Subject: Re: Computing derivative of det(A), A singular
From: mbk@caffeine.engr.utk.edu (Matt Kennel)
Date: 8 Dec 1996 20:44:38 GMT
John Hench (smshenc@rdg.ac.uk) wrote:
: I'd like to add to this discussion some references
: concerning matrix derivaties. They are "Matrix
: Derivatives" by Gerald Rogers and "Kronecker
: Products and Matrix Calculus" with Applications by
: A. Graham.
: BTW, Graham's book mentions that d|X|/dX is
: |X|X^{-1}, which is just the adj(X), so the
: question is how do you accurately compute the
: adjugate of a matrix, right?
Usually given matrix formulae with inverses the best way to
numerically solve them is to rewrite as a system of equations
in LAPACK-et-al-compatible form without explicit inverses.
e.g. if the desired quantity Q is
Q = |X| X^{-1}
rewrite as
Q*X = |X|
and solve for Q with whatever library routine is appropriate. Often
they want the unknown multiplied on the right, so take transposes:
X^T * Q^T = |X|^T
{ A * UNKNOWN = B } (standard LAPACK template)
solve for Q^T, giving Q after post processing.
--
Matthew B. Kennel/mbk@caffeine.engr.utk.edu/I do not speak for ORNL, DOE or UT
Oak Ridge National Laboratory/University of Tennessee, Knoxville, TN USA/
Subject: Re: Radius algorithm
From: Karine Lecarpentier
Date: Sun, 08 Dec 1996 13:26:18 -0800
> In article <329CB599.62CD@worldnet.att.net>,
> Marcus says:
>I have a number of x,y data points that form an arc of constant radius.
> Do you know a good technique for finding the center and radius of the arc?
>I've tried the following, but I'm not sure of it's reliability:
>take first data point, find line running through point but also tangent
>to the arc, take second data point and find line tangent again.
>Normalize both lines and find the intersection. This will give me the
>center, from here the radius is easy.
>
>Do you have any suggestions?
>Thanks, Marc
As some previous persons sugested, you shoud use an optimization method.
If C is your center, and M(i), the points of your trajectory,then
your radius is :
R = 1/N *sum_{i=1}^N {||M(i) - C ||},
You can find C by minimizing
m = sum_{k=1}^N {||M(k) - c|| - R}^2
= sum_{k=1}^N {||M(k) - c|| -
1/N *sum_{i=1}^N {||M(i) - C ||}^2
Or, if you are sure that your curve is a circle, you can use a linear
algorithm by setting :
R = sqrt (1/N * sum_{i=1}^N {||OM(i) - OC ||^2})
= sqrt (1/N * sum_{i=1}^N {|OM(i)|^2 - 2 t(OM(i)).OC + |OC|^2})
You want to minimize :
m = sum_{k=1}^N {||OM(k) - OC||^2 - R^2}^2
= sum_{k=1}^N { |OM(k)|^2 - 1/N*sum_{i=1}^N |OM(i)|^2 -
2*t(OM(k)-1/N*sum{i=1}^N OM(i)).OC}^2
You get then an easy to solve linear least squares paroblem where your
unknown is OC.
If you derive this expression with respect to OC, and set the derivation
to 0, you will get a simple equation A.OC = b, where A is a 2*2 matrix,
and b is a 2*1 vector.
I hope this helps. Feel free to contact me for more details.
Karine
----
Karine Lecarpentier, M.A.Sc student
UBC, Dept. of Electrical Engineering e-mail: karinel@ee.ubc.ca
2356 Main Mall phone: (604) 822-0532
Vancouver, B.C fax: (604) 822-5949
V6T 1Z4 Canada
Subject: Center of gravity...
From: Regis Mesnier
Date: Sun, 08 Dec 1996 20:00:27 -0500
Dear net users,
Having been recently asked by one of my student if (considering any
triangle ABC and G the center of gravity of that triangle) the angles
AGB, BGC, and CGA were all eguals (to 120 degres, of course), I came up
with an answer, but could not find the formal proof for it. If any of
you could indicate me where I'd be able find it, or even show it here, I
would sincerely appreciate it. Thanks in advance.
Regis Mesnier, e-mail:swann2@earthlink.net
Subject: Re: pointers on numerical analysis history?
From: jasonp@Glue.umd.edu (Jason Stratos Papadopoulos)
Date: 9 Dec 1996 05:25:46 GMT
Jean Debord (jdebord@MicroNet.fr) wrote:
: kamthan@cs.concordia.ca (KAMTHAN pankaj) wrote:
: >Could anybody suggest any pointers to articles on the historical
: >evolution of any of the various aspects of numerical analysis:
: >LU decompostion, Newton's method, Simpson's rule, etc.?
The more esoteric stuff like Gaussian quadrature, the Euler-Maclaurin
formula, continued fractions and such can be found in
Herman H. Goldstine, A History of Numerical Analysis from the 16th
Through the 18th Century. I think it mentions all the things you want,
except maybe the LU decomp (should have something about Gaussian elim.
though)
jasonp
