ABSTRACT
This chapter presents an algorithm for accomplishing the powerful and versatile singular-value decomposition. This allows the solution of a number of problems to be realised in a way which permits instabilities to be identified at the same time. This is a general strategy I like to incorporate into my programs as much as possible since I find succinct diagnostic information invaluable when users raise questions about computed answers—users do not in general raise too many idle questions! They may, however, expect the computer and my programs to produce reliable results from very suspect data, and the information these programs generate together with a solution can often give an idea of how trustworthy are the results. This is why the singular values are useful. In particular, the appearance of singular values differing greatly in magnitude implies that our data are nearly collinear. Collinearity introduces numerical problems simply because small changes in the data give large changes in the results. For example, consider the following two-dimensional vectors: A = ( 1 , 0 ) T B = ( 1 , 0 · 1 ) T C = ( 0 · 95 , 0 · 1 ) T . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315139784/ed4048a6-e22b-4325-b5dd-51851ea3ba32/content/eq110.tif"/>
