Basic Examples (2) 

Create ten Chebyshev nodes analytically:

In[1]:=

Out[1]=

Approximate the numbers:

In[2]:=

Out[2]=

Scope (2) 

Create the numbers on a domain from 0 to 2π:

In[3]:=

Out[3]=

Show on a number line:

In[4]:=

Out[4]=

Applications (6) 

Find points for fifth order approximation:

In[5]:=

Out[6]=

Perform the approximation of Cos[x]:

In[7]:=

Out[8]=

Represent the approximation as a polynomial:

In[9]:=

Out[9]=

Compare the functions:

In[10]:=

Out[10]=

Show the difference:

In[11]:=

Out[11]=

Compare to the error from a fifth-order series expansion evaluated at the midpoint:

In[12]:=

Out[12]=

Properties and Relations (2) 

The Chebyshev nodes are related to the roots of Chebyshev polynomials of the first kind:

In[13]:=

Out[14]=

Compare the zeros by finding the roots numerically and using the function ChebyshevNodes:

In[15]:=

Out[16]=

Chebyshev nodes match the coordinate components of CirclePoints:

In[17]:=

Out[17]=

Neat Examples (1) 

Show how Chebyshev nodes are related to equi-distantly spaced points on a semi-circle:

In[18]:=

Out[19]=