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Chebyshev Polynomials Reading

Problems

Differential Equation and Its Solution The Chebyshev differential equation is written as

(1

−

  d2 y x ) dx2 2

dy − x   dx + n

2

y = 0

 

n  = 0, 1, 2, 3, . . .

If we let  x  = cos t  we obtain

d2 y + n2 y   = 0 2 dt whose general solution is

y   =  A cos nt +  B sin nt or as

−1

y   =  A cos(n cos

−1

x) +  B sin(n cos

x)

 

|x|  1

The function  T n (x)  is a polynomial. For x <  1  we have

||

n

Tn (x) +  iUn (x) = (cos t +  i sin t) Tn (x)

− iU

n (x)

= (cos t

− i sin t)

n

n

    = x +  i 1 − x     = x−i 1−x 2

2

n

from which we obtain

 1 Tn (x) = 2

n

n

x + i 1 − x  + x − i 1 − x   2

2

||

For x >  1  we have nt

Tn (x) + Un (x) =   e Tn (x)

−U

2

nt

−

n (x) =   e

n

    = x± x −1     = x∓ x −1 2

n

The sum of the last two relationships give the same result for  T n (x).

2

 

Chebyshev Polynomials of the First Kind of Degree   n The Chebyshev polynomials  T n (x)  can be obtained by means of Rodrigue’s formula

Tn (x) =

  ( 2)n n!

−

(2n)!

 1 − x

2

dn dxn

x2 )n

−1

  (1

/2

n  = 0, 1, 2, 3, . . .

−

The first twelve Chebyshev polynomials are listed in Table 1 and then as powers of   x   in terms of  T  T n (x)  in Table 2.

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Table 1: Cheb Chebyshe yshev v Po Polynom lynomials ials of the First Kind T0 (x) = 1 T1 (x) =   x T2 (x) = 2x2 T3 (x) = T4 (x) = T5 (x) = T6 (x) = T7 (x) = T8 (x) = T9 (x) = T10 (x) = T11 (x) =

−1 4x − 3x 8x − 8x + 1 16x − 20x + 5 x 32x − 48x + 18 x − 1 64x − 112x + 56 x − 7x 128x − 256x + 160x − 32x + 1 256x − 576x + 432x − 120x + 9 x 512x − 1280x + 1120x − 400x + 50 x − 1 1024x − 2816x + 2816x − 1232x + 220x − 11x 3

4

2

5

3

6

4

2

3

5

7

8

6

4

9

7

5

10

11

2

3

7

9

4

2

4

6

8

5

3

 

Table 2: Po Powe wers rs of   x  as functions of   T n (x) 1 =   T 0 x   =   T 1 x2 =

 1 (T 0  +  T 2 ) 2

x3 =

 1 (3T 1  +  T 3 ) 4

4

x

 1 = 8 (3T 0  + 4T 2  +  T 4 )

x5 =

  1 (10T 1  + 5T 3  +  T 5 ) 16

x6 =

  1 (10T 0  + 15T 2  + 6T 4  +  T 6 ) 32

x7 =

  1 (35T 1  + 21T 3  + 7T 5  +  T 7 ) 64

x8 =

  1 (35T 0  + 56T 2  + 28T 4  + 8T 6  +  T 8 ) 128

x9 =

  1 (126T 1  + 84T 3  + 36T 5  + 9T 7  +  T 9 ) 256

x10 =

  1 (126T 0  + 210T 2  + 120T 4  + 45T 6  + 10T 8  +  T 10 10 ) 512

x11 =

  1 (462T 1  + 330T 3  + 165T 5  + 55T 7  + 11T 9  +  T 11 11 ) 1024

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Generating Function for   T n (x) The Chebyshev polynomials of the first kind can be developed by means of the generating function

∞

−

1 tx = 1 2tx +  t2

−

T  (x)t

n

n

n=0

Recurrence Formulas for   T n (x) When the first two Chebyshev polynomials   T 0 (x)   and   T 1 (x)  are known, all other polyno2  can be obtained by means of the recurrence formula mials  T n (x), n

≥

T n+1 (x) = 2xT n (x)

− T 

n−1 (x)

The derivative of  T   T n (x)  with respect to  x  can be obtained from

(1

2



− x )T  (x) = −nxT  (x) +  nT 

n−1 (x)

n

n

Special Values of   T n (x) The following special values and properties of  T n (x)  are often useful:

T n ( x) = ( 1)n T n (x)

−

−

T n (1) = 1

   

