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ExtendedLinearSolve[m] is an extended version of LinearSolve[m, v] which supports the one-argument form ExtendedLinearSolve[m]. This one-argument form yields a LinearSolveFunction object which when applied to a vector v gives the same result as LinearSolve[m, v].
ExtendedLinearSolve is particularly efficient when repeated solutions are needed for several different vectors v.
The matrix m can be given in the sparse form used by Developer`SparseLinearSolve.
See also: LinearSolve, Inverse, LUDecomposition, SparseLinearSolve.
Note: this is an experimental feature, and in future versions of Mathematica it may not be supported, or may have a different specification.