Midterm Review

# Midterm Review - EAD 115 Numerical Solution of Engineering...

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EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science

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Taylor’s Theorem • Can often approximate a function by a polynomial • The error in the approximation is related to the first omitted term • There are several forms for the error
() 2 ( 1) 1 ( 1) ''( ) ( ) ' ( ) ( ) ( ) 2! ! ! ( 1)! n n n x n n n a n n fa f a f x f a f a xa R n xt R f t dt n Rf n   2 1 ( 1) ''( ) ( ) ( ) ' ! ( n n n n n fx f x h f x h h h R n h n

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Series Truncation Error • In general, the more terms in a Taylor series, the less error • In general, the smaller the step size h , the less error • Error is O(h n+1 ), so halving the step size should result in a reduction of error that is on the order of 2 n+1 • In general the smoother the function the less the error
Numerical Differentiation 2 1 11 2 1 1 1 1 ( ) () ' ( ) ( ) ' ( ) ( ) ( ) ( ) '( ) ( ) '( ) ( ) i i i ii i i i i i i fx f x x x O x x f x x x O x x f x Ox x xx f f x Oh h         First Forward Difference

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2 1 1 ( ) () ' ( ) ( ) '( ) ( ) '( ) ( ) i ii i i i fx f xh O h f x Oh h f f x h   First Backward Difference
23 1 1 3 11 3 2 ( ) () ' 0 . 5' ' ( ) ( ) ' 0 . ' ( ) ( ) ( ) 2 '( ) ( ) ( ) ( ) '( ) ( ) 2 i ii i i i i i i fx f xh f O h f f O h f O h f O h f x Oh h     First Centered Difference

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Second Forward Difference       2 22 1 2 21 1 2 ' ' () / ( ) ''( ) ( ) ( ) ( ) ( ) ''( ) ( ) 2 ( ) ( ) i i ii i i i i i f x fx h h f x h f x h       

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1 1 1 2 22 () 2 2 ii i i ff f If f Ff f F If f f FI F IFF I I FF I   
1 22 2 2 12 11 2 2 2 () ( ) (2 ) 2 ( ) ) 2 i ii i i i i i i i i i f f f I Bf f I I B ff f f f f f F f F F F B f f     

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Propagation of Error • Suppose that we have an approximation of the quantity x , and we then transform the value of x by a function f( x ). • How is the error in f( x ) related to the error in x ? • How can we determine this if f is a function of several inputs?
2 ''( ) () ' 2! ' If the error is bounded ' If the error is random with standard deviation ( ) ' xx x x fx f x f x B f xB SD x SD f x f x     

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11 1 22 2 12 1 1 2122 1 1 1 1 212 (, ) ) ) ) ) ) ) ) If the errors are bounded ) ) ) ii xx x x x x fxx f f f f B f B f       2 ) B
Stability and Condition • If small changes in the input produce large changes in the answer, the problem is said to be ill conditioned or unstable • Numerical methods should be able to cope with ill conditioned problems • Naïve methods may not meet this requirement

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The error of the input is .
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Midterm Review - EAD 115 Numerical Solution of Engineering...

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