From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Jacobian of the SVD
Date: 29 Mar 2000 11:01:39 -0500
Newsgroups: sci.math.num-analysis
Summary: [missing]

My mathematical reply is found at the end of this article.

Here is a request: Please set up your e-mail editor to:

(1) break the lines after at most 75 characters, so that your text
    does not flow out of the screen, or does not break in a ragged
    fashion (I broke the lines myself without further editing),

(2) suppress html version of the message. Pretty-printing is somehow
    unnecessary in usenet.

[Math reply at the end, as I promised.]

In article <38E20DE5.3F1A5063@sophia.inria.fr>,
Theodore.Papadopoulo <Theodore.Papadopoulo@sophia.inria.fr> wrote:
:
:--------------F6F26C69EB93F2DCF6B3D254
:Content-Type: text/plain; charset=us-ascii
:Content-Transfer-Encoding: 7bit
:
:I'm interested in performing error propagation in some
:computations. Those computations involving singular value
:
:decompositions of some matrices, my current approach has been to derive
:the formulae that give the Jacobian of
:
:the decomposition with respect to the variables of the initial matrix.
:
:Ie if my matrix is decomposed as A = U D V' then I get the various
:dUij/dAij, dDi/dAij and dVij/dAij. These
:
:Jacobians are then used to obtain the covariance matrices of the overall
:computation.
:
:I have two questions:
:
:- Are there any bibliographic references relevant to that problem that I
:can use ?
:
:- Is there a better way to achieve this ?
:
:    Thanks a lot,
:
:            Theo.
:

Bad news first:
 The correspondence

 (input: a matrix) -> (output: its SVD)

is not a function to start with. The signs of the vectors in the
unitary factors can be changed, so some normalization is needed, and
this normalization may bring in extra instabilities (say you force the
first non-zero entry to be positive, but the number of preceding zero
entries is an integer-valued function, hence discontinuous as a function 
of small perturbations, if it is not constant by some lucky coincidence).
So, one can say goodbye to differentiability, especially near repeated
singular values.


Not-so-bad news:
 Literature about perturbations of singular values is quite rich, and
often presents results in the form of inequalities, which may be better
than first-order perturbation study using derivatives.

A long list of results and references can be found in

Nicholas J. Higham: Accuracy and Stability of Numerical Algorithms,
SIAM Philadelphia 1996,
ISBN 0-89871-355-2 (pbk).

More names connected with this topic: G.W. Stewart - dozens of
publications, but perhaps this one will address your concerns:

GWS + Ji-guang Sun: Matrix Perturbation Theory
Academic Press, London 1990, 
ISBN 0-12-670230-6

Good luck, ZVK(Slavek).