Exercise 47115 determine constants a b c and d that

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Exercise 4.7.11(5). Determine constants a, b, c, and d that will produce a quadrature formula Z 1 - 1 f ( x ) dx = af ( - 1) + bf (1) + cf 0 ( - 1) + d f 0 (1) (1) that has degree of precision 3. Solution. We obtain 4 equations in a, b, c, d by evaluating (1) with f ( x ) = 1 , x, x 2 , x 3 . The left hand side gives 2 , 0 , 2 / 3 , 0 for these functions, while the right hand side gives linear combinations of a, b, c, d . The linear system to solve is a + b = 2 , - a + b + c + d = 0 , a + b - 2 c + 2 d = 2 / 3 , - a + b + 3 c + 3 d = 0 , or, equivalently 1 1 0 0 - 1 1 1 1 1 1 - 2 2 - 1 1 3 3 a b c d = 2 0 2 / 3 0 . Using matlab to solve this system (syntax: x=A\b ), we obtain a = 1 , b = 1 , c = 1 / 3 , d = - 1 / 3. Finally, we check that the quadrature rule is not of higher degree of precision than 3 by observing that a + b - 4 c + 4 d = - 2 / 3, which is different than R 1 - 1 x 4 dx = 2 / 5.
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Exercise 4.7.13(7). Verify the entries for the values of n = 2 and 3 in Table 4.12 on page 232 by finding the roots of the respective Legendre polynomials, and use the equations preceding this table to find the coefficients associated with the values. Solution. The second and third (monic) Legendre polynomials are P 2 ( x ) = x 2 - 1 / 3, P 3 ( x ) = x ( x 2 - 3 / 5). The roots are therefore x (2) 1 = - 1 / 3 = - 0 . 5773502692 x (2) 2 = 1 / 3 = 0 . 5773502692 x (3) 1 = - p 3 / 5 = - 0 . 7745966692 x (3) 2 = 0 x (3) 3 = p 3 / 5 = 0 .
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  • Spring '08
  • Rieffel
  • Romberg's method

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