Midterm Review

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

Info iconThis preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
() 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 1 1 ( ) () ' ( ) ( ) '( ) ( ) '( ) ( ) i ii i i i fx f xh O h f x Oh h f f x h   First Backward Difference
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
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       
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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   
Background image of page 10
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     
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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?
Background image of page 12
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     
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 14
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
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The error of the input is .
Background image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 111

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

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online