## What are Chebyshev points?

In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the effect of Runge’s phenomenon.

**Why Chebyshev nodes reduces the Runge phenomenon?**

example of such a set of nodes is Chebyshev nodes, for which the maximum error in approximating the Runge function is guaranteed to diminish with increasing polynomial order. The phenomenon demonstrates that high degree polynomials are generally unsuitable for interpolation with equidistant nodes.

**How do you know which interpolate a polynomial?**

For example, suppose you are given the three points (x0,y0), (x1,y1) and (x2,y2). Then there is polynomial of degree 2, p2(x) = a0 +a1x+a2x2, such that p2(x0) = y0, p2(x1) = y1, and p2(x2) = y2. This is the quadratic interpolating polynomial through the three given points.

### Where are Chebyshev polynomials used?

The Chebyshev polynomials are used for the design of filters. They can be obtained by plotting two cosines functions as they change with time t, one of fix frequency and the other with increasing frequency: ( 2 π t ) , y ( t ) = cos

**Why are Chebyshev polynomials important?**

Chebyshev polynomials are important in approximation theory because the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. This approximation leads directly to the method of Clenshaw–Curtis quadrature.

**What is the maximum error if the interpolation is performed at the Chebyshev points?**

we find that has a maximum value of about 0.11. We then find Page 4 1. CHEBYSHEV POLYNOMIALS AND THE MINIMALIZATION OF ERROR 4 which has a maximum value of about 0.06, which is about half the value that we obtained for the simpler choice of points xi.

## What is the main reason that the Runge phenomenon appears for a higher degree polynomial?

Reason. Runge’s phenomenon is the consequence of two properties of this problem. The magnitude of the n-th order derivatives of this particular function grows quickly when n increases. The equidistance between points leads to a Lebesgue constant that increases quickly when n increases.