CSC_349_HANDOUT#22

CSC_349_HANDOUT#22 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 22 USE OF THE ERROR TERM OF POLYNOMIAL INTERPOLATION PROBLEM : produce a table of values of 1 , , 3 , 2 , , 0 for K h h h x e x = , such that linear interpolation between successive values in the table will yield an approximation with an absolute error < 10 6 . x o o o 0h = x 1 2h = x 2 f(x) = e x Let x k = kh , let ˜ x [0,1] and suppose that j is such that x j ˜ x x j + 1 . x 0 f(x) = e x o o P(x) x j = jh x ~ x j+1 = (j+1)h So, our problem can be stated as follows: if we estimate e ˜ x by P ( ˜ x ) , determine a value of h so that the error in this approximation, which is 6 ~ 10 is , ) ~ ( < x P e x for all values of ˜ x in [0, 1] .
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2 Solution : The error term for polynomial interpolation with n = 1 (linear interpolation) gives ) ~ )( ~ ( 2 ) ( ) ~ ( ) ~ ( 1 + = j j x x x x f x P x f ξ , and for our problem we have f ( x ) = e x , x j = jh and x j + 1 = ( j + 1) h . From above, (*) () h j x jh x e x P x f h j x jh x x j j ) 1 ( ~ ) ~ ( max max 2 1 ) ~ ( ) ~ ( ) 1 ( ~ 1 + + + .
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CSC_349_HANDOUT#22 - COMPUTER SCIENCE 349A Handout Number...

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