T 2n (0) = ( 1)n

−

T 2n+1 (0) = 0

T n ( 1) = ( 1)n

−

−

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Orthogonality Property of   T n (x) We can determine the orthogonality properties for the Chebyshev polynomials of the first kind from our knowledge of the orthogonality of the cosine functions, namely,



 0    cos(mθ) cos(n θ)   dθ   =  π/2

(m =  n )

 π

(m  =  n = 0)



0

 

π

(m  =  n  = 0)

Then substituting substituting

T n (x) = cos(nθ)

cos θ   =   x

to obtain the orthogonality properties of the Chebyshev polynomials:

 0   T  (x) T  (x) dx  = √ 1 − x  π/2 π  1

m



(m =  n )

n



(m  =  n = 0)

2

−1

 

(m  =  n  = 0)

We observe that the Chebyshev polynomials form an orthogonal set on the interval 1  with the weighting function   (1 x2 )−1/2

x

≤

−

−1 ≤

Orthogonal Series of Chebyshev Polynomials An arbitrary function   f (x)  which is continuous and single-valued, defined over the interval 1 x 1, can be expanded as a series of Chebyshev polynomials:

− ≤ ≤

f (x) =   A0 T 0 (x) +  A1 T 1 (x) +  A2 T 2 (x) +  . . . ∞

=

A T  (x) n

n

n=0

7

 

where the coefficients  A n  are given by

 1

 1 A0   = π

 

 2 An   = π

 

−1

f (x)  dx √  1−x

n  = 0

2

and

 1

f (x)  T n (x)  dx

−1

n  = 1, 2, 3, . . .

√ 1 − x

2

The following definite integrals are often useful in the series expansion of   f (x):  1

 

−1

√ 1 − x

2

 1

 

−1

−1

 

= π

−1

 1

x dx

√ 1 − x

= 0

x2 dx

 π = 2

2

 1

 

 1

dx

√ 1 − x

2

 

−1

 1

 

−1

x3 dx

√ 1 − x

2

x4 dx

√ 1 − x

2

x5 dx

√ 1 − x

2

= 0

=

  3π 8

= 0

Chebyshev Polynomials Over a Discrete Set of Points A continuous function over a continuous interval is often replaced by a set of discrete values of the functio function n at discrete discrete points. points. It can be shown shown that the Chebyshe Chebyshev v polynomi polynomials als   T n (x) are orthogonal over the following discrete set of  N   N   + 1  points   xi , equally spaced on  θ ,

θi   = 0,

  π   2π , , . . . (N  N  N 

π  − 1)  N   − ,

π

where

xi  = arccos  θ i We have 8

 

 0    1 1 T  (−1)T  (−1)+ T  (x )T  (x )+ T  (1)T  (1) =  N/2 2 2  N 



(m =  n )

N −1

m

n

m

i

n

i

m



(m  =  n = 0)

n

i=2

 

(m  =  n  = 0)

The  T m (x)  are also orthogonal over the following   N   points  t i  equally spaced,

θi   =

 (2 N  1)π   π   3π   5π , , , ..., 2N  2N  2N  2N 

 −  −

and

ti  = arccos  θ i

 0 T  (t )T  (t ) =  N/2  N 



(m =  n )

N 

m

i

n



(m  =  n = 0)

i

i=1

 

(m  =  n  = 0)

unequal spacing spacing of  The set of points  t i  are clearly the midpoints in   θ  of the first case. The unequal the points in   xi (N ti )  compensates for the weight factor

W (x) = (1

2

−x )

−1

/2

in the continuous case.

9

 

Additional Identities of Chebyshev Polynomials The Chebyshev polynomials are both orthogonal polynomials and the trigonometric  cos nx functions in disguise, therefore they satisfy a large number of useful relationships. The differentiation differentiation and integration integration properties are very important in analytical analytical and numerical numerical work. We begin with

−1

x]

−1

x]

T n+1 (x) = cos[(n + 1) cos cos and

T n

−1

(x) = cos[(n

− 1) cos

Differentiating both expressions gives

1 d[T n+1 (x)] = dx (n + 1)

−1

x

−1

x

 − sin[(n√   + 1) cos cos − 1−x 2

and

1

d[T n

−1

(x)]

 − sin[(n − 1) cos = −√ 1 − x 2

(n

− 1)

dx

Subtracting the last two expressions yields

1 d[T n+1 (x)] dx (n + 1)

− (n  −1 1) d[T  dx(x)] =   sin(n + 1)θsin−θsin(n − 1)θ n−1

or

T n+1 (x)

