Is MPI implementation of matrix inversion a standard library routine or good MPI implementation critically depends on your setup(software/hardware) so have to be coded by yourself.
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ScaLAPACK provides implementations of lots of linear algebra functionality, including the solution to least squares problems. See for example the ScaLAPACK users' guide. (I don't think ScaLAPACK has a general matrix inverse - hopefully your application needs the inverse only to solve a linear system - in which case, using the functionality to do that directly will be more accurate than attempting to find an inverse).
Edric
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