tailieunhanh - Tracking and Kalman filtering made easy P10

VOLTAGE LEAST-SQUARES ALGORITHMS REVISITED COMPUTATION PROBLEMS The least-squares estimates and minimum-variance estimates described in Section and and Chapter 9 all require the inversion of one or more matrices. Computing the inverse of a matrix can lead to computational problems due to standard computer round-offs [5, pp. 314–320]. To illustrate this assume that s¼1þ" ð10:1-1Þ Assume a six-decimal digit capability in the computer. Thus, if s ¼ 1:000008, then the computer would round this off to . If, on the other hand, s ¼ 1:000015, then the computer would round this off to . Hence, although the change in ". | Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright 1998 John Wiley Sons Inc. ISBNs 0-471-18407-1 Hardback 0-471-22419-7 Electronic 10 VOLTAGE LEAST-SQUARES ALGORITHMS REVISITED COMPUTATION PROBLEMS The least-squares estimates and minimum-variance estimates described in Section and and Chapter 9 all require the inversion of one or more matrices. Computing the inverse of a matrix can lead to computational problems due to standard computer round-offs 5 pp. 314-320 . To illustrate this assume that s 1 e Assume a six-decimal digit capability in the computer. Thus if s then the computer would round this off to . If on the other hand s then the computer would round this off to . Hence although the change in e is large a reduction of for the second case . the change in s is small 5 parts in 106 . . This small error in s would seem to produce negligible effects on the computations. However in carrying out a matrix inversion it can lead to serious errors as indicated in the example to be given now. Assume the nearly singular matrix 5 1 1 where s 1 e. Inverting A algebraically gives A-1 1 s 1 264 COMPUTATION PROBLEMS 265 If e then from we obtain the following value for A 1 without truncation errors V-1 A 1 . 104 However if e is truncated to then yields -1 A 1 104 10 -10 -10 10 Thus the 5 parts in 106 error in s results in a 50 error in each of the elements of A -1. Increasing the computation precision can help. This however can be costly in computer hardware and or computer time. There are however alternative ways to cope with this problem. When doing a LSE problem this involves the use of the voltage least-squares also called square-root algorithms which are not as sensitive to computer round-off errors. This method was introduced in Section and will