AME404SampleFinal

AME404SampleFinal - continuous second derivatives at all...

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AME 404 Final Exam GFS 106, December 8, 2004, 11 AM-1 PM Professor: Dravinski Name Instructions. Closed books and notes. No calculators. 1. (25 pts.) Lagrange coe cient polynomials of degree n for data ( x j ,y j ); j =1: m are de f ned by L ( n ) k ( x )= m Y j =1; j 6 = k ( x x j ) m Y j =1; j 6 = k ( x k x j ) ; k =1: m ; n = m 1 Draw a F owchart which evaluates all the Lagrange coe cient polynomials within MATLAB environ- mentandstorestheminamatr ix L , where each row r of L represents L ( n ) r ( x ) .H in t :U s eMATLAB functions "conv" and "poly". 2. (25 pts.) Consider a set of data ( x k ,y k ); k =1: n in which all abscissas are distinct. The data is to be f tted in a least-square sense by a cubic f ( x )= ax 3 + bx 2 + cx + d (a) Derive the corresponding normal equations for the coe cients a, b, c, and d so that the error E ( a, b, c, d )= n X k =1 ( f ( x k ) y k ) 2 is minimized. (b) Draw the corresponding F owchartthatw i l leva luatetheunknowncoe cients a d. Assume that the function "UpTrBk(A,b)" is given. 3. (20 pts.) Cubic splines which interpolate the data
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Unformatted text preview: continuous second derivatives at all interior nodes ( x i , y i ); i = 2 : n − 1 , are given by f i ( x ) = 1 6 h i £ ( x i +1 − x ) 3 f 00 i + ( x − x i ) 3 f 00 i +1 ¤ + · y i h i − h i 6 f 00 i ¸ ( x i +1 − x ) + · y i +1 h i − h i 6 f 00 i +1 ¸ ( x − x i ) x i ≤ x ≤ x i +1 ; i = 1 : n − 1 Assume that the second derivatives f 00 i ; i = 1 : n are known for a particular type of spline at all interior nodes. Draw a F owchart which evaluates the above splines at m locations z j ; j = 1 : m. 4. (30 pts.) Simpson’s quadrature is given by x 2 Z x f ( x ) dx ≈ h 3 ( f + 4 f 1 + f 2 ) (a) Determine the degree of precision for that quadrature. (b) Derive the corresponding composite quadrature in order to integrate f ( x ) over the range [ a, b ] . (c) Draw a F owchart which implements the results of Part (b) in MATLAB. 1...
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