This document discusses numerical errors and concepts related to error analysis. It defines significant digits and describes different types of errors like inherent errors, data errors, and numerical errors. It then explains absolute and relative errors, defining absolute error as the difference between the true and approximate values, and relative error as the absolute error divided by the true value. Finally, it introduces conditioning and stability, defining the condition number as the ratio of relative errors in the function and input. A large condition number means the problem is ill-conditioned and small changes in the input can lead to large changes in the output.

1. NumericalMethods
Lecture 2
Errors

2. Overview
– Significant Digits
– Types of Errors
– Absolute and Relative Errors
– Conditioning and Stability
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3. SignificantDigits
What are Significant Digits
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4. Significant Digits
– All non-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
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5. TypesofErrors
Different types of erros
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6. Types of Errors
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Errors
Inherent Errors
Data Errors
Conversion
Errors
Numerical Errors
Roundoff Errors
Chopping
Symmetric
Roundoff
Truncation
Errors
Modeling
Errors
Blunders

7. AbsoluteandRelative
Errors
Understanding some common error calculation
schemes
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8. Absolute and Relative Errors
– General Construction
– xt = xa + Error
– Error = xt - xa
– Absolute Error
– ea = |xt – xa|
– Relative Error
– er =
|xt – xa|
|xt|
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9. Conditioningand
Stability
Understand the concept of conditioning and stability
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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
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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
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12. Thank You
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