Numerical evaluation of polynomials in Chebyshev basis I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and -1, where Chebyshev polynomials change rapidly, is also required. 
I know that there exists Clenshaw algorithm for evaluation of Chebyshev polynomials, which is somewhat similar to the Horner scheme.
And this is all that I know...
I also saw the question about similar problematics. The proposed solution is to use higher precision. 
I wonder if there are some other methods for accurate evaluation of polynomials (particularly in Chebyshev basis) on the given intervals that don't heavily rely on extended precision calculations? Is it possible at least to improve the problem as much as possible, so that extended precision is needed only from really high polynomial orders? 
I also thought about interpolation of a high order polynomial by lower order ones on the intervals of interest. However, I don't know if there's any systematic procedure for this approach.

You can use Java code performing arbitrary calculations 
in Chebyshev (also in Legendre, Laguerre, Hermite) basis, see Appendix A of 
a paper http://arxiv.org/abs/1510.05510 
the source code is available from Ref [31] link in the paper.
There is little documentation available, but class Chebyshev.java
has a unit test, from which usage should be more or less clear.
Chebyshev B=new Chebyshev();
final double [] c=new double []{coef_T0,coef_T1,...}; // chebyshev basis coefs
for(double x=-1;x<1.0001;x+=0.1)
{
   System.out.println("x="+x+" F="+B.calculateSumD0(c.length-1,c,x));
}
