Math 128A - HW8 Solutions.pdf

# Exercise 47115 determine constants a b c and d that

• Homework Help
• 6

This preview shows page 5 - 6 out of 6 pages.

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.

Subscribe to view the full document.

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 .

#### You've reached the end of your free preview.

Want to read all 6 pages?

• Spring '08
• Rieffel
• Romberg's method

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern