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1% Quaternion Toolbox
2% Version 1.0             14-December-2006
3% ---------------------------------------------------------------------
4% Copyright © 2005-6  Stephen J. Sangwine         (S.Sangwine@IEEE.org)
5%                     Nicolas Le Bihan   (nicolas.le-bihan@lis.inpg.fr)
6%                     See the file Copyright.m for further details.
7% ---------------------------------------------------------------------
8%
9% quaternion is a Matlab class library designed to extend Matlab in as
10% natural a way as possible to handle quaternions. Many standard Matlab
11% features have been implemented, including standard operators such as
12% the arithmetic operators, matrix and elementwise products, indexing
13% and indexed assignment using the colon operator, concatenation, end
14% indexing, transpose and conjugate transpose, raising to a power, etc.
15%
16% A major objective has been to make it possible to write code that can
17% work unchanged for real, complex, quaternion, and even complexified
18% quaternion arrays, and for this reason standard Matlab functions have
19% been overloaded for quaternion arrays.
20%
21% A quaternion object, as implemented by this class, has a private
22% implementation based on a structure array storing the four components
23% of the quaternion. As with any standard Matlab type, a quaternion is by
24% default a matrix. A single quaternion is simply a matrix of one element.
25%
26% Quaternions may be pure or full. Pure quaternions have no scalar part
27% and operations that attempt to access the scalar part will result in an
28% error. The quaternions in a vector or matrix are either all pure or all
29% full -- it is not possible to construct a vector or matrix with a mix
30% of the two. A full quaternion with a scalar part which is zero differs
31% from a pure quaternion, which has no scalar part.
32%
33% Quaternion matrices can be constructed either by using a constructor
34% function, like this:
35%
36% q = quaternion(eye(5,5), zeros(5,5), randn(5,5), ones(5,5))
37%
38% or by using the three quaternion operators, named q1, q2 and q3, (these
39% three operators are also available under the names qi, qj and qk) like this:
40%
41% q = eye(5,5) + zeros(5,5) * q1 + randn(5,5) * q2 + ones(5,5) * q3
42%
43% The components of the quaternion so constructed will have a type determined
44% by the type supplied for the components (e.g. double, uint8, int16 ....).
45% The components may be REAL or COMPLEX. A complex quaternion can also be
46% constructed like this:
47%
48% q = complex(quaternion(1,2,3,4), 3)
49%
50% or using the Matlab value i:
51%
52% q = quaternion(1,2,3,4) + quaternion(5,6,7,8) .* i
53%
54% The components of a quaternion can be extracted using the five functions:
55%
56%   scalar         - Scalar part of a quaternion.
57%   vector, v      - Vector part of a quaternion (synonyms).
58%   s, x, y, z     - Components of a quaternion.
59%
60%   real           - Real part of a (complex) quaternion.
61%   imag           - Imaginary part of a (complex) quaternion.
62%   complex        - Construct a complex quaternion from real quaternions.
63%   quaternion     - Construct a quaternion from real or complex values.
64%
65% Other functions implemented are:
66%
67%   abs            - Modulus of a quaternion.
68%   conj           - Conjugate (quaternion, complex or total).
69%   unit           - Normalise a quaternion.
70%   sign           - Equivalent to unit, cf Matlab sign.m.
71%   inv            - Matrix and quaternion inverse.
72%   axis           - Axis of a quaternion.
73%   angle          - Angle or argument of a quaternion.
74%
75%   ceil, floor    - Round elements of quaternion towards plus or minus
76%   fix, round       infinity, zero, or nearest integer.
77%
78%   display        - Display array (does not show values)
79%   disp           - Display without array name.
80%   displayall     - Display components of quaternion.
81%   show           - Shorter synonym for displayall.
82%   char           - Convert quaternion to string.
83%   fprintf        - Output quaternions to file.
84%   write          - Write a quaternion array   to a text file.
85%   read           - Read  a quaternion array from a text file.
86%
87%   convert        - Convert components of quaternion to a different type.
88%
89%   dot            - Dot product.
90%   cross          - Vector product.
91%
92%   exp            - Exponential function.
93%   log            - Natural logarithm.
94%
95%   sqrt           - Square root.
96%
97%   sin, cos, tan  - Trigonometric functions.
98%   asin, acos,
99%         atan     - Inverse trigonometric functions.
100%   sinh, cosh,
101%         tanh     - Hyperbolic functions.
102%   asinh, acosh,
103%          atanh   - Inverse hyperbolic functions.
104%
105%   diag           - Extract or construct a diagonal.
106%   triu/tril      - Extract upper or lower triangular.
107%   norm           - Vector and matrix norms.
108%   sum            - Sum elements or columns.
109%   mean           - Mean of elements or columns.
110%
111%   ispure         - Test whether a quaternion (array) is pure.
112%   isempty        - Test whether a quaternion (array) is empty.
113%   isfinite       - Test whether a quaternion (array) is finite.
114%   isinf          - Test whether a quaternion (array) is infinite.
115%   isnan          - Test whether a quaternion (array) is NaN.
116%   isreal         - Test whether a quaternion (array) is real.
117%   ishermitian    - Test whether a quaternion (array) is Hermitian.
118%   isunitary      - Test whether a quaternion (array) is unitary.
119%
120%   length         - Length of a quaternion vector.
121%   size           - Size of a quaternion array.
122%   numel          - Number of elements in a quaternion array.
123%   repmat         - Replicate and tile a quaternion array.
124%   cat            - Concatenate arrays.
125%
126%   det            - Determinant.
127%   svd            - Singular value decomposition.
128%   eig            - Eigenvalue decomposition.
129%
130%   adjoint        - The complex adjoint of a quaternion array.
131%   unadjoint      - Construct a quaternion array from an adjoint array.
132%
133%   conv, conv2    - Convolution.
134%
135%   fft            - One dimensional (default) quaternion Fourier transform.
136%   fft2           - Two dimensional (default) quaternion Fourier transform.
137%   qfft           - One dimensional left or right one-dimensional QFFT.
138%   qdft           - One dimensional left or right one-dimensional QDFT.
139%   qfft2          - Two dimensional left or right two-dimensional QFFT.
140%   qdft2          - Two dimensional left or right two-dimensional QDFT.
141%   fftshift       - Quaternion overloading of the standard Matlab function.
142%   .............  - All of the above have inverses, prefixed with 'i'.
143%
144% The following builtin Matlab functions also work for quaternion arrays:
145%
146%   flipud, fliplr, rot90, ndims, trace, isequal, isvector
147%
148% The following Matlab functions also work for quaternion arrays:
149%
150%   cov, princomp, rank, var (there may be others)
151%
152% The following function is not yet usable: reshape.
153%
154% There are some auxiliary functions which are used to compute some of the
155% more elaborate functions above, such as svd, eig and qdft. These are:
156%
157% change_basis, orthonormal_basis, orthogonal.
158% householder_vector, householder_matrix, bidiagonalize, tridiagonalize.
159%
160% Some test code is provided in the directory 'test'. To run it set the
161% working directory to 'test' and type 'test'. This runs all the test
162% code.
163%
164% For more information, use help.
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