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To solve the \(\ell_1\) optimization problem of eq.~\ref{eq:bpdn} we rely on the spectral projected gradient \(\ell_1\) (SPGL1) algorithm developed by van~den~Berg and Friedlander~\cite{Berg2008} and their freely-available implementation. The matrix for the change of base operation is given by the Kronecker product of \(P\) with itself, however, we do not build the matrix and perform all the matrix multiplications as implicit operations in terms of P. This last approach has a much smaller numerical cost and memory requirements. Numerically the condition \(PAP^T = B\) is satisfied up to an error of \(10^{-7}\) in the Frobenius norm (vectorial 2-norm).  All quantum mechanical Hessian calculations were performed with the QChem 4.2~\cite{Shao_2006} software package, using density functional theory with the B3LYP exchange-correlation functional and the 6-31G* basis set. All molecular mechanics Hessian calculations were performed using freely-available open-source Tinker 6.2 6.2~\cite{http://dasher.wustl.edu/tinker/}  software package using the MM3 force field~\cite{Allinger_1989}.