MTH510_I Error Analysis - MTH 510 Numerical Analysis LF/S09...

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Unformatted text preview: MTH 510 Numerical Analysis LF/S09 I Error Analysis Objective Understand: Measures of Error used in numerical computation Causes of Error in Numerical Computation Some measure to reduce errors in computation Introduction Numerical Analysis:- Inexact mathematics Effective together with computer Source of Error Machine number representation Arithmetic Error Mathematical Approximation Challenges of NA Identification of Error Quantification of Error Control/limit within pre-specified error 1.1 Error Analysis Measures of True Error True Error Absolute Error True Relative Error ion approximat p value true p p p E t--- = ; ; ; p p E t- = p p p t - = Approximate Error True value (p)- not available in real problems (p)- available only when dealing analytically solvable function If (p) is not known, use the best approx. available Approximate Relative error; for 1 step process Approx. Error; for Iterative Process % 100 1 i i i a p p p-- = % 100 ion Approximat Error ion Approximat a = Error Control/Limitation in Numerical Analysis desired number t significan n value tolerable specified pre Others h Scarboroug Criteria s n s n s s a--- = = <-- / 2 / 10 % 10 5 . 2 MTH 510 Numerical Analysis LF/S09 Illustration 1.1 : Find the Error & Relative Error 25 . 000005 . ) 3 10 4 4 ) 2 10 1 . 9 002857 . ) 1 _______ __________ __________ , 000015 . , 00002 . ) 3 , 9996 , 10000 ) 2 14 . 3 , 142857 . 3 ) 1 4 4 = = = = = = = = = = = =-- t t t E E E p p p p p p Illustration 1.2 : Function Approximation Approximate e x for x=0.5 correct to 3 significant digits, p= e 0.5 =1.648721 Function Approx. Error Criteria ! ! 3 ! 2 1 3 2 n x x x x e n x K + + + = % 10 5 ....
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MTH510_I Error Analysis - MTH 510 Numerical Analysis LF/S09...

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