This file is part of the Independent JPEG Group's software.

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with respect to this software, its quality, accuracy, merchantability, or
fitness for a particular purpose.  This software is provided "AS IS", and
you, its user, assume the entire risk as to its quality and accuracy.

This software is copyright (C) 1994-1996, Thomas G. Lane.
All Rights Reserved except as specified below.

Permission is hereby granted to use, copy, modify, and distribute this
software (or portions thereof) for any purpose, without fee, subject to
these conditions:
(1) If any part of the source code for this software is distributed, then
this README file must be included, with this copyright and no-warranty
notice unaltered; and any additions, deletions, or changes to the original
files must be clearly indicated in accompanying documentation.
(2) If only executable code is distributed, then the accompanying
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of the Independent JPEG Group".
(3) Permission for use of this software is granted only if the user accepts
full responsibility for any undesirable consequences; the authors accept
NO LIABILITY for damages of any kind.

These conditions apply to any software derived from or based on the IJG
code, not just to the unmodified library.  If you use our work, you ought
to acknowledge us.

Permission is NOT granted for the use of any IJG author's name or company
name in advertising or publicity relating to this software or products
derived from it.  This software may be referred to only as "the Independent
JPEG Group's software".

We specifically permit and encourage the use of this software as the basis
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assumed by the product vendor.

This file contains a fast, not so accurate integer implementation of the
forward DCT (Discrete Cosine Transform).

A 2-D DCT can be done by 1-D DCT on each row followed by 1-D DCT
on each column.  Direct algorithms are also available, but they are
much more complex and seem not to be any faster when reduced to code.

This implementation is based on Arai, Agui, and Nakajima's algorithm for
scaled DCT.  Their original paper (Trans. IEICE E-71(11):1095) is in
Japanese, but the algorithm is described in the Pennebaker & Mitchell
JPEG textbook (see REFERENCES section in file README).  The following code
is based directly on figure 4-8 in P&M.
While an 8-point DCT cannot be done in less than 11 multiplies, it is
possible to arrange the computation so that many of the multiplies are
simple scalings of the final outputs.  These multiplies can then be
folded into the multiplications or divisions by the JPEG quantization
table entries.  The AA&N method leaves only 5 multiplies and 29 adds
to be done in the DCT itself.
The primary disadvantage of this method is that with fixed-point math,
accuracy is lost due to imprecise representation of the scaled
quantization values.  The smaller the quantization table entry, the less
precise the scaled value, so this implementation does worse with high-
quality-setting files than with low-quality ones.