Just to make sure I understand perfectly, if I am approximating for a 7th degree polynomial integration a 13 point Quadrature should give me the EXACT solution?
Only asking because you said multiple solutions are possible for the weights and arguments?
@sachinabey To get exact solution for integrating a 7th degree polynomial, you need 4 quadrature point rule. A n-point Gauss quad rule gives exact results for a polynomial of order (2n-1). Multiple solutions are possible for solving the nonlinear equations that result from deriving the rule, but when constrained that the quadrature points are in [-1,1], we get a unique solution.
I see a comment asking for the values of C1...Cn and X1...Xn, with a response that says the tables can be found online, but I have been looking for quite a while and there are no tables online that give the exact values, only decimal approximations. Where might one find the exact values for n=2 to n={some high value}? Better yet, how can we derive these values ourselves?
@jurrich I do not think you can find exact values as X1...Xn are zeros of a Legendre polynomial, Pn(x). They are given in irrational form as square-roots, etc. You can get some great details about it and write your own program to do this. Do a google search on Legendre-Gauss Quadrature wolfram and it will be the top search item.
@numericalmethodsguy Thanks, the Wolfram page was very helpful and I wrote a Mathematica program that generates the Ci and Xi values for people to use (although youtube won't let me link to it in this comment, nice), with a decimal precision of 16 on the page, and 256 in the downloadable data. Hopefully this can be of use for others who are also looking for high precision numbers. And of course: thank you for your lectures, they have very helpful!
@jurrich Wrote a blog abou this and a matlab program to go with it. Go to autarkaw(dot)wordpress(dot)com and go to July 7, 2011 entry on "A MATLAB program to find quadrature points and weights for Gauss-Legendre Quadrature rule"
@photobucketmagic they are given in a gauss quadrature table that you can find online. Use the values that corresponds to the number of points your quadrature is taking.
Just to make sure I understand perfectly, if I am approximating for a 7th degree polynomial integration a 13 point Quadrature should give me the EXACT solution?
Only asking because you said multiple solutions are possible for the weights and arguments?
Thanks in advance an fantastic lectures.
sachinabey 6 months ago in playlist More videos from numericalmethodsguy
@sachinabey To get exact solution for integrating a 7th degree polynomial, you need 4 quadrature point rule. A n-point Gauss quad rule gives exact results for a polynomial of order (2n-1). Multiple solutions are possible for solving the nonlinear equations that result from deriving the rule, but when constrained that the quadrature points are in [-1,1], we get a unique solution.
numericalmethodsguy 6 months ago
I see a comment asking for the values of C1...Cn and X1...Xn, with a response that says the tables can be found online, but I have been looking for quite a while and there are no tables online that give the exact values, only decimal approximations. Where might one find the exact values for n=2 to n={some high value}? Better yet, how can we derive these values ourselves?
jurrich 9 months ago
@jurrich I do not think you can find exact values as X1...Xn are zeros of a Legendre polynomial, Pn(x). They are given in irrational form as square-roots, etc. You can get some great details about it and write your own program to do this. Do a google search on Legendre-Gauss Quadrature wolfram and it will be the top search item.
numericalmethodsguy 9 months ago
@numericalmethodsguy Thanks, the Wolfram page was very helpful and I wrote a Mathematica program that generates the Ci and Xi values for people to use (although youtube won't let me link to it in this comment, nice), with a decimal precision of 16 on the page, and 256 in the downloadable data. Hopefully this can be of use for others who are also looking for high precision numbers. And of course: thank you for your lectures, they have very helpful!
jurrich 9 months ago
@jurrich Wrote a blog abou this and a matlab program to go with it. Go to autarkaw(dot)wordpress(dot)com and go to July 7, 2011 entry on "A MATLAB program to find quadrature points and weights for Gauss-Legendre Quadrature rule"
numericalmethodsguy 7 months ago
thank u so much!!! this is brilliant
24asiya 9 months ago
but I'm not given c1 and c2 or x1 x2....
photobucketmagic 1 year ago
@photobucketmagic they are given in a gauss quadrature table that you can find online. Use the values that corresponds to the number of points your quadrature is taking.
nighthawk2k3rsx 1 year ago