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I was wondering if anyone knew about any numerical methods specifically designed for integrating functions of the form $\sin(p_{m})$ and $\cos(p_{m})$ where $p_{m}$ is a polynomial of degree $m$. I don't think $m$ will be too large in the application, but it will probably be at least $4$. I was reading this thread about Laplace's method but I'm not really sure that's what I want.

Community
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Mr. G
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  • The only thing I can think of is an adaptive variant of a basic quadrature rule like Simpson's rule. – Ian Jul 12 '14 at 00:08
  • I think you should give more information. In particular, why are the standard adaptive integration routines not working for you? Is the integration region large enough that the integral is very oscillatory? In that case, why can't you use standard oscillatory integral algorithms? There could be some deeper issues here, but you should say why the standard approaches aren't working for you. – Kirill Jul 20 '14 at 02:39
  • @Kirill Actually, the integration range is small, but I need high accuracy. Standard methods may be fine, but I just wanted to check if there was anything specifically for this problem. – Mr. G Jul 21 '14 at 15:43
  • I see. I think this question is maybe a bit too vague, then. – Kirill Jul 21 '14 at 21:52

1 Answers1

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I think you can use Gaussian quadrature with weight functions $w(x)=\sin(p_m)$ and $w(x)=\cos(p_m)$. For any weight function there is a family of orthogonal polynomials related to that function in $[a,b]$. This family can be obtained by Gram-Schmidt orthogonalization process on $1,x,x^2,\ldots,x^n$.

Let $w(x)$ be a weight function in $[a,b]$ and $\{p_0,p_1,\ldots,p_n\}$ be a family of orthogonal polynomials with degree of $p_k=k$. Let $x_1,x_2,\ldots,x_n$ be the roots of $p_n(x)$ and $$ L_i(x)=\prod_{j=1, j\ne i}^{n}{\frac{x-x_j}{x_i-x_j}} $$ be the Lagrange polynomials of these roots. Then the Gaussian quadrature is defined by $$ \int_a^b{w(x)f(x)dx}\approx \sum_{i=1}^{n}{w_i f(x_i)}=\sum_{i=1}^{n}{\left(\int_a^b{w(x)L_i(x)dx} \right) f(x_i)} $$ The above formula is exact for all polynomials of degree $\le 2n-1$. This is better than Newton-Cotes formulas since the $(n+1)$ points Newton-Cotes rule is exact for polynomials of degree $\le n$ ($n$ odd) or $\le (n+1)$ ($n$ even).

Dante
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