exponentiation problem

[Sergei Steshenko-2]

Hello All,

examples:

"
octave:19> (-ones(1)) ^ 0.25
ans =  0.70711 + 0.70711i
octave:20> (-ones(2)) ^ 0.25
ans =

   0.42045 + 0.42045i   0.42045 + 0.42045i
   0.42045 + 0.42045i   0.42045 + 0.42045i

octave:21> (-ones(3)) ^ 0.25
ans =

   0.31024 + 0.31022i   0.31018 + 0.31017i   0.31018 + 0.31021i
   0.31018 + 0.31017i   0.31025 + 0.31025i   0.31018 + 0.31019i
   0.31018 + 0.31021i   0.31018 + 0.31019i   0.31024 + 0.31021i

octave:22> (-ones(3)) .^ (0.25 * ones(3))
ans =

   0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
   0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
   0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i

octave:23> (ones(3)) ^ 0.25
ans =

   0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
   0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
   0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
"

-

"
octave:19> (-ones(1)) ^ 0.25
ans =  0.70711 + 0.70711i
",

"
octave:22> (-ones(3)) .^ (0.25 * ones(3))
ans =

   0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
   0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
   0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
"

are correct IMHO;

"
octave:23> (ones(3)) ^ 0.25
ans =

   0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
   0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
   0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
"

is especially worrysome because imaginary part appears out of nowhere.

Regards,
  Sergei.

Applications From Scratch: http://appsfromscratch.berlios.de/


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[John W. Eaton]

On 15-Jan-2008, Sergei Steshenko wrote:

| examples:
|
| "
| octave:19> (-ones(1)) ^ 0.25
| ans =  0.70711 + 0.70711i
| octave:20> (-ones(2)) ^ 0.25
| ans =
|
|    0.42045 + 0.42045i   0.42045 + 0.42045i
|    0.42045 + 0.42045i   0.42045 + 0.42045i
|
| octave:21> (-ones(3)) ^ 0.25
| ans =
|
|    0.31024 + 0.31022i   0.31018 + 0.31017i   0.31018 + 0.31021i
|    0.31018 + 0.31017i   0.31025 + 0.31025i   0.31018 + 0.31019i
|    0.31018 + 0.31021i   0.31018 + 0.31019i   0.31024 + 0.31021i
|
| octave:22> (-ones(3)) .^ (0.25 * ones(3))
| ans =
|
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|
| octave:23> (ones(3)) ^ 0.25
| ans =
|
|    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
|    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
|    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
| "
|
| -
|
| "
| octave:19> (-ones(1)) ^ 0.25
| ans =  0.70711 + 0.70711i
| ",
|
| "
| octave:22> (-ones(3)) .^ (0.25 * ones(3))
| ans =
|
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
| "
|
| are correct IMHO;
|
| "
| octave:23> (ones(3)) ^ 0.25
| ans =
|
|    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
|    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
|    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
| "
|
| is especially worrysome because imaginary part appears out of nowhere.

It is not surprising that the matrix cases are faulty for singular
matrices or matrices with repeated eigenvalues given that they are
implemented using a simple similarity transform.  I think that fixing
that problem is not trivial.  Perhaps we should just try to provide a
warning in these cases.  I'd certainly consider a patch for that.

jwe
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[Sergei Steshenko-2]

--- Sergei Steshenko <sergstesh@...> wrote:

> Hello All,
>
> examples:
>
> "
> octave:19> (-ones(1)) ^ 0.25
> ans =  0.70711 + 0.70711i
> octave:20> (-ones(2)) ^ 0.25
> ans =
>
>    0.42045 + 0.42045i   0.42045 + 0.42045i
>    0.42045 + 0.42045i   0.42045 + 0.42045i
>
> octave:21> (-ones(3)) ^ 0.25
> ans =
>
>    0.31024 + 0.31022i   0.31018 + 0.31017i   0.31018 + 0.31021i
>    0.31018 + 0.31017i   0.31025 + 0.31025i   0.31018 + 0.31019i
>    0.31018 + 0.31021i   0.31018 + 0.31019i   0.31024 + 0.31021i
>
> octave:22> (-ones(3)) .^ (0.25 * ones(3))
> ans =
>
>    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
>    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
>    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
>
> octave:23> (ones(3)) ^ 0.25
> ans =
>
>    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
>    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
>    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
> "
>
> -
>
> "
> octave:19> (-ones(1)) ^ 0.25
> ans =  0.70711 + 0.70711i
> ",
>
> "
> octave:22> (-ones(3)) .^ (0.25 * ones(3))
> ans =
>
>    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
>    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
>    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> "
>
> are correct IMHO;
>
> "
> octave:23> (ones(3)) ^ 0.25
> ans =
>
>    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
>    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
>    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
> "
>
> is especially worrysome because imaginary part appears out of nowhere.
>
> Regards,
>   Sergei.
>
> Applications From Scratch: http://appsfromscratch.berlios.de/
>
>
>       ____________________________________________________________________________________
> Never miss a thing.  Make Yahoo your home page.
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>

octave-2.9.14:

