Bratu's problem, which is the nonlinear eigenvalue equation Au + 2 exp(u)= 0 with u = 0 on the walls of the unit square and 2 as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance of symmett:v: because of invariance under the C 4 rotation group and parity in both x and y, one can slash the size of the basis set by a...

When a function f(x) is holomorphic on an interval x ∈ [a, b], its roots on the interval can be computed by the following three-step procedure. First, approximate f(x) on [a, b] by a polynomial fN (x) using adaptive Chebyshev interpolation. Second, form the Chebyshev– Frobenius companion matrix whose elements are trivial functions of the Chebyshev coefficients of the interpolant fN (x). Third, ...

We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values the variable when E$ of can be erroneous. Our perform exact arithmetic in field scalars $\mathsf{K}$ and standard powers or Chebyshev polynomials, which case characteristic is $\ne 2$. return list valid interpolants support points run polynomial-time. For power basis...

We present a Python module named PyCheb, to solve the ordinary differential equations by using spectral collocation method. PyCheb incorporates discretization using Chebyshev points, barycentric interpolation and iterate methods. With this Python module, users can initialize the ODEsolver class by passing attributes, including the both sides of a given differential equation, boundary conditions...