(x)   2cos nθ   sin θ − = = 2T  (x) (n + 1) (n − 1) sin θ 

T n 

−1

n

10

 

n

≥2

 

Therefore 

T 2 (x) = 4T 1 

T 1 (x) =   T 0 

T  (x) = 0 0

We have the formulas for the differentiation of Chebyshev polynomials, therefore these formulas can be used to develop integration for the Chebyshev polynomials:

      1 T  (x) T  (x) − (n − 1)  + C   T  (x)dx   = 2 (n + 1)    1   T  (x)dx   = T  (x) +  C  4   n−1

n+1

n

1

n

≥2

2

  T 0 (x)dx   =   T 1 (x) +  C 

The Shifted Chebyshev Polynomials For analytical and numerical work it is often convenient to use the half interval   0 1 x instead of the full interval 1 1. For this purpose the shifted Chebyshev Chebyshev polynomials x are defined:

≤ ≤

− ≤ ≤

∗

T n (x) =  T n

∗ (2x − 1)

Thus we have for the first few polynomials ∗

T 0   = 1 ∗

−1 8x − 8x + 1 32x − 48x + 18 x − 1 128x − 256x + 160x − 32x + 1

T 1   = 2x ∗

T 2   = ∗

T 3   = ∗

T 4   =

2

2

3

4

3

2

11

 

and the following powers of  x  x  as functions of  T   T n∗(x);

∗

1 =   T 0

 1 x   = 2 (T 0   + T 1 ) ∗

∗

x2 =

 1 (3T 0   + 4 T 1   + T 2 ) 8

x3 =

  1 (10T 0   + 15 T 1   + 6 T 2   + T 3 ) 32

x4 =

  1 (35T 0   + 56T 1   + 28T 2   + 8 T 3   + T 4 ) 128

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

The recurrence relationship for the shifted polynomials is:

∗

T n+1 (x) = (4x

∗

∗

− 2)T  (x) − T 

n−1 (x)

n

 

∗

T 0 (x) = 1

or

∗

xT n (x) =

 1  1  1 T n+1 (x) + T n (x) + T n 4 2 4 ∗

∗

∗ −1

(x)

where

∗



−1

T n (x) = cos n cos

(2x

− 1)  =  T  (2x − 1) n

Expansion of   xn in a Series of   T n (x) A method of expanding   xn in a series of Chebyshev polynomials employes the recurrence relation written as

xT n (x) =   1 [T n+1 (x) +  T n 2

−1

(x)]

 

xT 0 (x) =   T 1 (x) 12

n  = 1, 2, 3 . . .

 

To illustrate the method, consider  x 4

4

x

  x2  x =   x (xT 1 ) = [T 2  +  T 0 ] = [T 1  +  T 3  + 2T 1 ] 2 4 2

=

 1

[3xT 1  +  xT 3 ] =

 1

[3T 0  + 3T 2  +  T 4  +  T 2 ]

4 8  1  1  3 = T 4  + T 2  + T 0 8 2 8

This result is consistent with the expansion of  x 4 given in Table 2.

Approximation of Functions by Chebyshev Polynomials Sometimes when a function  f (x)  is to be approximated by a polynomial of the form

∞

 a x f (x) = n

n

 

+ E N  N (x)

n=0

|x| ≤ 1

where E n (x)   does not exceed an allowed limit, it is possible to reduce the degree of the polynomi polyn omial al by a process process called called economiza economization tion of power power seri series. es. The procedure procedure is to con conver vertt the polynomial to a linear combination of Chebyshev polynomials:

|

 |

N 

N 

a x = b T  (x) n

n=0

n

n

n

 

n  = 0, 1, 2, . . .

n=0

It may be possible to drop some of the last terms without permitting the error to exceed   1, the number of terms which can be omitted is the prescri prescribed bed limit. limit. Since Since T n (x) determined by the magnitude of the coefficient  b .

 |

|≤

The Chebyshev polynomials are useful in numerical work for the interval because

 |

1. T n (x)]

≤ 1  within −1 ≤ x ≤ 1

2. The maxima and minima are of comparable comparable value. value. 13

 −1   ≤   x   ≤   1

 

3. The maxima and minima are spread reasonably uniformly uniformly over the interval interval

−1 ≤ x ≤ 1 4. All Chebyshev Chebyshev polynomials satisfy satisfy a three term recurrence relation. 5. The They y are easy to compute compute and to convert convert to and from a power series series form.

