# $Author: saulius $
# $Revision: 9750 $
# $Date: 2022-02-08 06:24:00 +0000 (Tue, 08 Feb 2022) $

Compute the inverse octonion
============================

PROGRAM
========

Write a Perl program that would compute and inverse octonion for each
input octonion.

Program Name:
oinverse

Calling the program:
oinverse octonions.dat octonions_*. dat

When input files are specified, each of their data lines is records
one octonion. Input files are optional; if they are not provided, the
program must read data from its standard input.

ALGEBRA OF THE OCTONIONS
========================

Octonion algebra and its properties are described in Wikipedia [1] and
in most algebra textbooks (e.g. [2]).

Because octonion algebra is non-associative, there are several
multiplications table variants [1]; select multiplication table for
the program implementation that coincides with the the table in the
source [2] (it should also be the same as the table given in [1]).

DATA FORMATS
===============

Input format
------------

The input format is a vector format for octonion components,
described in general form in the file [3].

Example of input flow:

# First line of the file 'octonions.dat'; data:
#@ 1 i j k l il jl kl
   1 1 1 1 1 1 1 1 1
# Last line of the file 'octonions.dat'.

Each line in the file is a single octonion component listed
one after the other.

Output format
-------------

Data must be output to standard output in the same format as data in
input files (format described in [1]), indicating one octonion of the
result. The first line MUST be a comment listing the SVN of the
program used (or other version control systems) Id:

# Id: oinverse 1 2011-09-28 07: 41: 19Z author

Note that there should be no dollar symbols ('$') around the Id:
string, unlike in the SVN keywords (but SVN keywords SHOULD be used to
generate the string!)

In the second row, the program MUST output a row of column names:

#@ 1 i j k l il jl kl

Data must be provided below. Each line must contain one octonion, the
reverse octonion for the one in the corresponding data row.

Example output file:

# Id: oinverse 1 2011-09-28 07: 41: 19Z author
#@ 1 i j k l il jl kl
0.125 -0.125 -0.125 -0.125 -0.125 -0.125 -0.125 -0.125

REFERENCES
================ 

1. Wikipedia (2020) Octonion. https://en.wikipedia.org/wiki/Octonion
   [last accessed: 2020-02-15 19:33:49 EET]

2. H.-B. Ebbinghaus et al. (1995) Numbers. Springer, ISBN
   3-540-97497-0

3. Saulius Gražulis (2020) Hiperkompleksinių skaičių vektorinis formatas.
   https://saulius.grazulis.lt/~saulius/paskaitos/VU/bioinformatika-III/užduotys-praktikai/1-užduotis/formatai/hiperkompleksinių-skaičių-vektorinis-formatas.txt
   [last accessed: 2020-02-16 12:37:46 EET]