📂Numerical Analysis

First kind Chebyshev polynomials

Definition ¹

$T_{n} (x) = \cos \left( n \cos^{-1} x \right)$ is called the first kind Chebyshev polynomial.

Basic Properties

Recurrence Formula

• [0]: $$T_{n+1} (x) = 2x T_{n} (x) - T_{n-1} (X)$$

Orthogonal Set

• [1] Inner product of functions: Given the weight $w$ as $\displaystyle w(x) := {{1} \over { \sqrt{1 - x^2} }}$, $\left\{ T_{0} , T_{1}, T_{2}, \cdots \right\}$ forms an orthogonal set.

Chebyshev Nodes

• [2]: The roots of $T_{n} (x)$ are the Chebyshev nodes for $k=1, \cdots , n$. $$x_{k} = \cos \left( {{2k-1} \over {2n}} \pi \right)$$

Even and Odd Functions

• [3]:

$$T_{n} (-x) = (-1)^{n} T_{n} (x)$$

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• Typically, $0 \le \theta \le \pi$ is defined as $\theta := \cos^{-1} x $.

See Also

• First kind Chebyshev polynomial
• Second kind Chebyshev polynomial
• Relationship between first and second kind Chebyshev polynomials
• Chebyshev polynomials as solutions to Chebyshev’s differential equation

Description

The first kind Chebyshev polynomial for $n = 0, \cdots , 3$ is expressed as follows:

$$ \begin{align*} T_{0} (x) =& 1 \\ T_{1} (x) =& x \\ T_{2} (x) =& 2x^{2} - 1 \\ T_{3} (x) =& 4x^{3} - 3x \end{align*} $$

The first kind Chebyshev polynomial is very useful not only in numerical analysis but also in applied mathematics in general, and together with the second kind Chebyshev polynomial, it has a wide range of interesting properties.

Meanwhile, the first kind Chebyshev polynomial can also be defined using the inverse $T_{0} (x) = 1$, $T_{1} (x) = x$, and the recurrence formula [0]. This is also true for the second kind Chebyshev polynomial, so it is reasonable to refer to the first and second kinds for reasons $T_{1} (x) = 1 \cdot x$ and $U_{1} (x) = 2 \cdot x$.

Proof

[0]

Since $T_{n} (x) = \cos \left( n \theta \right)$, by the addition theorem of trigonometric functions $$ T_{n \pm 1} (x) = \cos (n \pm 1) \theta = \cos (n \theta ) \cos \theta \mp \sin ( n \theta ) \sin \theta $$ Adding both sides $$ T_{n+1} (x) + T_{n-1} (x) = 2 \cos (n \theta ) \cos \theta = 2 T_{n} (x) x $$ Rearranging $T_{n-1} (x)$ gives $$ T_{n+1} (x) = 2x T_{n} (x) - T_{n-1} (x) $$

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[1]

Since $dx = - \sin \theta d \theta = - \sqrt{1 - x^2} d \theta$, $$ \begin{align*} \displaystyle \left< T_{n}, T_{m} \right> =& \int_{-1}^{1} T_{n} (x) T_{m} (x) {{1} \over { \sqrt{1 - x^2} }} dx \\ =& - \int_{\pi}^{0} \cos n \theta \cos m \theta d \theta \\ =& \int_{0}^{\pi} \cos n \theta \cos m \theta d \theta \\ =& \begin{cases} \pi/2 &, n=m \\ 0 &, n \ne m \end{cases} \end{align*} $$ Therefore, $\left\{ T_{0} , T_{1}, T_{2}, \cdots \right\}$ forms an orthogonal set.

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[2]

Trivial by definition.

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[3]

Case 1. $n=0,1$

$$ T_{0} (-x) = 1 = T_{0} (x) $$

$$ T_{1} (-x) = (-x) = -x = - T_{1} (x) $$

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Case 2. $n \ge 2$ is even

Since all terms of $T_{n}(x)$ have even degrees except the coefficient $0$, $T_{n}(-x) = T_{n}(x)$

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Case 3. $n \ge 2$ is odd

Since all terms of $T_{n}(x)$ have odd degrees except the coefficient $0$, $T_{n}(-x) = - T_{n}(x)$

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Implementation

Below is the code for the Chebyshev polynomial written in R.

Since the polynomial itself is returned, it can be directly used for calculations. n is the degree, kind specifies the type, and if the print option is true, it outputs the coefficients.

The coefficients are output from the constant term to the high-order term, and since the first kind Chebyshev polynomial is $T_{3} (x) = 4x^{3} - 3x$, it is correctly obtained. The function value is also accurately calculated as $T_{3} (3) = 4 \cdot 3^{3} - 3 \cdot 3 = 108-9 = 99$.

Chebyshev<-function(n,kind=1,print=F)
{
  p<-NA
  
  if((round(n)-n)!=0 | n<0) {stop("Wrong Degree!!")} #degree must be nonnegative integer
  if(!kind%in%(1:2)) {stop("Wrong Kind!!")} #kind must be 1 or 2
  
  if(n==0)
  {
    if(print) {print(1)}
    
    p<-function(x) {return(1)}
    return(p)
  }
  
  if(n==1)
  {
    if(print) {print(c(0,kind))}
    
    p<-function(x) {return(kind*x)}
    return(p)
  }
 
  coef0<-c(1)
  coef1<-c(0,kind)
  
  for(i in 1:(n-1))
  {
    coef2<- ( c(0,2*coef1) - c(coef0,0,0) )
    coef0<-coef1
    coef1<-coef2
  }
  
  if(print) {print(coef2)}
  
  p<-function(x)  {return(sum(coef2*x^(0:n)))}
  return(p)
}
 
p<-Chebyshev(1,1); p(2)
p<-Chebyshev(3,1,T); p(3)