From bug-octave-request at bevo dot che dot wisc dot edu  Fri Dec 22 10:49:25 2000
Subject: inner product in complex n-space
From: Rolf Fabian <fabian at tu-cottbus dot de>
To: "'bug-octave UWISC'" <bug-octave at bevo dot che dot wisc dot edu>
Date: Fri, 22 Dec 2000 10:49:23 -0600 (CST)


Hi

last spring, I've sent a bug report to this list
( see http://www.octave.org/mailing-lists/bug-octave/1999/44 )
and the proposed changes were adopted to ,dot.m'.

Now, I'd to realize, that the mixing of complex conjugation and transposition
operations is still incorrect  and  that the function dot( x,y ) V2.1.32 -script may
still fail if complex input vectors x,y are involved.
( Real vectors are handled correctly )

So I've completely rewritten the function script  with respect to the following points

1) give a clear definition of the used  ,dot'-product in complex n-space.

I gave preference to the following definition, given by most textbooks
e.g.  by  Seymour Lipschutz ,Linear Analysis''  (Schaum)

		    length
	dot ( x, y ) = SUM  x(k) * conj( y(k) )		<==
		      k=1

The inner product in complex Euklidian n-space must
be defined asymmetrically concerning x and y vectors
(see e.g. also given ref).


NOTE that an alternate possible definition might be 

		    length
	dot ( x, y ) = SUM  conj(x(k)) * x (k) 
		      k=1

	which is quite different from the one given above.

	So, in order to avoid any confusion, the preferenced definition (<== )
	appears in the help-header of the script

2) all script calls removed  e.g. ,is_vector' ,  only builtin -funcs used
3) all if clauses removed, instead  x( : )   y( : )  feature used inside
an unwind_protect block, because setting global
,do_fortran_indexing=1' is required.
Then the central function module simplifies considerably to

	 Z =  x( : ).'  *  conj (  y( : )  )

This always computes the correct answer complying to the
choosen definition of the dot operation.  It does no more depend
on any of the 4 possible combinations of x,y - vector
orentations as was the case in previous versions of ,dot.m'.

Check-by 
:>x=(t=randn(6,1) + randn(6,1)*i ) / norm(t)
# this constructs a complex random norm(x)==1 column vector
# next four lines use the revised script from below
:> dot( x , x   )	|- ans = 1
:> dot( x , x.' )	|- ans = 1
:> dot( x.',x  )	|- ans = 1
:> dot( x.',x.' )	|- ans = 1
so, all 4 combinations of orientations OK !
Note, that this check fails with current V2.1.32 dot-script (also all previous versions).

Bye Rolf



--------------------------------------- snip ------------------------------------------------
## Copyright (C) 1998 John W. Eaton
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2, or (at your option)
## any later version.
##
## This program is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this file.  If not, write to the Free Software Foundation,
## 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

## -*- texinfo -*-
##  at deftypefn {Function File} {} dot (@var{x}, @var{y})
## Computes the dot product of two vectors defined as
##	 at var{x}(1)*conj(@var{y}(1)) + .. + @var{x}(N)*conj(@var{y}(N))
##  Both vectors may have any orientation, but their length N must be the same.
##  at end deftypefn

## Author: jwe
## completely rewritten by Rolf Fabian <fabian at tu-cottbus dot de>  001220

function z = dot(x,y)

if	(nargin != 2)
	usage ("dot( x,y )");
endif

save_do_fortran_indexing = do_fortran_indexing;

unwind_protect
	do_fortran_indexing = 1;

	if	( any(size(x)==1) && any(size(y)==1) && length(x)==length(y) )
		z = x(:).' * conj( y(:) );
	else
		error ("dot: both arguments must be vectors of the same length");
	endif

unwind_protect_cleanup

	do_fortran_indexing = save_do_fortran_indexing;

end_unwind_protect

endfunction
--------------------------------------- snip ------------------------------------------------



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