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MATH2051_07 - Durham University EXAMINATION PAPER date exam...

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Durham University EXAMINATION PAPER date May/June 2007 exam code MATH2051/01 description NUMERICAL ANALYSIS II Time allowed: 3 hours Examination material provided: None Instructions: Credit will be given for the best FOUR answers from Section A and the best THREE answers from Section B. Questions in Section B carry TWICE as many marks as those in Section A. Approved electronic calculators may be used. ED01/2007 Durham University Copyright continued
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2 MATH2051/01 SECTION A 1. Consider the following two rearrangements for the equation x 3 + 14 x 37 = 0: (a) x = 37 x 3 14 ; (b) x = (37 14 x ) 1 / 3 . The equation has a root p in the range [2 . 0 , 2 . 1]. Determine whether each rearrange- ment may or may not be used successfully as the basis for an iterative method to find p . 2. (a) We need to evaluate p 2 (0 . 75) where p 2 ( x ) = x 2 + αx + β, but we don’t know the values of the constants α and β . However, we do know that x 0 0 . 5 p 2 ( x ) 2 3 . Using divided differences, find p 2 (0 . 75). (b) We first obtained p 2 ( x ) by interpolating a more complicated function f using its values at x = 0 , 0 . 5 and 1. We know that | f (3) ( x ) | ≤ 0 . 2 for x [0 , 1]. Using the interpolation error formula f ( x ) p 2 ( x ) = ( x x 0 )( x x 1 )( x x 2 ) 3! f (3) ( c ) , how accurately does p 2 (0 . 75) approximate f (0 . 75)? 3. (i) A differentiable function takes the values f ( 0 . 1) = 1 . 629 , f (0) = 1 . 312 and f (0 . 1) = 1 . 043 . Using the central and backward difference formulae f ( x 2 ) f ( x 0 ) x 2 x 0 and f ( x 1 ) f ( x 0 ) x 1 x 0 estimate f (0). Which do you think will be the more accurate estimate, and why? (ii) Without using a calculator evaluate ( 1 + 10 4 ) 6 ( 1 + 10 6 ) 4 correct to eight significant digits. ED01/2007 Durham University Copyright continued
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3 MATH2051/01 4. Let F s R n × n be a real Frobenius matrix of index s . State what this means.
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