Re: Problem with Singular Value Decomposition Algorithm

["Jim Love" <Jim.Love@asml.com>]

In every text that I have read dealing with the process of SVD, they have always placed the eigenvalues in a decreasing order.  The stated reason is to provide a standard solution and methodology.  Why would anybody build an API that does something different??  If you are using the SVD to find the least squares fit of a plane or a cylinder for example, order matters (if you want the right answer hehehe).  Yes I can add to my code to re-order the output, but so will most everybody else that uses this function.  So it should be done in the API.  Just a suggestion.

James A. Love
X4477
Pager 1-800-286-1188 Pin# 400659

>>> James Theiler <jt@lanl.gov> 09/12/01 10:51AM >>>
On Wed, 12 Sep 2001, Jim Love wrote:

] I have downloaded the latest beta release and the SVD algorithm
] produces the wrong answers.  It appears that columns are swapped
] in the output and possibly a sign problem.
]
] Here was my test array:
]
] 1 1 0.975
] 1 -1 0.975
] -1 -1 -0.925
] -1 1 -1.025
]
] The correct answer for the S vector is:  2.7940 2.0000 0.0358
] The gls output was: 2.0000 2.7940 0.0358
]
] The Correct Q matrix is:
]
] -0.7155    0.0256   -0.6981
]     0.0183    0.9997    0.0179
]    -0.6983   -0.0000    0.7158
]
] The gls output was:
]
] -0.025633 -0.715538 -0.698103
] -0.999671 0.018347 0.017900
] -0.000000 -0.698332 0.715774
]
] A similar problem was seen in the U matrix.
]
] Any ideas?  Is this caused by my implementation or is it a real bug?

I don't believe this is a bug at all.
Both answers are correct.  For SVD, the algorithm is
asked to find matrices U,S,Q such that U.S.Q' equals
the original matrix.  If you swap the columns of Q
and swap the equivalent rows of U, and also swap the
corresponding elements of S, you have another solution.

Also if you multiply one of the columns of Q by -1,
and multiply the corresponding row of U by -1, you
will get another equivalent solution.

Sometimes you want the solution with the eigenvalues (the
diagonal elements of the S matrix -- represented as a vector)
sorted numerically, as the one you cite as "correct" but
that is a convenience that can be performed after the fact.
And in fact, I notice that the documentation says that
eigenvalues are "generally chosen" to form a non-decreasing
sequence.  Maybe worth a sentence to say that these algorithms
do not always do that, or else add a utility that does this
postprocessing step.

jt

---------------------------------------------
James Theiler                     jt@lanl.gov 
MS-D436, NIS-2, LANL        tel: 505/665-5682
Los Alamos, NM 87545        fax: 505/665-4414
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