Interpolation - formula to nd p 2 ( x ) and use it to...

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1 MAT 2384-Practice Problems on Interpolation Methods 1. Calculate the Lagrange polynomial of degree 2, p 2 ( x ), to 4 decimal places that fits the following three data points ( x i , f ( x i )) for a certain unknown function f : (1 . 01 , 1) , (1 . 02 , 0 . 9888) , (1 . 04 , 0 . 9784) and from it interpolate a value of f at x = 1 . 035 and extrapolate a value of f at x = 1 . 055. Give error bounds on your estimation of f (1 . 035) if 0 . 251 f 000 ( t ) 0 . 45 for any t [1 , 1 . 04]. 2. The error function is given by erf( x ) = 2 π R t 0 e - t 2 dt . Note that it hard to get an exact value for erf( x ) since we don’t know an antiderivative for e - t 2 . Given that erf(0 . 25) = 0 . 27633, erf(0 . 5) = 0 . 52050 and erf(1) = 0 . 84270, calculate the Lagrange polynomial of degree 2, p 2 ( x ), to 5 decimal places that approximates erf( x ). Use it to approximate erf(0 . 75). Using the error formula for Lagrange Interpolation, give bounds on your estimation of erf(0 . 75). [Hint. The following fact from Calculus is useful: If f ( x ) = R x a g ( t ) dt , then f 0 ( x ) = g ( x )] 3. Given that f (0) = 0, f (1) = 0 . 9461, f (2) = 1 . 6054, use Newton’s forward difference
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Unformatted text preview: formula to nd p 2 ( x ) and use it to estimate the value at x = 1 . 5. 4. Given that f (0 . 5) = 0 . 479, f (1) = 0 . 841 and f (2) = 0 . 909 for some unknown function f , esti-mate f (0 . 8) and f (0 . 9) by quadratic interpolation via Newtons divided dierence polynomial (with coecients rounded to 5 decimal places). 5. Given the four data points ( x i , f ( x i )) from an unknown function f : (1 ,-3 . 02) , (2 , 1 . 25) , (3 , 3 . 1487) , (4 ,-2 . 546) : (a) estimate f (2 . 5) and f (3 . 5) by cubic interpolation via Newtons divided dierence poly-nomial (with coecients rounded to 5 decimal places). (b) estimate f (2 . 5) and f (3 . 5) by cubic interpolation via Newtons forward dierence formula. (c) Given that 1 | f (4) ( t ) | 2 for any t [1 , 4] give error bounds for your estimates in part ( a )....
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