MAD4401 Test1 - Name MAD 4401, Numerical Analysis Keesling...

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Unformatted text preview: Name MAD 4401, Numerical Analysis Keesling Test 1 9/21/11 Do all problems. Show your work and explain your answers. Each problem is worth 20 points. 1. 1 dx using Romberg Integration. Use up to 32 −4 1 + x 2 subdivisions in the Composite Trapezoidal Rule. Circle the best estimate of the integral. Use five digits in the first four columns. Use twelve digits for entries in the last two columns. How many digits can you expect to be correct in this answer? Explain. Estimate the ∫ 4 n=1 n=2 n=4 n=8 n = 16 n = 32 2. Give the Newton-Cotes coefficients for n = 1, 2, 3, 4 . n =1 a0 = a1 = n=2 a0 = a1 = a2 = n=3 a0 = a1 = a2 = a3 = n=4 a0 = a1 = a2 = a3 = a4 = 3. The nth Legendre polynomial is given by the derivative formula. 1 dn ( x 2 − 1)n ) n n( 2 ⋅ n! dx Use this to determine the polynomial formulas for the Legendre polynomials for n = 0, 1, 2, 3, 4, 5 , and n = 10. 4. Find the roots of the third degree Legendre polynomial on the interval [-1,1]. What coefficients would be used with these roots in Gaussian Quadrature? 5. The following describes the Open Newton Cotes formula for the positive integer n on the interval [0,1]. The points used are 2 3 n⎫ ⎧1 , , ,…, ⎨ ⎬ . The coefficients are chosen so that the formula n + 1⎭ ⎩n +1 n +1 n +1 ∫ 1 0 n i p( x ) dx = ∑ ai ⋅ p ( n+1 ) i =1 is exact for polynomials up to degree n – 1. Find the coefficients {a1, a2, a3, a4, a5, a6 } for n = 6 . Give the value for the as fractions. 0 a1 = a2 = a3 = a4 = a5 = a6 = 1 n +1 2 n +1 3 n +1 n n +1 1 ...
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This note was uploaded on 11/12/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.

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MAD4401 Test1 - Name MAD 4401, Numerical Analysis Keesling...

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