Chebyshev pseudospectral method

<p>The Chebyshev pseudospectral method for <a href="/facts/Optimal_control/9SmcQHjJ">optimal control</a> problems is based on <a href="/facts/Chebyshev_polynomials/HoBINqgC">Chebyshev polynomials of the first kind</a>. It is part of the larger theory of <a href="/facts/Pseudospectral_optimal_control/Gv0XADRR">pseudospectral optimal control</a>, a term coined by <a href="/facts/I._Michael_Ross/nV8tGD3W">Ross</a>. Unlike the <a href="/facts/Legendre_pseudospectral_method/XXJapl1u">Legendre pseudospectral method</a>, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions of the method have been proposed: one by Elnagar et al., and another by Fahroo and Ross.  The two versions differ in their quadrature techniques.  The <a href="/facts/Ross%25E2%2580%2593Fahroo_pseudospectral_method/4eYsAHNI">Fahroo–Ross method</a> is more commonly used today due to the ease in implementation of the <a href="/facts/Clenshaw%25E2%2580%2593Curtis_quadrature/fFNmUf4G">Clenshaw–Curtis quadrature</a> technique (in contrast to Elnagar–Kazemi's cell-averaging method). In 2008, Trefethen showed that the Clenshaw–Curtis method was nearly as accurate as <a href="/facts/Gauss_quadrature/MLmU1Dgc">Gauss quadrature</a>.
  This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods.  A complete mathematical theory for Chebyshev PS methods was finally developed in 2009 by Gong, Ross and Fahroo.
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Chebyshev pseudospectral method open-in-new

Chebyshev pseudospectral method