I'd like to improve the current state of the linear algebra code over CDF (and by extension, over RDF). The purpose would be to make Sage more usable for teaching various topics involving matrices with complex entries and orthogonal vectors (thus introducing square roots). You can actually do quite a bit of this over the rationals, where (to my surprise) the square roots get approximated by rationals, and the accumulated errors are quite small. But it makes more sense to me to actually work over CDF and liberally employ the .zero_at() method when showing students how to interpret the results.
In any event, the goal would be to work up to things like QR decompositions, unitary matrices, orthogonal projections, etc. A lot of this is present, but not always documented well. I have not done a comprehensive survey yet, but for example, everything about QR decomposition talks about (and doctests) real matrices, even though the calls to NumPy are doing the right thing with complex entries and returning Q as unitary (not just orthogonal). But there needs to be some prerequiste gaps filled for students to study these objects, the algorithms, etc (such as there is not yet a function to take the conjugate of a vector). The inner product especially has me stumped. The following does not seem to be what we want at all: sage: v = vector(CDF, [I, I]) sage: v.inner_product(v) -2.0 I'd like to have the "usual" version with a conjugate of one of the vectors prior to the dot product (the sesquilinear form). I can't see that setting a custom inner product matrix can make this happen, but maybe I'm missing something. Should the inner product be over-ridden for vector spaces over the complex numbers? Or, I hate to ask, should there be something new, like .complex_inner_product(), or .sesquilinear_inner_product()? (Just kidding about that last one.) Or maybe this inner product is lurking somewhere and I'm not noticing it. I know Jason Grout has done a lot of work to integrate this with NumPy. I'm trolling for anything else folks can send me that might help: advice, informative Trac tickets, potential pitfalls, secret desires. Thanks! Rob -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
