assignment
  Error, (in shake) Not implemented evalr/Im

I guess, that it is not trivial to expand it to Complex
and be aware, that it should work as input for functions.

Say 'lousy' is along the complex axis or the Negatives
and one wants a logarithm of it will even stumble into
branch cuts.

May be a reason, why it is avoided up to now?

@pagan Thx, did not try it, though it is the mathematical way to show,
that those trigometrics are algebraic :-(

For Devore's code: I have given up to decipher it to usual coding.

And do not see, how one can get (the) minimal polynomial.

Edited: may be something in the spirit of Watkins, W. and Zeitlin, J.
"The Minimal Polynomial of ...  http://www.jstor.org/pss/2324301

Edited2: yes, from http://arxiv.org/abs/1001.5192v2, where [20] is
the article above; thus having the degree one can write it down in
terms of the roots (or multiplies until 'done' ?)

@pagan Thx, did not try it, though it is the mathematical way to show,
that those trigometrics are algebraic :-(

For Devore's code: I have given up to decipher it to usual coding.

And do not see, how one can get (the) minimal polynomial.

Edited: may be something in the spirit of Watkins, W. and Zeitlin, J.
"The Minimal Polynomial of ...  http://www.jstor.org/pss/2324301

Edited2: yes, from http://arxiv.org/abs/1001.5192v2, where [20] is
the article above; thus having the degree one can write it down in
terms of the roots (or multiplies until 'done' ?)

Pagan, thank you!

Why is it a minimal polynomial (where I will have to resolve the condensed way of coding)?

Hm, the name of the routine was AlgTrig or similar ...

PS: how you got cos(Pi/13) = as above ?

Pagan, thank you!

Why is it a minimal polynomial (where I will have to resolve the condensed way of coding)?

Hm, the name of the routine was AlgTrig or similar ...

PS: how you got cos(Pi/13) = as above ?

Yes, that is around that post (especially because of your reply there)

I want it for cases, where Maple refuses to answer (for what reason ever
it does, have not looked into Maple's code).

There was a routine by Carl Devore, which was better in some cases,
but I do not know a public link on it.

I will be musing with that the next days, after work, sometimes one
can find nice results, cos(Pi/192) for example :-)

Not in all cases Maple can find a solution for RootOf, like the sextic.

Yes, that is around that post (especially because of your reply there)

I want it for cases, where Maple refuses to answer (for what reason ever
it does, have not looked into Maple's code).

There was a routine by Carl Devore, which was better in some cases,
but I do not know a public link on it.

I will be musing with that the next days, after work, sometimes one
can find nice results, cos(Pi/192) for example :-)

Not in all cases Maple can find a solution for RootOf, like the sextic.

Thx ... I have to upgrade my version :-)

Do you see a way to use that option for the following (which
is the answer I got with Maple 12 and hesitate to re-code)?

1/2*RootOf(-1+3*_Z+3*_Z^2-4*_Z^3-_Z^4+_Z^5,index = 3)

Edited: I withdraw, allvalues does it ...

Thx ... I have to upgrade my version :-)

Do you see a way to use that option for the following (which
is the answer I got with Maple 12 and hesitate to re-code)?

1/2*RootOf(-1+3*_Z+3*_Z^2-4*_Z^3-_Z^4+_Z^5,index = 3)

Edited: I withdraw, allvalues does it ...

Thanks, providing free stuff is certainly a good idea and appreciated.

However you provide it in Russian, so its value is somewhat limited,
most in the western world do not speak that language (nor are used
to the alphabet), including me.

@hirnyk 

Seems I have to fresh up my Analysis ...

Why? Is it the same book you cited last?
And why the single-sided derivative exists?

Have to take some coffee, at least ...

Edited: @Preben: was your answer for me or for hirnyk?

@hirnyk 

Seems I have to fresh up my Analysis ...

Why? Is it the same book you cited last?
And why the single-sided derivative exists?

Have to take some coffee, at least ...

Edited: @Preben: was your answer for me or for hirnyk?