New in Quaternion toolbox for Matlab 2.0 (Apr 3, 2013)
- This major new release adds an additional class to the toolbox, for computing with octonions, making maximum use of existing quaternion functionality to implement the new octonion functions.
- The octonion code operates using the Cayley-Dickson construction, that is the internal representation of an octonion is an ordered pair of quaternions.
- For the moment, the octonion class should be regarded as experimental, and the functionality is limited compared to the quaternion class. We have not listed all the new octonion functions here, because most are overloadings of existing Matlab or quaternion functions. However, we have implemented the LU decomposition, and left and right matrix inverses that will work for small arrays (linv and rinv).
- We have also added two new functions called associator and commutator which take arbitrary argument types, but make most sense with quaternions and octonions.
- Functions for creating random and zero octonions follow the naming conventions used for the corresponding quaternion functions (e.g. rando corresponds to randq).
- Although some other functions will work for octonion matrices, their results may not be correct.
- Test code for the octonion functions is somewhat limited at present, but basic functionality is tested.
- CAUTION: Since the octonion algebra is not associative, multiplication of octonion matrices is not associative. The documentation will be updated gradually to add the octonion functionality, but it is likely that coverage will be sketchy for some time, since many existing quaternion documentation files need to be updated. We do not plan to change the name of the toolbox, since it remains first and foremost a quaternion toolbox. The octonions are a bonus.
- A new function, part, for octonions and quaternions provides indexed access to the components. For the quaternions this provides similar functionality to the existing s/scalar/x/y/z functions, but for the octonions it provides the only means to access the individual components of the vector part - we did not want to create seven named functions corresponding to the existing quaternion functions x/y/z.
- A new function scatter4 provides a way to visualize distribution of quaternions in four dimensions using six orthographic projections. scatter3 and image now support an output parameter (a graphics handle) just as the corresponding Matlab functions.
- A new function, frenet, computes the Frenet-Serret frames of a quaternion sequence or curve.
- The private function isoquaternion has been altered to fix a problem with the sqrt function in which sqrt(quaternion(-1)) returned zero rather than the correct result (0+1i) + 0 * I + 0 * J + 0 * K (the complex root of -1). Since isoquaternion is used in many other functions, this may also have an impact elsewhere, but this remains to be studied.
- The checking of Matlab version numbers in the test code has been updated.
- Note that with Matlab version 8 the Start menu has been removed. The QTFM documentation is accessible from the Documentation window, under Supplemental Software. The other functionality of the Start button seems to have vanished, but the test code can be run by typing qtfm_test in the command window.