I was reading a 1960's article by Edsger Dijkstra in which he lamented
absence of triangular arrays (or ability to specify different shapes for
arrays) in Algol 60.

What are some uses of triangular arrays (or other non rectangular shaped
arrays)?

What might a fortran-compatible/based declaration syntax look like?

What might an element access syntax look like?

--

Gary Scott
mailto:garylscott@sbcglobal dot net

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Gary Scott wrote:

As in the storage of a symmetric matrix, I suppose.

In traditional Fortran, a single subscripted array big enough to hold
the matrix elements.

Paraphrasing from the Levine-Callahan-Dongarra vectors benchmark
(netlib.org)

On 2007-08-11 01:00:00 -0300, Gary Scott sbcglobal.net> said:

Matrix inversion is for intoductory algebra books. Real men (to use a
curious metaphore) use TRIANGULAR decompositions to allow easy back
solving in a numerical stable fashion.

Gordon Sande wrote:

Gary Scott wrote:

Once upon a time memory was scant and expensive. With memory to spare,
it is simpler to store a second factor, or the original of a symmetric
matrix, in the other half of a square matrix. The additional memory
serves a potentially useful purpose and the array indexing remains
simple and routine.

Charles Russell bellsouth.net> wrote:

Yep, Been there. Done that. Wrote my own versions of some of those
in-place algorithms. (Not that mine were anything but mundane; in those
days one mostly wrote one's own code unless what you needed happened to
be in a library package that your site owned.)

Richard Maine wrote:

On 2007-08-11 11:38:04 -0300, Tim Prince sbcglobal.net> said:

Gary Scott sbcglobal.net> wrote:

Oh. I see. Well, triangular arrays are indeed used a lot. There are some
cases where triangular arrays are directly used. There are probably even
more cases where symmetric arrays are used, and a symmetric (or
Hermetian, if one is talking complex numbers) matrix can be stored as
(or reduced to) a triangular one. Many, many problems inherently involve
symmetric matrices. For example, your basic quadratic form uses a
symmetric matrix, and thus also does any algorithm that uses a quadratic
form as an approximation (that would be a lot of algorithms).

Many libraries have special procedures for handling symmetric or
triangular matrices. it isn't just for the storage benefit either. There
are cases where you can use far "better" algorithms that apply only to
symmetric matrices. That gets off into the "math stuff".

There is excellent commantary above on the maths and usfulness of
Triangular Matrices.
I won't (can't?) compete.
However, once upon a time you used triangula methods, but wrote out
full matrix rows as temporary storage, so the theoretical part was
internal and triangular and the practical part was external and square
(and usually tape). So you had no need for triangular-to-square and
back operations for internal work other than non-general private
solving of problems.

Now you just use general square internal operations and let the
operating system automatically handle the temporary storage problem
should it be needed and just call on your huge rows as (if ever?)
needed.

It's called "Moot".