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Re: Matrices and "type tags"

 From: Jon Wilkening Subject: Re: Matrices and "type tags" Date: Thu, 9 Dec 1999 12:42:50 -0800 (PST)

```There are all sorts of matrices it would be nice to be able to
recognize:  triangular, tridiagonal, SPD, orthogonal, nearly singular...
In matlab I think they try solve A\b by back-substitution if A
is triangular, by cholesky if it is SPD, and otherwise by Gaussian
elimination with partial pivoting -- but you never really know
what it decided to do.  Perhaps instead of type-tags and the
all-purpose backslash there should be a family of less elegant
commands that give more control over how the system is solved.
(Maybe there are such commands already?)

Jon Wilkening

On Thu, 9 Dec 1999, Michael Hanke wrote:

> On Thu, 09 Dec 1999, Thomas Hoffmann wrote:
> >In the recent talk about a "backsub" function the idea of automatically
> >recognizing
> >tridiagonal matrices surfaced.
> >
> >If a matrix has not a special type "tridiagonal", then it has to have a
> >"tag", which says
> >that the matrix has this property (RLaB uses such "tags", matrices are lists
> >there) or
> >one has to check the matrix for this property (which takes a lot of time).
> [snip]
> >Any opinions?
> I was thinking about that subject a little bit. I do not think that it is
> necessary to introduce a special tag for tridiagonal matrices. I even do not
> like the idea of having a special backsub function. Instead, it would be
> better
> to overload the backslash operator. If this is done on the fortran level, it
> does not seem to be expensive.
>
> The lu decomposition is an order n^3 process, using a tridiagonal matrix
> (backsubstitution) is an order n^2 process. Testing for tridiagonality (even
> permuted tridiagonality as obtained after [L,U]=lu(A)) has exactly the same
> complexity. If the rows of L must be permuted first, an O(n log n) sorting
> process is additionally involved. So the loss in efficiency is probably less
> than a factor of 2. I think that this is tolerable.
>
> Michael
>
> --
> +---------------------------------------------------------------+
> |  Michael Hanke                Royal Institute of Technology   |
> |                               S-10044 Stockholm               |
> |                               Sweden                          |
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> |  Visiting address:            Lindstedtsvaegen 3              |
> |  Phone:                       + (46) (8) 790 6278             |
> |  Fax:                         + (46) (8) 790 0930             |
> +---------------------------------------------------------------+
>
>
>
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