1. Errors in Numerical Methods
Chapter 2
Errors in Numerical Methods
and Their Impacts

2. Objectives Know the difference between accuracy&precision
Understand round-off error
Understand approximation error and know how to apply

3. Content Introduction Errors Round-off errors Approximate errror
Total errors
Conclusion

4. Introduction Why we need to know ?
Computers are great tools, however, without fundamental understanding of engineering problems, they will be useless.

5. Errors
We ask for numerical methods since we cannot get exact solution !!
Numerical methods only provide approximate results, not exact ones.
So how we confident our results obtained from
numerical methods ????
See in next slide how can we cope this with?

6. Errors (cont’d)
Accuracy. How close is a computed or measured value to the true value
Precision (or reproducibility). How close is a computed or measured value to previously computed or measured values.
Inaccuracy (or bias). A systematic deviation from the actual value.
Imprecision (or uncertainty or variance). Magnitude of scatter.

7. Errors (cont’d)

8. Errors (cont’d)
Number of “significant figures” indicates precision. Significant digits of a number are those that can be used with confidence, e.g., the number of certain digits plus one estimated digit.
53,800 How many significant figures?
5.38 x
5.380 x
5.380 x
Zeros are sometimes used to locate the decimal point not significant figures.

9. Errors (cont’d) Error Definitions True Value = Approximation + Error
Et = True value – Approximation (+/-)
MATLAB Example
True error

10. Errors (cont’d)
What u can see is we can’t estimate the true error for all cases !! (why ?) So we use the following error definition instead.
Approximation error …

11. Errors (cont’d)
Apply approximation error to numerical approach (iterative)
(+ / -)
Meaning that the result
is correct at least n
significant figures
Define criteria :-
Compute until

12. Errors (cont’d) DIY: MATLAB (Parachutist problem)
From your previous assignment, compare the approximation errors at t = 1,2,..,12 seconds for two cases, Δt = 0.5 and 0.1 respectively. Crticize why the approximation errors from these two cases are different !!

13. Round-off … Why round-off errors occur ?
1) There are numbers that can’t be expressed by a fixed number of significant figures
2) Base-2 number can’t precisely represent base-10 number (completely).
3) Fraction number in computer is represent using a floating point form, e.g.
Integer part
exponent
mantissa
Base of the number system used
where

14. Round-off … (cont’d) How floating numbers ‘re stored in a computer ??
Integer part
exponent
mantissa
Base of the number system used
11 bits
52 bits

15. Round-off … (cont’d) How floating numbers ‘re stored in a computer
(base-2 number) ??

16. Round-off … (cont’d)
Examples:  x103
Suppose only 4 decimal places to be stored
0.1567x103
Rounding/Chopping
0.1568x103
Now u can see how the round-off error occurs due to
the limited room for mantissa !!!!

17. Round-off … (cont’d) Examples: MATLAB Double precision case
Type format long
a= what you expect
Now try more example:
learn round(0.5)
Type round(0.75*0.3/0.01) what you expect

18. Approxi … Example: To get the cos(x) for small x: If x=0.5
From the supporting theory, for this series, the error is no greater than the first omitted term.

19. Approxi …
Using Taylor’s series approximation

20. Approxi … Example: f(x) = -0.1x4-0.15x3-0.5x2-0.25x+1.2
(estimate this function at x = 1 with h = 1, given that
x(0)=1.2)
Try to derive your own and
also write a program to show
for number of order
n =1,2,3,…,5

21. Approxi … U can use Taylor series to avoid the round off errors
For example: try to calculate
ex at x = 0 X

22. Total …
There is a trade-off

23. Others… Blunders Human mistakes Model errors Incomplete mathematical
Data uncertainty
Bias, variance from measurements