principles and applications of microearthquake networks

118 5. In versiori nnd Optimization
numbers in computers rarely correspond to those on real numbers. Since
floating-point numbers have a finite number of digits, roundoff error occurs
in representing any number that has more significant digits than the
computer can hold. An arithmetic operation on floating-point numbers
may introduce roundoff error and can result in underflow or overflow
error. For example, if we multiply two floating-point numbers x and y ,
each oft significant digits, the product is either 2t or (2t - 1) significant
digits, and the computer must round off the product to t significant digits.
If x and y has exponents of p, and p,, it may cause an overflow if (p, + p,)
exceeds the upper exponent range allowed, and may cause an underflow if
(p, + p,) is smaller than the lower exponent range allowed.
An effective way to minimize roundoff errors is to use double precision
floating-point numbers and operations whenever in doubt. Unfortunately,
the exponent range for single precision and double precision floating-point
numbers for most computers is the same, and one must be careful in
handling overflow and underflow errors. In addition, there are many other
pitfalls in computer computations. Readers are referred to textbooks on
computer science, such as Dahlquist and Bjorck (1974) and Forsythe at af.
(1977), for details.
5.3.2.
Computational Aspects of Generalized Inversion
The purpose of generalized inversion is to help us understand the nature
of the problem Ax = b better. We are interested in getting not only a
solution to the problem Ax = b, but also answers to the following questions:
(1) Do we have sufficient information to solve the problem? (2) Is
the solution unique? (3) Will the solution change a lot for small errors in b
and/or A? (4) How important is each individual observation to our solution?
In Section 5.2 we solved our problem Ax = b by seeking a matrix H
which serves as an inverse and satisfies certain criteria. The singular value
decomposition described in Section 5.1 permits us to construct this inverse
quite easily. Let A be a real rn X n matrix. The n X rn matrix H is
said to be the Moore-Penrose generalized inverse of A if H satisfies the
following conditions (Ben-Israel and Greville, 1974, p. 7):
(5.52)
AHA = A, HAH = H, (AH)T = AH
(HA)T = HA
The unique solution of this generalized inverse of A is commonly denoted
as A-. It can be verified that if we perform singular value decomposition