  Incremental singular value decomposition (SVD) was proposed by Brand to
efficiently compute the SVD of a matrix. The algorithm needs to compute
thousands or millions of orthogonal matrices and to multiply them together.
However, the multiplications may corrode the orthogonality. Hence many
reorthogonalizations are needed in practice. In [Linear Algebra and its
Applications 415 (2006) 20-30], Brand asked "It is an open question how often
this is necessary to guarantee a certain overall level of numerical precision;
it does not change the overall complexity." In this paper, we answer this
question and the answer is we can avoid computing the large amount of those
orthogonal matrices and hence the reorthogonalizations are not necessary by
modifying his algorithm. We prove that the modification does not change the
outcomes of the algorithm. Numerical experiments are presented to illustrate
the performance of our modification.