"
octave:1> ones(3) ^ 0.25
ans =

   NaN - NaNi   NaN - NaNi   NaN - NaNi
   Inf - NaNi   Inf - NaNi   Inf - NaNi
   NaN - NaNi   NaN - NaNi   NaN - NaNi
";

octave-2.9.16:

"
octave:1> ones(3) ^ 0.25
ans =

   0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
   0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
   0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
".

I do not have octave-2.9.15 to check.

Regards,
  Sergei.

Applications From Scratch: http://appsfromscratch.berlios.de/


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[Sergei Steshenko-2]

--- "John W. Eaton" <jwe@...> wrote:

> On 15-Jan-2008, Sergei Steshenko wrote:
>
> | examples:
> |
> | "
> | octave:19> (-ones(1)) ^ 0.25
> | ans =  0.70711 + 0.70711i
> | octave:20> (-ones(2)) ^ 0.25
> | ans =
> |
> |    0.42045 + 0.42045i   0.42045 + 0.42045i
> |    0.42045 + 0.42045i   0.42045 + 0.42045i
> |
> | octave:21> (-ones(3)) ^ 0.25
> | ans =
> |
> |    0.31024 + 0.31022i   0.31018 + 0.31017i   0.31018 + 0.31021i
> |    0.31018 + 0.31017i   0.31025 + 0.31025i   0.31018 + 0.31019i
> |    0.31018 + 0.31021i   0.31018 + 0.31019i   0.31024 + 0.31021i
> |
> | octave:22> (-ones(3)) .^ (0.25 * ones(3))
> | ans =
> |
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |
> | octave:23> (ones(3)) ^ 0.25
> | ans =
> |
> |    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
> |    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
> |    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
> | "
> |
> | -
> |
> | "
> | octave:19> (-ones(1)) ^ 0.25
> | ans =  0.70711 + 0.70711i
> | ",
> |
> | "
> | octave:22> (-ones(3)) .^ (0.25 * ones(3))
> | ans =
> |
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> | "
> |
> | are correct IMHO;
> |
> | "
> | octave:23> (ones(3)) ^ 0.25
> | ans =
> |
> |    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
> |    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
> |    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
> | "
> |
> | is especially worrysome because imaginary part appears out of nowhere.
>
> It is not surprising that the matrix cases are faulty for singular
> matrices or matrices with repeated eigenvalues given that they are
> implemented using a simple similarity transform.  I think that fixing
> that problem is not trivial.  Perhaps we should just try to provide a
> warning in these cases.  I'd certainly consider a patch for that.
>
> jwe
>

I am not sure, but should, say,

(ones(3)) ^ 0.25

be equivalent to

(-ones(3)) .^ (0.25 * ones(3))

?

I mean, that was my expectation, but it could be wrong.

Thanks,
  Sergei.

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[John W. Eaton]

On 15-Jan-2008, Sergei Steshenko wrote:

| I am not sure, but should, say,
|
| (ones(3)) ^ 0.25
|
| be equivalent to
|
| (-ones(3)) .^ (0.25 * ones(3))
|
| ?
|
| I mean, that was my expectation, but it could be wrong.

Your expectation is wrong.

  A^2 == A*A

  A^(1/2) == sqrtm (A)

etc.

jwe
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[Sergei Steshenko-2]

--- "John W. Eaton" <jwe@...> wrote:

> On 15-Jan-2008, Sergei Steshenko wrote:
>
> | examples:
> |
> | "
> | octave:19> (-ones(1)) ^ 0.25
> | ans =  0.70711 + 0.70711i
> | octave:20> (-ones(2)) ^ 0.25
> | ans =
> |
> |    0.42045 + 0.42045i   0.42045 + 0.42045i
> |    0.42045 + 0.42045i   0.42045 + 0.42045i
> |
> | octave:21> (-ones(3)) ^ 0.25
> | ans =
> |
> |    0.31024 + 0.31022i   0.31018 + 0.31017i   0.31018 + 0.31021i
> |    0.31018 + 0.31017i   0.31025 + 0.31025i   0.31018 + 0.31019i
> |    0.31018 + 0.31021i   0.31018 + 0.31019i   0.31024 + 0.31021i
> |
> | octave:22> (-ones(3)) .^ (0.25 * ones(3))
> | ans =
> |
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |
> | octave:23> (ones(3)) ^ 0.25
> | ans =
> |
> |    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
> |    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
> |    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
> | "
> |
> | -
> |
> | "
> | octave:19> (-ones(1)) ^ 0.25
> | ans =  0.70711 + 0.70711i
> | ",
> |
> | "
> | octave:22> (-ones(3)) .^ (0.25 * ones(3))
> | ans =
> |
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> |    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
> | "
> |
> | are correct IMHO;
> |
> | "
> | octave:23> (ones(3)) ^ 0.25
> | ans =
> |
> |    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
> |    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
> |    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
> | "
> |
> | is especially worrysome because imaginary part appears out of nowhere.
>
> It is not surprising that the matrix cases are faulty for singular
> matrices or matrices with repeated eigenvalues given that they are
> implemented using a simple similarity transform.  I think that fixing
> that problem is not trivial.  Perhaps we should just try to provide a
> warning in these cases.  I'd certainly consider a patch for that.
>
> jwe
>

