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However, there is a more effective way which takes into account the fact that we already have A and the input matrix was modified slightly. The Sherman-Morrison formula serves as the basis of an algorithm: Here A is a square Nx N matrix whose inverse matrix we know, u and v are the columns of height N defining the matrix modification, (u These cases have a different computational complexity. A variety of methods for computing the Moore-Penrose inverse of a rank-one modified matrix have been developed so far. Relationships between various generalized inverses of an arbitrary matrix and corresponding generalized inverses of its rank-one modifications were investigated in [19].The leading idea in [15, 16] was successive computation of the symmetric rank-one (SR1) updates .A procedure for implementing this algorithm in existing computer code is presented. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

The incorporated metric called Convex SIMilarity (CSIM) index is a simplified version of the Structural SIMilarity (SSIM) index which is convex and error-sensitive.

The authors of the paper [15] introduced a computational procedure for the Moore-Penrose inverse of a symmetric rank-one perturbed matrix.

Using this method, the authors of [16] proposed a finite method for computing the minimum-norm least-squares solution of the linear system .

The fastest is an inverse matrix update where only one element of the source matrix is changed. The inverse matrix update by row or column is only 2 times slower (thus, if three or more elements of one row or column were modified, it would better to use this algorithm than the previous one).

The inverse matrix update with arbitrary vectors u and v is three times slower than the simple update. At least, one step is stable enough if A and B are well-conditioned.




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