Page 54 - Compact Numerical Methods For Computers

44 Compact numerical methods for computers
Example 3.1. The generalised inverse of a rectangular matrix via the singular-value
decomposition
Given the matrices U, V and S of the singular-value decomposition (2.53), then by
the product
+ + T
A = VS U (2.56)
the generalised (Moore-Penrose) inverse can be computed directly. Consider the
matrix

A Hewlett-Packard 9830 operating in 12 decimal digit arithmetic computes the
singular values of this matrix via algorithm 1 to six figures as

13·7530, 1·68961 and 1·18853E-5
with

and

+
The generalised inverse using the definition (2.57) of S is then (to six figures)

However, we might wonder whether the third singular value is merely an
approximation to zero, that is, that the small value computed is a result of
+
rounding errors. Using a new definition (3.37) for S , assuming this
singular value is really zero gives

If these generalised inverses are used to solve least-squares problems with
T
b = (1, 2, 3, 4)