Matlab pseudo-inverse

As stated earlier, Matlab s philosophy is to read everything as a matrix. Consequently, the basic operators for multiplication, right division, left division, power (, /,, A) automatically perform corresponding matrix operations (A will be introduced shortly in the context of square matrices, / and will be discussed later, in the context of linear regression and the calculation of a pseudo inverse, see The Pseudo-Inverse, p.117). 

In general, non-linear problems cannot be resolved explicitly, i.e. there is no equation that allows the computation of the result in a direct way. Usually such systems can be resolved numerically in an iterative process. In most instances, this is done via a truncated Taylor series expansion. This downgrades the problem to a linear one that can be resolved with a stroke of the brush or the Matlab / and commands see The Pseudo-Inverse (p.ll 7). 

Matlab is, of course, aware of the fundamental importance of the pseudoinverse and created its own notation for it. In Matlab we could write a=inv (F F) F y but it is numerically much more efficient to use the appropriate Matlab back-slash command as in a=F y. It is to be read from the right to the left as y divided by F, implying, of course, the multiplication of the left pseudo-inverse of F with y as given in equation (4.30). 

For the computation of the pseudo-inverse, it is crucial that the vectors f j are not parallel, or more correctly, that they are linearly independent. Otherwise, the matrix FlF is singular and cannot be inverted. Matlab issues a warning. We can gain a certain level of understanding by adapting Figure 4-10 ... 

Referring back to Matlab, it is very important to use the correct slash operator or / for the left and right pseudo inverse. Applying the wrong one will invariably result in an error message or worse, in a potentially undetected error. 

Here V is an m x m matrix, Lf an n x n matrix and 5 an n x m diagonal matrix. When A is rank deficient, then S S cannot be inverted because of very small or zero singular values. In this case, we only take the r non-zero singular values so that S becomes an r x r matrix where r is the rank of A. The MATLAB function pinv computes the pseudo-inverse. 

The next MATLAB session shows the use of the pinv (pseudo-inverse function). 

To solve equations (7-9), an approach with direct solution of the system (8) at each step of the iterative process by the pseudo- (or generalized) inversion method is used. It is based on singular value decomposition (SVD). It is well-known that the SSVDC procedure in the Linpack library is used to calculate SVD [12]. Paper [13] presents a standard SVD procedure in Fortran-IV used in the present paper. The current MATLAB system versions have a built-in function svd(A) implementing this decomposition for an arbitrary nxm matrix A. The calculation scheme of the SVD procedure is in decomposing the matrix (9) at each step of the iterative process into the product of three matrices. 

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