gaussian quadrature - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:43:09Z http://mathoverflow.net/feeds/question/51092 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51092/gaussian-quadrature gaussian quadrature someone 2011-01-04T08:38:15Z 2011-03-17T23:40:16Z <p>Gaussian quadrature allows us to integrate polynomials up to order $2 n-1$ using only $n$ function values.</p> <p>$\int_{x_0}^{x_1} ( \sum_{i=0}^{2 n-1} a_i x^i ) dx = f(a_0, \dots , a_{2 n-1})$</p> <p>thus, the function $f(a_0, \dots , a_{2 n-1})$, which naively has $2 n$ parameters, can actually be calculated using only $n$ parameters. Or, there exists a coordinate transform $a_i\rightarrow a_i^{\star}$</p> <p>$f(a_0, \dots , a_{2 n-1}) \rightarrow f^{\star}(a_0^{\star}, \dots , a_n^{\star} , \dots , a_{2 n-1}^{\star})$</p> <p>such that $\frac{\partial f^{\star} }{a_i^{\star}} \equiv 0$ if $i>n$</p> <p>I understand that this is true, and that the construction of gaussian quadrature has $2n$ free parameters, i.e. sampling positions and weights, which seemingly explains why it can integrate polynomials up to order $2 n-1$.</p> <p>Nevertheless, I wonder if there is a "geometric" argument that makes it clear that the function $f(a_0, \dots , a_{2 n-1})$, in $2n$-dimensional space, can be calculated using only $n$ parameters.</p> http://mathoverflow.net/questions/51092/gaussian-quadrature/58797#58797 Answer by Andrei MF for gaussian quadrature Andrei MF 2011-03-17T23:40:16Z 2011-03-17T23:40:16Z <p>Look at it this way: what you are computing is a PROJECTION. In this sense, you are not surprised that in order to compute the value of the function $f(x_1, x_2)=x_1$ you need only 1 parameter. Something similar happens here. Instead of taking as parameters the coefficients $a_j$ of the expansion of the given polynomial in the basis of monomials, take the $a_j$ to be the expansion of this polynomial in terms of the orthogonal polynomials with respect to the integration weight. Then the quadrature formula is given by the linear combination of the coefficients $a_n, \dots, a_{2n-1}$ only.</p>