These properties together produce an approximating polynomial which minimizes error in its applicati application. on. This This is different different from the least least squares squares approxim approximatio ation n where where the sum of  the squares squares of the errors errors is mini minimi mized zed;; the the maxim maximum um error itself itself can be quite quite la large rge.. In the Cheby Chebyshev shev approxima approximatio tion, n, the average average error error can be large large but the maximum maximum error error is minimi min imized zed.. Chebyshe Chebyshev v appro approxima ximatio tions ns of a fun functio ction n are sometim sometimes es said said to be mini-m mini-max ax approximations of the function. The following table gives the Chebyshev polynomial approximation of several power series.

14

 

Table 3: Po Powe werr Seri Series es and its Cheb Chebyshev yshev Appro Approximat ximation ion 1.   f (x) =  a 0

f (x) =  a 0 T 0

2.   f (x) =  a 0  +  a1 x

f (x) =  a 0 T 0  +  a1 T 1

3.   f (x) =  a 0  +  a1 x +  a2 x2

f (x) =

a0  +

  a2

a2

T 0  +  a1 T 1  +

 2  T 

2

 2



4.   f (x) =  a 0  +  a1 x +  a2 x2 + a3 x3

f (x) =



  a2 a0  + 2



T 0  +



  3a3 a1  + 4



T 1  +

a  2

2

T 2  +

a 

a2

  a4

3

4

T 3

5.   f (x) =  a 0  +  a1 x +  a2 x2 + a3 x3 + a4 x4

f (x) =

a0  +

  a2

2



+

  a3

T 0  +

 8

a1  +

  3a3

T 1  +

 4



+

T 2  +

 2  2  a 4

+

8

T 4

a3

 8  T 

3

6.   f (x) =  a 0  +  a1 x +  a2 x2 + a3 x3 + a4 x4 + a5 x5

f (x)

=



  a2   3a4 + a0  + 2 8

+



T 0   +



  3a3   5a5 + a1  + 4 8

  5a5 + 4 16

a

3

15



T 3  +

a  4

8



T 1   +

T 4  +

a

a  5

16

  a4 + 2 2 2

T 5



T 2

 

Table 4: Formulas for Economization of Power Series x   =   T 1 x2 =

 1 (1 +  T 2 ) 2

x3 =

 1 (3x +  T 3 ) 4

x4 =

 1 (8x2 8

x5 =

  1 (20x3 16

− 5x + T  )

x6 =

  1 (48x4 32

− 18x

x7 = x8

  1

− 1 +  T  ) 4

5

2

+ 1 + T 6 )

56x3 + 7 x +  T 7 )

(112x5

− − 160x

64   1 = (256x6 128

4

x9 =

  1 (576x7 256

x10 =

  1 (1280x8 512

x11 =

  1 (2816x9 1024

5

− 432x

+ 32 x2

− 1 +  T  )

+ 120x3 6

8

− 9x + T  )

+ 400x4

− 1120x

7

− 2816x

9

2

− 50x

+ 1232x5

+ 1 + T 10 10 ) 3

− 220x

+ 11 x +  T 11 11 )

For easy reference the formulas for economization of power series in terms of Chebyshev are given in Table 4.

16

 

Assigned Problems Problem Set for Chebyshev Polynomials 1. Obt Obtain ain the first three Cheby Chebyshev shev polynomial polynomialss   T 0 (x), T 1 (x)   and   T 2 (x)  by means of  the Rodrigue’s formula.

2. Show Show that the Chebyshev polynomial polynomial   T 3 (x)  is a solution of Chebyshev’s equation of  order  3 .

3. By means of the recurrence formula formula obtain Chebyshev Chebyshev polynom p olynomials ials   T 2 (x)   and   T 3 (x) given  T 0 (x)   and  T 1 (x).

4. Sho Show w that T n (1) = 1   and  T n ( 1) = ( 1)n

−

− 5. Sho Show w that T  (0) = 0   if   n  is odd and   (−1)

n/2

n

if  n  n  is even.

6. Sett Setting ing x  = cos θ  show that

 1 T n (x) = 2 where  i  =

n

n

x + i 1 − x  + x − i 1 − x   2

2

√  1. −

7. Find the general solution solution of Chebyshev’s Chebyshev’s equation for  n  = 0.

8. Obt Obtain ain a series series expansion expansion for  f (x) =  x 2 in terms of Chebyshev polynomials  T n (x),

3

x2

A T  (x) = n

n

n=0

9. Exp Express ress  x4 as a sum of Chebyshev polynomials of the first kind.

17