MaplePrimes - answers and comments on Question, interval arithmetics: only for Reals?

Re and Im

hirnyk — Fri, 04 Mar 2011 01:58:18 Z

How about the following:

> x := (1/5)*Pi:
>map(shake, [Re(I*x+3-I), Im(I*x+3-I)]);



               [INTERVAL(2.9999999970 .. 3.0000000030),

INTERVAL(-0.371681471539 .. -0.371681467025)] ?

hm ... yes&no

Axel Vogt — Fri, 04 Mar 2011 02:37:10 Z

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?

Syntax mistake

hirnyk — Fri, 04 Mar 2011 02:50:54 Z

Of course, after the syntax mistake

> x := I*x+3-I;

Error, recursive assignment

the map(shake, [Re(I*x+3-I), Im(I*x+3-I)]); command does not work. Why don't you invent a correct example?

Syntax mistake, sorry

Axel Vogt — Sat, 05 Mar 2011 00:05:00 Z

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.

Slight modification

hirnyk — Sat, 05 Mar 2011 01:10:15 Z

@Axel Vogt

How about this?
> z := (1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3);

                                    (1/3)
               1 /         (1/2)\                  1
               - \4 + 4 I 3     /      + ---------------------
               4                                              (1/3)
                                         /            (1/2)\
                                         \4 + 4 I 3      /
> zz := evalr(evalf(z));

0.9396926209 - 1* 10^(-10)*I
> shake(op(1, zz));

INTERVAL(0.939692619960 .. 0.939692621840)

> shake(op(2, zz));

INTERVAL(-1.0000000010* 10^(-10) .. -9.999999990* 10 ^(-11))

Concerning range arithmetic for complexvalued functions one can look at

http://www2.math.uni-wuppertal.de/~xsc/software/intpakX/ , indicated by Thomas Richard.

route to follow

Axel Vogt — Sat, 05 Mar 2011 02:47:13 Z

Hirnyk, hmhmhm ...

Yes, through evalf = numerics is ok, but I miss the estimates.
May be one has to use the suggested package & its theory.

---

off topic:

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