Re: Roots
Posted by
Jon Peck on
Nov 16, 2019; 6:07pm
URL: http://spssx-discussion.165.s1.nabble.com/Roots-tp5738554p5738570.html
As I said earlier, if the exponent is a rational number, you (the user or the software) can in some cases rewrite the expression in order to do this calculation.
E.g.,
x**(b/c) = (x**b)**(1/c)
but this only works for even c if x is negative if you are staying away from complex numbers, which are not generally supported in statistical software.
But if the expression is
x**y
you would have to find the b/c form to do this, and in general this is mathematically impossible (irrational number) or very difficult, and even if you find it, that only works sometimes.
So this is a complex (pun intended) problem, and the behavior would be very confusing to most users. That means a lot of work for a small and not very useful improvement in functionality.
Even statistical packages can't do everything, so IMO there are much better uses for scarce resources.
On Sat, Nov 16, 2019 at 10:09 AM Stan Gorodenski <
[hidden email]> wrote:
Jon,
Yesterday I joined the SAS (Statistical Analysis Software) discussion
group to ask the question (I used to program in SAS in my previous job
and that is how I know about it). SAS cannot do it either. This really
surprises me because SAS is even more expensive to use than SPSS. One
cannot own a package, like SPSS. It is leased annually. It is directed
toward large businesses, educational institutions, and large
organizations that can afford it. I am so surprised that SPSS, and
especially SAS, cannot do this computation, given the money they cost.
Someone in the SAS discussion group thought Mathematica might be able to
do it.
Stan
On 11/16/2019 7:32 AM, Jon Peck wrote:
> I thought it would be interesting to explore negative numbers
> exponentiated to fractional powers in other systems. Here is what I
> found. (NaN is a value for not a number. Equivalent to SYSMIS.)
>
>
> SPSS Statistics
> compute z = (-1)**(1/3).
> SYSMIS
> Warning # 523
> >During the execution of the indicated command, an attempt was made to
> raise a
> >non-positive number to a fractional power, e.g. (-3)**2.5.
>
> golang
> output := math.Pow(-1, 1./3.)
> NaN
> No warning
>
> Python
> (-1)**(1./3.)
> no result - calculation is stopped
> ValueError: negative number cannot be raised to a fractional power
>
> R
> (-1)^(1/3)
> NaN
> No warning
>
> Although I have not seen the SPSS internal transformation code, I
> expect that it uses logs to calculate exponentiation with a fractional
> power, hence the error.
>
> The expression could be refactored to get a result in the case where
> the denominator of the exponent is odd but not as a general solution.
> Since Statistics, as with most statistical packages, does not support
> complex numbers in expressions, that solution is not available.
>
> From Wikpedia...
>
> The powers of negative real numbers are not always defined and are
> discontinuous even where defined. In fact, they are only defined when
> the exponent is a rational number with the denominator being an odd
> integer.
>
> If the definition of exponentiation of real numbers is extended to
> allow negative results then the result is no longer well-behaved.
>
> Neither the logarithm method nor the rational exponent method can be
> used to define /b/^/r/ as a real number for a negative real number
> /b/ and an arbitrary real number /r/. Indeed, /e/^/r/ is positive for
> every real number /r/, so ln(/b/) is not defined as a real number for
> /b/ ≤ 0.
>
> The rational exponent method cannot be used for negative values of
> /b/ because it relies on continuity
> <https://en.wikipedia.org/wiki/Continuous_function>. The function
> /f/(/r/) = /b/^/r/ has a unique continuous extension^[15]
> <https://en.wikipedia.org/wiki/Exponentiation#cite_note-Denlinger-15>
> from the rational numbers to the real numbers for each /b/ > 0. But
> when /b/ < 0, the function /f/ is not even continuous on the set of
> rational numbers /r/ for which it is defined.
>
> For example, consider /b/ = −1. The /n/th root of −1 is −1 for every
> odd natural number /n/. So if /n/ is an odd positive integer,
> (−1)^(/m///n/) = −1 if /m/ is odd, and (−1)^(/m///n/) = 1 if /m/ is
> even. Thus the set of rational numbers /q/ for which (−1)^/q/ = 1 is
> dense <https://en.wikipedia.org/wiki/Dense_set> in the rational
> numbers, as is the set of /q/ for which (−1)^/q/ = −1. This means
> that the function (−1)^/q/ is not continuous at any rational number
> /q/ where it is defined.
>
> On the other hand, arbitrary complex powers
> <https://en.wikipedia.org/wiki/Exponentiation#Powers_of_complex_numbers> of
> negative numbers /b/ can be defined by choosing a /complex/ logarithm
> <https://en.wikipedia.org/wiki/Complex_logarithm> of /b/.
>
>
>
>
> On Fri, Nov 15, 2019 at 10:26 AM Stan Gorodenski
> <[hidden email] <mailto:[hidden email]>> wrote:
>
> I don't know if there is any demand, but I think the square root
> of -1
> is used in electronics. I'm not into this so I really don't know.
> I did
> not pose this question because I have an application for it. I was
> just
> curious since it seems that sophisticated software packages like SPSS
> and SAS should be able to do it. I just sent an email to join the SAS
> discussion group and will ask them if SAS can do it. I suppose one
> could
> write a routine to return a -1 if the the denominator of the
> exponent is
> an odd number.
> Stan
>
> On 11/15/2019 10:05 AM, Rich Ulrich wrote:
> > I wonder - Is there any demand for the exception-coding
> > that would be necessary? How many people write code
> > where they want to take the fractional root of a negative
> > number, where the fraction is the reciprocal of an odd integer?
> > ( Note, the fraction cannot be expressed EXACTLY on a binary
> > computer. How is that accommodated?)
> >
> > The natural programming solution to non-integer roots is
> > to use logs. I suppose if there is an area where the problem
> > comes up, specialized programs for that area might do it.
> > I suspect the efficient solution might use a special subroutine
> > call rather than an in-line expression.
> >
> > --
> > Rich Ulrich
> >
> >
> ------------------------------------------------------------------------
> > *From:* SPSSX(r) Discussion <[hidden email]
> <mailto:[hidden email]>> on behalf of
> > Bruce Weaver <[hidden email]
> <mailto:[hidden email]>>
> > *Sent:* Friday, November 15, 2019 11:35 AM
> > *To:* [hidden email] <mailto:[hidden email]>
> <[hidden email] <mailto:[hidden email]>>
> > *Subject:* Re: Roots
> >
> https://www.ibm.com/support/knowledgecenter/en/SSLVMB_26.0.0/statistics_reference_project_ddita/spss/base/syn_transformation_expressions_domain_errors.html
> >
> >
> >
> > Kirill Orlov wrote
> > > See DOMAIN ERRORS paragraph in Command Syntax Reference.
> > >
> > >
> > > 15.11.2019 4:53, Stan Gorodenski пишет:
> > >> compute z = (-1)**(1/3).
> > >
> > > =====================
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> >
> >
> > -----
> > --
> > Bruce Weaver
> > [hidden email] <mailto:[hidden email]>
> > http://sites.google.com/a/lakeheadu.ca/bweaver/
> >
> > "When all else fails, RTFM."
> >
> > NOTE: My Hotmail account is not monitored regularly.
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> >
> > --
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> --
> Jon K Peck
> [hidden email] <mailto:[hidden email]>
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