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Lecture 2 - Numerical Methods
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- 2. Overview – Significant Digits –Types of Errors – Absolute and Relative Errors – Conditioning and Stability Tashreef Muhammad 2
- 3. SignificantDigits What are SignificantDigits Tashreef Muhammad 3
- 4. Significant Digits – Allnon-zero digits are significant – All zeros occurring between non-zero digits are significant – Trailing zeros after decimal are significant – Zeros between decimal point and preceding a non-zero digit is significant – Trailing zeros are not significant in absence of decimal point Tashreef Muhammad 4
- 5. TypesofErrors Different types oferros Tashreef Muhammad 5
- 6. Types of Errors TashreefMuhammad 6 Errors Inherent Errors Data Errors Conversion Errors Numerical Errors Roundoff Errors Chopping Symmetric Roundoff Truncation Errors Modeling Errors Blunders
- 7. AbsoluteandRelative Errors Understanding some commonerror calculation schemes Tashreef Muhammad 7
- 8. Absolute and RelativeErrors – General Construction – xt = xa + Error – Error = xt - xa – Absolute Error – ea = |xt – xa| – Relative Error – er = |xt – xa| |xt| Tashreef Muhammad 8
- 9. Conditioningand Stability Understand the conceptof conditioning and stability Tashreef Muhammad 9
- 10. Conditioning and Stability –Describes sensitivity to uncertainty – Condition Number = 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑖𝑖𝑖𝑖 𝑓𝑓(𝑥𝑥) 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑖𝑖𝑖𝑖 𝑥𝑥 – Relative Error in f(x) = Δ𝑓𝑓 𝑓𝑓 𝑥𝑥 = 𝑓𝑓′ 𝑥𝑥 Δ𝑥𝑥 𝑓𝑓(𝑥𝑥) – Relative Error in x = Δ𝑥𝑥 𝑥𝑥 – Hence, Condition Number = 𝑥𝑥𝑓𝑓′ 𝑥𝑥 𝑓𝑓 𝑥𝑥 Tashreef Muhammad 10
- 11. Conditioning and Stability –Significance of condition number – Large condition number means “ill-conditioned” – It dictates that for small change x, large change in f(x) – In other words, “very much sensitive” – “ill-conditioned” is “ill-defined”, no fixed values Tashreef Muhammad 11
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