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

2. Objectives Know finite-word length effect
Know computing errors, their causes and impacts to numerical calculation
Know how to compute the errors
Know the effect of error propagation
Know how to avoid large errors

3. Content
Introduction
Finite word-length effect (Floating-point number representation)
Error
How to avoid error
Conclusion

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

5. Finite wordlength effect
How floating numbers are stored in a computer ?
Sign bit
Exponent
Mantissa
Base of the number system used
IEEE 64-bit floating-point number representation

6. Finite wordlength effect

7. Finite wordlength effect
Ex: 402C S
Exponent
Mantissa 4 2 C 1
S = 0
E = Exp-1023= =3
M =
Check with MATLAB command: “hex2num”
F = +(1.11)2x23
= (1110)2=14

8. Finite wordlength effect(cont’d)
Step to convert from decimal number to IEEE stand.
Check with MATLAB
Use num2hex command S
Exponent
Mantissa 1 4 8

9. Finite wordlength effect(cont’d)
How to add 2 numbers :-
This is done by bit alignment (only mantissa) with refer to the biggest number. Finally, the Mantissa words are added in 2’s system.
Example : 1015
Ans: x
Example : 1
Ans: x
1015
Example : (show only Mantissa)
Ans: 1 1 1
Right shift 49 bits 1

10. Finite wordlength effect(cont’d)1 1
430C6BF
How to substract 2 numbers :-
Complement and done by bit alignment with refer to the biggest number
1015
Example : 1 1 1
2’s of 1 1
1015-1 1

11. Finite wordlength effect(cont’d)1 1
430C6BF52633FFF8
Will u try with the following problems?

12. Errors Error from numerical algorithm Truncation error Round-off error
Overflow/Underflow
Loss of significance
Negligible addition
Error magnification

13. Errors: Numerical algorithm
Numerical integration on
Error from exact solution =
Error from exact solution =

14. Errors: Numerical algorithm
Numerical integration on

15. Errors: Truncation error
Truncation error is resulted from truncation series.
Ex. Say x = 1 find exp(x)

16. Error: Loss of significance
Bad subtraction in finite wordlength can create
loss of significance.
See the following example:
Try with x= 1,10,100,…1015

17. Error: Propagation error
Errors are propagated with the four arithmetic
Operations. Let the exact values are X and Y, where
their related values are x and y, respectively.

18. Errors
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.

19. Errors (cont’d)

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

21. 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 …

22. 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

23. Errors: Practical