Non-normalized version of the Direct Inversion in the Spectral Subspace method. A new formulation of the method without Lagrangian

Recently, a novel method, called the Direct Inversion in the Spectral Subspace (DISS, J. Math. Chem. 47, 1085-1105 (2010)), has been developed for the quantitative (and, in a limited sense, qualitative) analysis of homogeneous chemical mixtures. The method belongs to the “supervised classification” methods because (beyond the mixture's spectrum) it needs the knowledge of the components' spectra, either experimental or calculated. Two different versions of the DISS method were established: the normalized (approximate) and the non-normalized (accurate) methods. In the present work, the revised non-normalized version is discussed in a general and elegant way together with the normalized variant. The original DISS method (with the use of a sole restriction by the Lagrange multiplier method) leads to an iterative solution of a system of linearized equations. A new formulation of the method (abbreviated as DISS_Magar) is presented without Lagrange multipliers and iteration. The complete equivalence of the DISS_Magar and the original DISS methods has been proved for both the normalized and the non-normalized versions in an elegant and simple way. The DISS method leads to a much smaller system of linear equations than that in the “multiwavelength spectroscopic method,” and it is simpler than the MCR─ALS or ICA algorithms, with the DISS requiring less mathematical operations. Further mathematical proofs are presented for the principles underlying the DISS method along with applications to experimental and simulated data sets.