Hi Durga,
There’s an error in your calculations here. You mention that for the SVD
of a symmetric matrix, we must have U=V, but this is not a correct
statement. The unitary matrices are only equivalent if the matrix A is
positive semidefinite.
In your example, you provide the matrix {{1,4},{4,1}}, which has
eigenvalues 5 and -3. This is not positive semidefinite, and thus there's
no requirement that the unitary matrices be equivalent.
If you verify your example with something like wolfram alpha, you’ll find
that R’s solution is correct.
-Aidan
-----------------------
Aidan Lakshman (he/him) <https://www.ahl27.com/>
Doctoral Fellow, Wright Lab <https://www.wrightlabscience.com/>
University of Pittsburgh School of Medicine
Department of Biomedical Informatics
ah...@pitt.edu
(724) 612-9940
------------------------------
*From:* R-devel <r-devel-boun...@r-project.org> on behalf of Durga Prasad
G me14d059 <me14d...@smail.iitm.ac.in>
*Sent:* Tuesday, August 1, 2023 4:18:20 AM
*To:* Martin Maechler <maech...@stat.math.ethz.ch>; r-devel@r-project.org
<r-devel@r-project.org>; profjcn...@gmail.com <profjcn...@gmail.com>
*Subject:* Re: [Rd] Concerns with SVD -- and the Matrix Exponential
Hi Martin, Thank you for your reply. The response and the links provided by
you helped to learn more. But I am not able to obtain the simple even
powers of a matrix: one simple case is the square of a matrix. The square
of the matrix using direct matrix multiplication operations and svd (A = U
D V') are different. Kindly check the attached file for the complete
explanation. I want to know which technique was used in building the svd in
R-Software. I want to discuss about svd if you schedule a meeting.
Thanks and Regards
Durga Prasad
On Mon, Jul 17, 2023 at 2:13 PM Martin Maechler <
maech...@stat.math.ethz.ch>
wrote:
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.
> JN
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://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1137%2F1020098&data=05%7C01%7Cahl27%40pitt.edu%7C8575b77db32345ca544b08db927ceae0%7C9ef9f489e0a04eeb87cc3a526112fd0d%7C1%7C0%7C638264837816871329%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=Y4mlFL%2FggLKd7FoIoY62esiFGUwukRG0YmELsJj7nd0%3D&reserved=0
<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://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1137%2FS00361445024180&data=05%7C01%7Cahl27%40pitt.edu%7C8575b77db32345ca544b08db927ceae0%7C9ef9f489e0a04eeb87cc3a526112fd0d%7C1%7C0%7C638264837817183809%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=5%2FlssUGC6q7SUy0PY7gZCqWi0%2BXbNwZD0FaAgIcOWdY%3D&reserved=0
<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://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Fcran.r-project.org%2Fpackage%3Dexpm&data=05%7C01%7Cahl27%40pitt.edu%7C8575b77db32345ca544b08db927ceae0%7C9ef9f489e0a04eeb87cc3a526112fd0d%7C1%7C0%7C638264837817183809%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=LlAzmeRZd5tNqAgTuYTTSsSakWEj85%2B%2F%2FP%2FM0DnZNLk%3D&reserved=0
<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://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.rdocumentation.org%2Fpackages%2Fexpm%2Fversions%2F0.999-7%2Ftopics%2Fexpm&data=05%7C01%7Cahl27%40pitt.edu%7C8575b77db32345ca544b08db927ceae0%7C9ef9f489e0a04eeb87cc3a526112fd0d%7C1%7C0%7C638264837817340048%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=ZrT%2FYvklccqLCORBMf6nGop5o3n7O2thkknG1UAS2Gc%3D&reserved=0
<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.
