Simple Matrix class

[Paul McGuire]

I've posted a simple Matrix class on my website as a small-footprint
package for doing basic calculations on matrices up to about 10x10 in
size (no theoretical limit, but performance on inverse is exponential).
Includes:
- trace
- transpose
- conjugate
- determinant
- inverse
- eigenvectors/values (for symmetric matrices)
- addition and multiplication (with constant or other matrix)

Matrices are easily built from formatted strings, as in:

m = Matrix( """1 2 3 4
5 11 20 3
2 7 11 1
0 5 3 1""")

Pretty much a no-strings-attached license, just don't hassle me about
little things like warranty, support, merchantability , accuracy, etc.
See it at
http://www.geocities.com/ptmcg/pytho...html#matrix_py

-- Paul

[Paul McGuire]

On Jan 24, 1:47 pm, Robert Kern <robert.k...@gm ail.comwrote:

Paul McGuire wrote:

And the purpose/motivation for "reimplemen ting it better" would be
what, exactly? So I can charge double for it?

So you can have accurate results, and you get a good linear solver out of the
process. The method you use is bad in terms of accuracy as well as efficiency.

Dang, I thought I was testing the results sufficiently! What is the
accuracy problem? In my test cases, I've randomly created test
matrices, inverted, then multiplied, then compared to the identity
matrix, with the only failures being when I start with a singular
matrix, which shouldn't invert anyway.

One of the main reasons I wrote this was as a simple linear solver for
small order problems, so I would like this to at least be able to do
_that_. Efficiency aside, I thought I was at least getting the right
answer...

-- Paul

[Paul Rubin]

"Paul McGuire" <pt***@austin.r r.comwrites:

Dang, I thought I was testing the results sufficiently! What is the
accuracy problem? In my test cases, I've randomly created test
matrices, inverted, then multiplied, then compared to the identity
matrix, with the only failures being when I start with a singular
matrix, which shouldn't invert anyway.

There's a lot of ill-conditioned matrices that you won't hit at
random, that aren't singular, but that are nonetheless very stressful
for numerical inversion. The Hilbert matrix a[i,j]=1/(i+j+1) is a
well known example.

If you're taking exponential time to invert matrices (sounds like
you're recursively using Cramer's Rule or something) that doesn't
begin to be reasonable even for very small systems, in terms of
accuracy as well as speed. It's a pure math construct that's not of
much practical value in numerics.

You might look at the Numerical Recipes books for clear descriptions
of how to do this stuff in the real world. Maybe the experts here
will jump on me for recommending those books since I think the serious
numerics crowd scoffs at them (they were written by scientists rather
than numerical analysts) but at least from my uneducated perspective,
I found them very readable and well-motivated.

[Robert Kern]

Paul McGuire wrote:

On Jan 24, 1:47 pm, Robert Kern <robert.k...@gm ail.comwrote:

>Paul McGuire wrote:

>>And the purpose/motivation for "reimplemen ting it better" would be
what, exactly? So I can charge double for it?

So you can have accurate results, and you get a good linear solver out of the
process. The method you use is bad in terms of accuracy as well as efficiency.

Dang, I thought I was testing the results sufficiently! What is the
accuracy problem? In my test cases, I've randomly created test
matrices, inverted, then multiplied, then compared to the identity
matrix, with the only failures being when I start with a singular
matrix, which shouldn't invert anyway.

Ill-conditioned matrices. You should grab a copy of _Matrix Computations_ by
Gene H. Golub and Charles F. Van Loan.

For example, try the Hilbert matrix n=6.

H_ij = 1 / (i + j - 1)

http://en.wikipedia.org/wiki/Hilbert_matrix

While all numerical solvers have issues with ill-conditioned matrices, your
method runs into them faster.

--
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless enigma
that is made terrible by our own mad attempt to interpret it as though it had
an underlying truth."
-- Umberto Eco

[Robert Kern]

Paul Rubin wrote:

You might look at the Numerical Recipes books for clear descriptions
of how to do this stuff in the real world. Maybe the experts here
will jump on me for recommending those books since I think the serious
numerics crowd scoffs at them (they were written by scientists rather
than numerical analysts) but at least from my uneducated perspective,
I found them very readable and well-motivated.

As a scientist (well, former scientist) and programmer, I scoff at them for
their code. The text itself is decent if you need a book that covers a lot of
ground. For any one area, though, there are usually better books. _Matrix
Computations_, which I mentioned elsewhere, is difficult to beat for this area.

--
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless enigma
that is made terrible by our own mad attempt to interpret it as though it had
an underlying truth."
-- Umberto Eco

[Paul McGuire]

On Jan 24, 3:18 pm, Robert Kern <robert.k...@gm ail.comwrote:

Ill-conditioned matrices. You should grab a copy of _Matrix Computations_ by
Gene H. Golub and Charles F. Van Loan.

For example, try the Hilbert matrix n=6.

H_ij = 1 / (i + j - 1)

Sure enough, this gets ugly at n=6.

Thanks for the reference to Matrix Computations; I'll try to track down
a copy next time I'm at Half-Price Books.

Meanwhile, it's back to the day job...

-- Paul