implement norms for matrices

[aghitza]

Add functionality to implement various types of norms on matrices; for starters, see
http://en.wikipedia.org/wiki/Matrix_norm

CC: @haraldschilly

Component: linear algebra

Issue created by migration from https://trac.sagemath.org/ticket/1763

[dfdeshom]

Attachment: trac_1763via2512.patch.gz

[dfdeshom]

comment:2

This patch adds the most common norm used for matrices: 1,2,\inf and Frobenius (and "entry-wise norm"). 2 points:

• I don't believe there's a general formula for the p-norm, although I hear estimations can be computed. I could be wrong though.
• I stuck this in matrix2.pyx and arbitrarily chose to use convert anything to RDF matrices since RDF matrices actually have a SVD method (which is needed for the 2-norm). I'd like to hear people comments on that.

[aghitza]

apply after the above patch

[aghitza]

comment:4

Attachment: 1763-2.patch.gz

Hi Didier,

I have a few concerns about the patch:

1. as it is, it only returns an RDF if p=2, and some other type depending on the coefficient of the matrix otherwise; it would be preferable to have a uniform behavior

2. for vectors, the syntax for the inf-norm is norm(Infinity), whereas here it is norm('inf'); it would be better to be consistent about this

3. there are a bunch of trivial typos in the docstring and comments

4. being a number theorist, to me 'frob' is much more suggestive of Frobenius than 'fro'.

Since all of this is fairly straightforward, I went ahead and put up an add-on patch.

[robertwb]

comment:5

This doesn't handle matrices with negative or complex entries correctly, it should be taking absolute values in several places.

+1 to AlexGhitza's changes though.

[aghitza]

apply instead of the other patches

[aghitza]

comment:6

Attachment: 1763-replacement.patch.gz

I modified the previous patches so that norm() works correctly on matrices with negative and/or complex entries. I also added a conjugate() function for conjugating a complex matrix.

1763-replacement.patch should be applied instead of the previous patches.

[mwhansen]

comment:7

Looks good to me.

[sagetrac-mabshoff]

comment:8

Merged in Sage 3.0.alpha5