J C Nash
on Sun, 16 Jul 2023 13:30:57 -0400 writes:
> Better check your definitions of SVD -- there are several
> forms, but all I am aware of (and I wrote a couple of the
> codes in the early 1970s for the SVD) have positive
> singular values.
Indeed.
More generally, the decomposition A = U D V'
(with diagonal D and orthogonal U,V)
is not at all unique.
There are not only many possible different choices of the sign
of the diagonal entries, but also the *ordering* of the singular values
is non unique.
That's why R and 'Lapack', the world-standard for
computer/numerical linear algebra, and others I think,
make the decomposition unique by requiring
non-negative entries in D and have them *sorted* decreasingly.
The latter is what the help page help(svd) always said
(and you should have studied that before raising such concerns).
-----------------------------------------------------------------
To your second point (in the document), the matrix exponential:
It is less known, but still has been known among experts for
many years (and I think even among students of a class on
numerical linear algebra), that there are quite a
few mathematically equivalent ways to compute the matrix exponential,
*BUT* that most of these may be numerically disastrous, for several
different reasons depending on the case.
This has been known for close to 50 years now:
Cleve Moler and Charles Van Loan (1978)
Nineteen Dubious Ways to Compute the Exponential of a Matrix
SIAM Review Vol. 20(4)
https://doi.org/10.1137/1020098
Where as that publication had been important and much cited at
the time, the same authors (known world experts in the field)
wrote a review of that review 25 years later which I think (and
hope) is even more widely cited (in R's man/*.Rd syntax) :
Cleve Moler and Charles Van Loan (2003)
Nineteen dubious ways to compute the exponential of a matrix,
twenty-five years later. \emph{SIAM Review} \bold{45}, 1, 3--49.
\doi{10.1137/S00361445024180}
i.e. https://doi.org/10.1137/S00361445024180
It is BTW also cited on the Wikipedia page on the matrix
exponential:
==> For this reason, Professor Douglas Bates, the initial
creator of R's Matrix package (which comes with R) has added the
Matrix exponential very early to the package:
------------------------------------------------------------------------
r461 | bates | 2005-01-29
Add expm function
------------------------------------------------------------------------
Later, I've fixed an "infamous" bug :
------------------------------------------------------------------------
r2127 | maechler | 2008-03-07
fix the infamous expm() bug also in "Matrix" (duh!)
------------------------------------------------------------------------
Then, Vincent Goulet and Christophe Dutang wanted to provide more
versions of expm() and we collaborated, also providing expm()
for complex matrices and created the CRAN package {expm}
--> https://CRAN.R-project.org/package=expm
which already provided half a dozen different expm algorithms.
But research and algorithms did not stop there. In 2008, Higham
and collaborators even improved on the best known algorithms
and I had the chance to co-supervise a smart Master's student
Michael Stadelmann to implement Higham's algorithm and we even
allowed to tweak it {with optional arguments} as that was seen
to be beneficial sometimes.
See e.g.,
https://www.rdocumentation.org/packages/expm/versions/0.999-7/topics/expm
> On 2023-07-16 02:01, Durga Prasad G me14d059 wrote:
>> Respected Development Team,
>>
>> This is Durga Prasad reaching out to you regarding an
>> issue/concern related to Singular Value Decomposition SVD
>> in R software package. I am attaching a detailed
>> attachment with this letter which depicts real issues
>> with SVD in R.
>>
>> To reach the concern the expressions for the exponential
>> of a matrix using SVD and projection tensors are obtained
>> from series expansion. However, numerical inconsistency
>> is observed between the exponential of matrix obtained
>> using the function(svd()) used in R software.
>>
>> However, it is observed that most of the researchers
>> fraternity is engaged in utilising R software for their
>> research purposes and to the extent of my understanding
>> such an error in SVD in R software might raise the
>> concern about authenticity of the simulation results
>> produced and published by researchers across the globe.
>>
>> Further, I am very sure that the R software development
>> team is well versed with the happening and they have any
>> specific and resilient reasons for doing so. I would
>> request you kindly, to guide me through the concern.
>>
>> Thank you very much.