Page 42 - Compact Numerical Methods For Computers

32 Compact numerical methods for computers
constructed such that they are orthogonal to the columns of U computed via
equation (3.5) and to each other. Thus equations (3.6) and (3.7) are also satisfied.
An alternative approach is to set the columns of U corresponding to zero singular
values to null vectors. By choosing the first k of the singular values to be the
non-zero ones, which is always possible by simple permutations within the matrix
T
V, this causes the matrix U U to be a unit matrix of order k augmented to order n
with zeros. This will be written

(3.8)

While not part of the commonly used definition of the svd, it is useful to require
the singular values to be sorted, so that

S > S > S > . . . > S > . . . > S .
11
nn
33
kk
22
This allows (2.53) to be recast as a summation
(2.53a)
Partial sums of this series give a sequence of approximations

à , à , . . . , à .
1
2
n
where, obviously, the last member of the sequence
à = A
n
since it corresponds to a complete reconstruction of the svd. The rank-one matrices

T
u S v j
j jj
can be referred to as singular planes, and the partial sums (in order of decreasing
singular values) are partial svds (Nash and Shlien 1987).
A combination of (3.1) and (3.6) gives
AV = US (3.9)

or, using (3.3), the orthogonality of V,
T
A = USV (2.53)
which expresses the svd of A.
The preceding discussion is conditional on the existence and computability of a
suitable matrix V. The next section shows how this task may be accomplished.

3.3. ORTHOGONALISATION BY PLANE ROTATIONS
The matrix V sought to accomplish the orthogonalisation (3.1) will be built up as