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40                Compact numerical methods for computers
                           Singular values
                             3.3658407311E+00 1.0812763036E+00 6.7431328720E-01 5.3627598567E-01
                             3.3658407311E+00 1.0812763036E+00 6.7431328701E-01 5.3627598503E-01



                           Hilbert segment:

                           Column orthogonality of U
                           Largest inner product is 5, 5 =  -1.44016460160157E-006
                           Largest inner product is 3, 3 = 5.27355936696949E-016

                           Singular values
                           1.27515004411E+000 4.97081651063E-001 1.30419686491E-001 2.55816892287E-002
                           1.27515004411E+000 4.97081651063E-001 1.30419686491E-001 2.55816892259E-002

                           3.60194233367E-003
                           3.60194103682E-003



                            3.6. USING THE SINGULAR-VALUE DECOMPOSITION TO SOLVE
                                                LEAST-SQUARES PROBLEMS
                           By combining equations (2.33) and (2.56), the singular-value decomposition can
                           be used to solve least-squares problems (2.14) via
                                                               +  T
                                                         x = VS U b.                         (3.36)
                             However, the definition (2.57) of S +  is too strict for practical computation,
                           since a real-world calculation will seldom give singular values which are identi-
                           cally zero. Therefore, for the purposes of an algorithm it is appropriate to define

                                                                                             (3.37)


                           where q is some tolerance set by the user. The use of the symbol for the tolerance
                           is not coincidental. The previous employment of this symbol in computing the
                           rotation parameters and the norm of the orthogonalised columns of the resulting
                           matrix is finished, and it can be re-used.
                             Permitting S +  to depend on a user-defined tolerance places upon him/her the
                           responsibility for deciding the degree of linear dependence in his/her data. In an
                           economic modelling situation, for instance, columns of U corresponding to small
                           singular values are almost certain to be largely determined by errors or noise in
                           the original data. On the other hand, the same columns when derived from the
                           tracking of a satellite may contain very significant information about orbit
                           perturbations. Therefore, it is not only difficult to provide an automatic definition
                               +
                           for S , it is inappropriate. Furthermore, the matrix B = US contains the principal
                           components (Kendall and Stewart 1958-66, vol 3, p 286). By appropriate
                           choices of q in equation (3.37), the solutions x corresponding to only a few of the
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