Page 103 - Compact Numerical Methods For Computers

92 Compact numerical methods for computers

has a lower triangle

-1 2
-1 0 3
-1 0 1 4
-1 0 1 2 5

which gives the Choleski factor
1
-1 1
-1 -1 1
-1 -1 -1 1
-1 -1 -1 -1 1.

Example 7.2. Solving least-squares problems via the normal equations

Using the data in example 3.2, it is straightforward to form the matrix B from the
last four columns of table 3.1 together with a column of ones. The lower triangle
T
of B B is then (via a Hewlett-Packard 9830 in 12 decimal digit arithmetic)
18926823
6359705 2164379
10985647 3734131 6445437
3344971 1166559 2008683 659226
14709 5147 8859 2926 13

Note that, despite the warnings of chapter 5, means have not been subtracted, since
the program is designed to perform least-squares computations when a constant
(column of ones) is not included. This is usually called regression through the
origin in a statistical context. The Choleski factor L is computed as

4350·496868
1461·834175 165·5893864
2525·147663 258·3731371 48·05831416
768·8710282 257·2450797 14·66763457 40·90441964
3·380993125 1·235276666 0·048499519 0·194896363 0·051383414