Example 1.   Form several Chebyshev polynomials of degree  n = 1,2, 3, 4, and 5  for the function  [Graphics:Images/ChebyshevPolyMod_gr_203.gif]  over the interval  [Graphics:Images/ChebyshevPolyMod_gr_204.gif]  using Chebyshev's nodes.

Solution 1.

1 (a).  Construct the Chebyshev interpolation polynomial  [Graphics:../Images/ChebyshevPolyMod_gr_205.gif],  of degree n = 1.

[Graphics:../Images/ChebyshevPolyMod_gr_206.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_207.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_208.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_209.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_210.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_211.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_212.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_213.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_214.gif]


1 (b).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_215.gif],  of degree n = 2.

[Graphics:../Images/ChebyshevPolyMod_gr_216.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_217.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_218.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_219.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_220.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_221.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_222.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_223.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_224.gif]


1 (c).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_225.gif], of degree n = 3.

[Graphics:../Images/ChebyshevPolyMod_gr_226.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_227.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_228.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_229.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_230.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_231.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_232.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_233.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_234.gif]


1 (d).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_235.gif], of degree n = 4.

[Graphics:../Images/ChebyshevPolyMod_gr_236.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_237.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_238.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_239.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_240.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_241.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_242.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_243.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_244.gif]


1 (e).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_245.gif], of degree n = 5.

[Graphics:../Images/ChebyshevPolyMod_gr_246.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_247.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_248.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_249.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_250.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_251.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_252.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_253.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_254.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004