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ExtendedLUBackSubstitution[data, b, f] is an extended version of LUBackSubstitution[data, b] which allows the function f to be effectively applied to the original matrix.
The possible values for f are Identity, Transpose and Conjugate.
The matrix m can be given in the sparse form used by Developer`SparseLinearSolve.
See also: LUBackSubstitution, SparseLinearSolve.
Note: this is an experimental feature, and in future versions of Mathematica it may not be supported, or may have a different specification.