Is Gauss Quadrature Better than Clenshaw--Curtis? TeX Export

@article{trefethen:67, abstract = {We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of \$\log((z+1)/(z-1))\$ in the complex plane. Gauss quadrature corresponds to Pad\'{e} approximation at \$z=\infty\$. Clenshaw–Curtis quadrature corresponds to an approximation whose order of accuracy at \$z=\infty\$ is only half as high, but which is nevertheless equally accurate near \$[-1,1]\$.}, author = {Trefethen, Lloyd N.}, citeulike-article-id = {4893339}, citeulike-linkout-0 = {http://dx.doi.org/10.1137/060659831}, citeulike-linkout-1 = {http://link.aip.org/link/?SIR/50/67/1}, doi = {10.1137/060659831}, journal = {SIAM Review}, keywords = {approximation, chebyshev, clenshawcurtis, expansion, fft, gauss, methods, newtoncotes, quadrature, rational, spectral}, number = {1}, pages = {67--87}, posted-at = {2009-06-18 15:31:52}, priority = {2}, publisher = {SIAM}, title = {Is Gauss Quadrature Better than Clenshaw--Curtis?}, url = {http://dx.doi.org/10.1137/060659831}, volume = {50}, year = {2008} }

TY - JOUR ID - trefethen:67 L3 - citeulike-article-id:4893339 N2 - We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of $\log((z+1)/(z-1))$ in the complex plane. Gauss quadrature corresponds to Padé approximation at $z=\infty$. Clenshaw–Curtis quadrature corresponds to an approximation whose order of accuracy at $z=\infty$ is only half as high, but which is nevertheless equally accurate near $[-1,1]$. IS - 1 JF - SIAM Review EP - 87 TI - Is Gauss Quadrature Better than Clenshaw--Curtis? PB - SIAM VL - 50 SP - 67 KW - approximation KW - chebyshev KW - clenshawcurtis KW - expansion KW - fft KW - gauss KW - methods KW - newtoncotes KW - quadrature KW - rational KW - spectral AU - Trefethen, Lloyd PY - 2008/// UR - http://dx.doi.org/10.1137/060659831 ER -