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MTH510_I Error Analysis

# MTH510_I Error Analysis - MTH 510 Numerical Analysis...

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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 - - - = ˆ ; ; ˆ ; ˆ p p E t - = p p p t ˆ - = ε 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 ˆ ˆ ˆ 1 i i i a p p p - - = ε % 100 ion Approximat Error ion Approximat a = ε 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 . 0 2 ε ε ε ε ε

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MTH 510 Numerical Analysis LF/S09 Illustration 1.1 : Find the Error & Relative Error 25 . 0 000005 . 0 ) 3 10 4 4 ) 2 10 1 . 9 002857 . 0 ) 1 _______ __________ __________ , 000015 . 0 ˆ , 00002 . 0 ) 3 , 9996 ˆ , 10000 ) 2 14 . 3 ˆ , 142857 . 3 ) 1 4 4 = = × = = × = = = = = = = = - - ε ε ε 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 . 0 ; 10 5 . 0 2 / 10 1 3 - - - × = × = = s n s ε ε % 100 ) ( ) ( ) ( ) ( %; 100 648721 . 1 ) ( 648721 . 1 ) ( 1 i x i x i x i a i x i t e e e e - - = - = ε ε With 6 terms ε a < ε s Result accurate to 5 significant digits considering 6 terms # of Terms (e x ) i ε t %
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MTH510_I Error Analysis - MTH 510 Numerical Analysis...

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