0

I have this question for a Numerical Analysis class. However, we've just finished chapter four in class, and this question relates to the end of chapter 5. I'm not quite sure why it's on this assignment. If someone could point me in the right direction, I would really appreciate it! Here is the question:

Using the method of undetermined coefficients to derive the Gauss quadrature rule of order 2n with quadrature points xi and weights $w_i$ = , i = 1, 2, . . . , n, we obtain the following systems of 2n equations:

$\sum_{i=1}^n w_i*x_i^{2j} = 2/(2*j + 1)$

$\sum_{i=1}^n w_i*x_i^{2j+1} = 0$

where j = 0, 1, ...., n-1

Write out and Solve the following systems for n = 1.

End of question. Thank you for the read, and I'd appreciate any help!

Devon Puchailo
  • 41

1 Answers1

1

You are just supposed to explicitly write out the $n=1$ case of the equations. There is only one term in each sum and only $x_1, w_1$ to determine and as $j$ ranges up to $n-1$ you only have $j=0$ to consider. You get two equations in two unknowns.

Ross Millikan
  • 374,822
  • Am I just supposed to make an augmented Matrix and solve for my two variables then? – Devon Puchailo Mar 12 '17 at 21:25
  • 1
    If you write out the two equations they should be easy to solve by inspection. – Ross Millikan Mar 12 '17 at 21:31
  • I was seriously over thinking this, thank you very much for your help! – Devon Puchailo Mar 12 '17 at 21:33
  • so, in the case of n = 2

    I would get the equations:

    $w_1 * x_1^2 + w_2*x_2^2 = 2/3$

    and: $w_1x_1^3 + w_2x_2^3 = 0$

     – Devon Puchailo Mar 12 '17 at 21:53
  • For $n=2$ you get four equations, two from the first sum and two from the second. One of each is with $j=0$, the other with $j=1$. There are two $w$s and two $x$s to solve for. Those are the two $j=1$ ones. The idea will eventually be that with $n$ points you can make the quadrature formula exact for polynomials up to degree $2n-1$. The four equations for the $n=2$ case represent the fact that the quadrature will be exact for $x^0,x^1,x^2,x^3$ and linearity takes care of the rest. You will see that all next chapter. This exercise is a lead-in. – Ross Millikan Mar 12 '17 at 23:58
  • Sorry, I feel like I'm being dense here.

    So for my four equations, I'm going to have two for the case that j = 0: (IE the first ones I solved in the above part), and two for the case where j = 1 (IE the ones I posted above).

    Correct?

    I don't need to solve the case for n = 2. I'll need to do Newton's method for until a convergence of $10^-10$ is reached.

     – Devon Puchailo Mar 13 '17 at 00:21
  • That is right, though for $n=2$ the sums will have two terms even in the first two equations. You then get four equations in four unknowns. I haven't written them out, but I think you can get an analytic solution. There are tables online of the weights and nodes for Gaussian integration. – Ross Millikan Mar 13 '17 at 02:24