Inverse problem the source composition is known

The mxn matrix A of phase compositions is defined by its current element atj, such as 

From the concentration of each element in the liquid and in the source, we can retrieve the degree of melting and the residual mineralogy. We assume that the liquid contains 676 ppm Ni, 426 ppm Cr, 0.763 ppm Yb and 0.1 ppm Rb, whereas its source composition y vector is (2500, 1500, 0.2, 0.01) in ppm. We will test the assumption that the residuum is composed of olivine and clinopyroxene with the partition coefficients given above. Phase abundances Jt will be ordered as liquid, olivine and clinopyroxene. Let us compute, as an example, the element in the third row and second column of the matrix A 

The whole matrix A is built in a similar manner, which gives 

This result reproduces the original melting conditions with 10 percent liquid while the residue contains 0.54/(0.54 + 0.36), i.e., 60 percent olivine and 0.36/(0.54+0.36), i.e., 40 percent clinopyroxene. Since we knew that the model was perfectly obeyed, we could expect that phase abundances sum up to unity. o= 

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