Do the following two examples also represent
"singular matrices or matrices with repeated eigenvalues":

"
octave:1> matr = [1,2,3;4,5,6;7,8,9]
matr =

   1   2   3
   4   5   6
   7   8   9

octave:2> matr ^ 0.25
ans =

   0.748810 + 0.524341i   0.418020 + 0.142211i   0.087323 - 0.239825i
   0.566187 + 0.057852i   0.640363 + 0.015765i   0.714353 - 0.026508i
   0.383656 - 0.408544i   0.862519 - 0.110868i   1.341476 + 0.186902i
",

"
octave:6> rand_matr = rand(3,3)
rand_matr =

   0.99802   0.48525   0.71332
   0.16917   0.97283   0.27022
   0.93073   0.65589   0.33549

octave:7> rand_matr ^ 0.25
ans =

   0.926082 + 0.130202i   0.085460 + 0.046235i   0.234396 - 0.182054i
   0.020287 + 0.037973i   0.971381 + 0.013484i   0.081891 - 0.053095i
   0.312304 - 0.252407i   0.181693 - 0.089630i   0.685046 + 0.352925i
"

?

Thanks,
  Sergei.

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[John W. Eaton]

On 15-Jan-2008, Sergei Steshenko wrote:

| Do the following two examples also represent
| "singular matrices or matrices with repeated eigenvalues":
|
| "
| octave:1> matr = [1,2,3;4,5,6;7,8,9]
| matr =
|
|    1   2   3
|    4   5   6
|    7   8   9

This matrix is obviously singular (and my personal favorite).  Do you
not know what the term singular matrix means?

| octave:6> rand_matr = rand(3,3)
| rand_matr =
|
|    0.99802   0.48525   0.71332
|    0.16917   0.97283   0.27022
|    0.93073   0.65589   0.33549
|
| octave:7> rand_matr ^ 0.25
| ans =
|
|    0.926082 + 0.130202i   0.085460 + 0.046235i   0.234396 - 0.182054i
|    0.020287 + 0.037973i   0.971381 + 0.013484i   0.081891 - 0.053095i
|    0.312304 - 0.252407i   0.181693 - 0.089630i   0.685046 + 0.352925i

That matrix is not singular, nor does it have repeated eigenvalues.
Furthermore, the result for ^0.25 that you show seems to me to be
correct, at least to a reasonable degree of numerical accuracy.
Couldn't you just check these things for yourself (hint: rank(x),
eig(x), abs(x - (x^0.25)^4) )?

jwe
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[Sergei Steshenko-2]

--- "John W. Eaton" <jwe@...> wrote:

> On 15-Jan-2008, Sergei Steshenko wrote:
>
> | Do the following two examples also represent
> | "singular matrices or matrices with repeated eigenvalues":
> |
> | "
> | octave:1> matr = [1,2,3;4,5,6;7,8,9]
> | matr =
> |
> |    1   2   3
> |    4   5   6
> |    7   8   9
>
> This matrix is obviously singular (and my personal favorite).  Do you
> not know what the term singular matrix means?
>
> | octave:6> rand_matr = rand(3,3)
> | rand_matr =
> |
> |    0.99802   0.48525   0.71332
> |    0.16917   0.97283   0.27022
> |    0.93073   0.65589   0.33549
> |
> | octave:7> rand_matr ^ 0.25
> | ans =
> |
> |    0.926082 + 0.130202i   0.085460 + 0.046235i   0.234396 - 0.182054i
> |    0.020287 + 0.037973i   0.971381 + 0.013484i   0.081891 - 0.053095i
> |    0.312304 - 0.252407i   0.181693 - 0.089630i   0.685046 + 0.352925i
>
> That matrix is not singular, nor does it have repeated eigenvalues.
> Furthermore, the result for ^0.25 that you show seems to me to be
> correct, at least to a reasonable degree of numerical accuracy.
> Couldn't you just check these things for yourself (hint: rank(x),
> eig(x), abs(x - (x^0.25)^4) )?
>
> jwe
>

The examples I gave here had been created before you explained me that my assumption
on exponentiation was wrong, so please disregard them.

