102
edits
Mfasi
Joined 28 February 2014
→Your task
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* implement a function to find a confluent permutation (a very specific task). | * implement a function to find a confluent permutation (a very specific task). | ||
The functions that should be addressed after that one are expm (), logm () and sqrtm (). The implementation of the first function is stable as it exploits Padé approximants, but could be improved using an heuristic that switches to some other kind of evaluation when such rational functions are ill-conditioned. The logm () functions is a recent implementation and uses a scaling and squaring method. Maybe some | The functions that should be addressed after that one are expm (), logm () and sqrtm (). The implementation of the first function is stable as it exploits Padé approximants, but could be improved using an heuristic that switches to some other kind of evaluation when such rational functions are ill-conditioned. The logm () functions is a recent implementation and uses a scaling and squaring method. Maybe some improvements can be done, since it computes just the principal matrix logarithm, not allowing the user to choose different branches. I would not touch sqrtm () instead, since it seems to be a good C++ implementation of an algorithm by Higham that should be very similar to the one in Matlab. I would instead focus on developing the p-th root, as there is a [http://poisson.phc.unipi.it/~maxreen/bruno/pdf/B.%20Iannazzo%20and%20C.%20Manasse%20-%20A%20Schur%20logarithmic%20algorithm%20for%20fractional%20powers%20of%20matrices%20-%20SIMAX.pdf recent work] that seems promising, though being rather involved it could be tricky to implement in a robust way. | ||
The hyperbolic and trigonometric functions have already been written down straightforwardly in thfm (), from the linear-algebra package. As it relies on sqrtm (), expm () and logm (), it will benefit of all the improvements to such functions. | The hyperbolic and trigonometric functions have already been written down straightforwardly in thfm (), from the linear-algebra package. As it relies on sqrtm (), expm () and logm (), it will benefit of all the improvements to such functions. | ||