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| Materialtyp: | Preprint |
| Publicerad: |
1999
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| Länkar: | https://arxiv.org/abs/math/9908095 |
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| _version_ | 1866909853548544000 |
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| author | Horwitz, Alan |
| author_facet | Horwitz, Alan |
| contents | Let M(f) denote the Midpoint Rule and T(f) the Trapezoidal Rule for estimating integral_a^b f(x) dx. Then Simpson's Rule = tM(f) + (1-t)T(f), where t = 2/3. We generalize Simpson's Rule to multiple integrals as follows.
Let D be some polygonal region in R^n, let P_0,...,P_m denote the vertices of D, and let P_(m+1) = center of mass of D.
Define the linear functionals M(f) = Vol(D)f(P_(m+1)), which generalizes the Midpoint Rule, and T(f) = Vol(D)(1/(m+1))sum(f(P_j), j = 0,...,m), which generalizes the Trapezoidal Rule. Finally, our generalization of Simpson's Rule is given by the cubature rule(CR) L_t = tM(f) + (1-t)T(f), for t in [0,1]. We choose t, depending on D, so that L_t is exact for polynomials of as large a degree as possible. In particular we derive CRs for the n simplex and unit n cube. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_9908095 |
| institution | arXiv |
| publishDate | 1999 |
| record_format | arxiv |
| spellingShingle | A Version of Simpson's Rule for Multiple Integrals Horwitz, Alan Numerical Analysis Classical Analysis and ODEs 65D32 Let M(f) denote the Midpoint Rule and T(f) the Trapezoidal Rule for estimating integral_a^b f(x) dx. Then Simpson's Rule = tM(f) + (1-t)T(f), where t = 2/3. We generalize Simpson's Rule to multiple integrals as follows. Let D be some polygonal region in R^n, let P_0,...,P_m denote the vertices of D, and let P_(m+1) = center of mass of D. Define the linear functionals M(f) = Vol(D)f(P_(m+1)), which generalizes the Midpoint Rule, and T(f) = Vol(D)(1/(m+1))sum(f(P_j), j = 0,...,m), which generalizes the Trapezoidal Rule. Finally, our generalization of Simpson's Rule is given by the cubature rule(CR) L_t = tM(f) + (1-t)T(f), for t in [0,1]. We choose t, depending on D, so that L_t is exact for polynomials of as large a degree as possible. In particular we derive CRs for the n simplex and unit n cube. |
| title | A Version of Simpson's Rule for Multiple Integrals |
| topic | Numerical Analysis Classical Analysis and ODEs 65D32 |
| url | https://arxiv.org/abs/math/9908095 |