Thanks,
  Sergei.




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[Rolf Fabian]

John W. Eaton wrote:

On 15-Jan-2008, Sergei Steshenko wrote:

| examples:
|
| "
| octave:19> (-ones(1)) ^ 0.25
| ans =  0.70711 + 0.70711i
| octave:20> (-ones(2)) ^ 0.25
| ans =
|
|    0.42045 + 0.42045i   0.42045 + 0.42045i
|    0.42045 + 0.42045i   0.42045 + 0.42045i
|
| octave:21> (-ones(3)) ^ 0.25
| ans =
|
|    0.31024 + 0.31022i   0.31018 + 0.31017i   0.31018 + 0.31021i
|    0.31018 + 0.31017i   0.31025 + 0.31025i   0.31018 + 0.31019i
|    0.31018 + 0.31021i   0.31018 + 0.31019i   0.31024 + 0.31021i
|
| octave:22> (-ones(3)) .^ (0.25 * ones(3))
| ans =
|
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|
| octave:23> (ones(3)) ^ 0.25
| ans =
|
|    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
|    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
|    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
| "
|
| -
|
| "
| octave:19> (-ones(1)) ^ 0.25
| ans =  0.70711 + 0.70711i
| ",
|
| "
| octave:22> (-ones(3)) .^ (0.25 * ones(3))
| ans =
|
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
|    0.70711 + 0.70711i   0.70711 + 0.70711i   0.70711 + 0.70711i
| "
|
| are correct IMHO;
|
| "
| octave:23> (ones(3)) ^ 0.25
| ans =
|
|    0.43874 + 0.00001i   0.43865 + 0.00000i   0.43869 - 0.00002i
|    0.43865 + 0.00000i   0.43876 + 0.00000i   0.43867 - 0.00001i
|    0.43869 - 0.00002i   0.43867 - 0.00001i   0.43872 + 0.00002i
| "
|
| is especially worrysome because imaginary part appears out of nowhere.

It is not surprising that the matrix cases are faulty for singular
matrices or matrices with repeated eigenvalues given that they are
implemented using a simple similarity transform.  I think that fixing
that problem is not trivial.  Perhaps we should just try to provide a
warning in these cases.  I'd certainly consider a patch for that.

jwe

Dear jwe

Concerning the first criterion (singularity of base matrix b)
I strongly agree.

On the other hand 'repeated eigenvalues' of base matrix b
is definitely a bad criterion for failure of  b^scalar.
As a counter example consider simply b=c*eye(n).
(scalar c). Then b pocesses n repeated eigenvalues
but b^scalar is well defined.

You need to check whether  b is 'defective'. That means
that correct indicator for failure of b^scalar  is rank deficiency
of the eigensysstem matrix which is formed from eigenvectors
of b.

Rolf Fabian

< r dot fabian at jacobs-university dot de>

[Rolf Fabian]

John W. Eaton wrote:

It is not surprising that the matrix cases are faulty for singular
matrices or matrices with repeated eigenvalues given that they are
implemented using a simple similarity transform.  I think that fixing
that problem is not trivial.  Perhaps we should just try to provide a
warning in these cases.  I'd certainly consider a patch for that.

jwe

Dear jwe

Concerning the first criterion (singularity of base matrix b)
I strongly agree.

On the other hand 'repeated eigenvalues' of base matrix b
is definitely a bad criterion for problems with calculation
of  b^scalar.
As a counter example consider simply b=c*eye(n).
(scalar c). Then b pocesses n repeated eigenvalues
but b^scalar is still well defined and can be easily calculated.

You need to check whether  b is 'defective'. That means
that correct indicator for problems with calculation of
b^scalar  is rank deficiency of the eigensysstem matrix
which is formed from eigenvectors of b.

But even if  b is found to be 'defective' this does
not mean that b^scalar  is undefined.
It can be calculated but this requires a
robust 'logm' function which is unfortunately
not available in Octave.


Rolf Fabian

< r dot fabian at jacobs-university dot de>

[John W. Eaton]

On 16-Jan-2008, Rolf Fabian wrote:

| But even if  b is found to be 'defective' this does
| not mean that b^scalar  is undefined.
| It can be calculated but this requires a
| robust 'logm' function which is unfortunately
| not available in Octave.

So, since you seem to be an expert in this area, how about
contributing one?

jwe
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