Moore-Penrose pseudo-inverse

Solution of the system of equations The system of Eq. (3), whose equations combine numerical values, theoretical expressions, and covariances, can be solved for the adjusted variables Z best estimates of their values can thus be calculated. The method used in [2,3] consists in using a sequence of linear approximations to system (3), around a numerical vector Z that converges toward the solution of the full, non-linear system (this is akin to Newton s method—see, e.g. [23]). Each of the successive linear approximations to system (3) is solved through the Moore-Penrose pseudo-inverse [20] (see, also. Ref. [2, App. E]). The numerical solution for Z as found in CODATA 2002 can be found on the web . These values are such that the equations in system (3) are satisfied, as a whole, as best as possible [3, App. E]). 

Alternatively, a weighted generalized inverse G may be used instead of the Moore-Penrose pseudo-inverse ... 

The easiest way to solve Eq. (2.3.15a) is to use directly the representation of the Moore Penrose pseudo-inverse G" " = G (GG ) This corresponds to the solution of the normal equations. 

The Moore-Penrose pseudo-inverse then reads... 

Computation of the regression coefficients b vectorwise is carried out by formation of the pseudo-inverse matrix X" " (Moore-Penrose matrix) according to... 

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