Explicit Computation of the Concentration Profiles

As we have seen with the previous iterative refinement and ITTFA, convergence generally is very sluggish. Even with moderately complex systems, it is often too slow to be useful. There are alternative, non-iterative methods that compare favourably with the above iterative algorithms. 

We start the derivation with the standard equations Y=CA and Y=USV which we combine, see equation (5.44). Post-multiplication with V AV1)-1 results in 

To compute C, the only unknown is A. It is advantageous to regard the product S(A ) 1 as the unknown. Dimensional analysis shows that it is a ncxnc square matrix (nc =number of components), we call it a transformation matrix T  

This is a very interesting and useful equation and we will return to it several times in later parts of this chapter. Equation (5.49) relates the concentration matrix C to the matrix U of eigenvectors. It is worthwhile representing the equation graphically. 

For an nc= 2 component system the square transformation matrix T has only 4 elements, for a 3 component system there are 9 elements, etc. 

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The program Ma in ALS, m forms the backbone of the ALS algorithm. It reads in the data set Data Chrom2a (p.251) which simulates an overlapping chromatogram of three components. It is the data set we used previously in Chapter 5.3.3 to demonstrate the concepts of iterative and explicit computation of the concentration profiles, based on the window information from EFA. 

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