  Singular Value Decomposition (SVD) is the basic body of many statistical
algorithms and few users question whether SVD is properly handling its job.
  SVD aims at evaluating the decomposition that best approximates a data
matrix, given some rank restriction. However often we are interested in the
best components of the decomposition rather than in the best approximation .
This conflict of objectives leads us to introduce {\em Total SVD}, where the
word "Total" is taken as in "Total" least squares.
  SVD is a least squares method and, therefore, is very sensitive to gross
errors in the data matrix. We make SVD robust by imposing a weight to each of
the matrix entries. Breakdown properties are excellent.
  Algorithmic aspects are handled; they rely on high dimension fixed point
computations.
