NumericalMethodsDr.LaiShengJianResearch Building 704#, UESTC
Dr. Lai Sheng Jian Research Building 704#, UESTC
1 Preliminaries03-1.3ErrorAnalysis
1 Preliminaries 03-1.3 Error Analysis
ErrorAnalysisDefinition1.7absoluteerror.relativeerrorLP-PRE,=p-ppDefinition 1.8 significant digits d, it is the largest nonnegative integer[p-P]10l-a2PExample1.15classroomwork32 - bit precision : 2-23 =- 1.2e - 764 - bit precision: 2-52 = 2.2e -169/22/20265:57AM
9/22/2026 5:57 AM 3 Definition 1.7 absolute error, relative error Definition 1.8 significant digits d, it is the largest nonnegative integer Example 1.15 classroom work = − ˆ E p p p − = ˆ p p p R p 1 10 2 − − ˆ d p p p 23 32 2 1 2 7 − − = − bit precision e : . 52 64 2 2 2 16 − − = − bit precision e :
ErrorAnalysis-TruncationErrorThenotionoftruncationerrorJX8Ex6y=Ps(x)42Figure1.7The graphs of yf(x) = et, y = Pg(x), and the0.00.51.01.5underthe curvefor0≤x≤2.Round-off Error,the computer hardware works with only alimitednumber ofdigitsinmachinenumbers,roundingerrorsareintroduced andpropagated insuccessive computation9/22/20265:57AM
9/22/2026 5:57 AM 4 The notion of truncation error Example 1.16 the truncated Taylor series P8 Round-off Error, the computer hardware works with only a limited number of digits in machine numbers, rounding errors are introduced and propagated in successive computation. 2 4 6 8 2 2 1 2 3 4 = + + + + + + + . . ! ! ! ! n x x x x x e x n 1 2 2 0 = = 0 544987104184 / . x e dx p 4 6 8 1 2 2 0 1 0 544986720817 2 3 4 + + + + = = / . ! ! ! x x x x dx p 7 7 03442 10− − = ˆ . , ? p p significant digits p
Error Analysis-ChoppingOffversusRounding-offConsideranyrealnumberpp=±0.d,d,d...d,dk+...×10",1≤d,≤9Real numberp flchop(p) is called chopped floating-point representationflchop(p)=±0.d,d,d...d, ×10"An alternative k-digit representation is called the rounded chopped floatingpointrepresentationflrouna(p)=±O.d,d,d,...rk×1o"The last digit rk is obtained by rounding the number dkdk +1 ... to the nearestinteger.p=22/7=3.14285714285714..flchop(p)= 0.314285×10lflrouna (p) = 0.314286×109/22/20265:57AM
9/22/2026 5:57 AM 5 Consider any real number p Real number p 𝒇𝒍𝒄𝒉𝒐𝒑(𝒑) is called chopped floating-pointrepresentation An alternative k-digit representation is called the rounded chopped floatingpoint representation The last digit 𝒓𝒌 is obtained by rounding the number 𝒅𝒌𝒅𝒌 + 𝟏 . to the nearest integer. 1 2 3 1 1 0 10 1 9 = + . . . , n k k p d d d d d d 1 2 3 ( ) . . = 0 10n chop k fl p d d d d 1 2 3 ( ) . . = 0 10n round k fl p d d d r 1 1 22 7 3 14285714285714 0 314285 10 0 314286 10 / . . ( ) . ( ) . chop round p fl p fl p = = = =