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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