Example 3.   Form several Chebyshev polynomials of degree  n = 1,2, 3, 4, and 5  for the function  [Graphics:Images/ChebyshevPolyMod_gr_315.gif]  over the interval  [Graphics:Images/ChebyshevPolyMod_gr_316.gif]  using Chebyshev's abscissas.

Solution 3.

3 (a).  Construct the Chebyshev interpolation polynomial  [Graphics:../Images/ChebyshevPolyMod_gr_317.gif],  of degree n = 1.

[Graphics:../Images/ChebyshevPolyMod_gr_318.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_319.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_320.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_321.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_322.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_323.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_324.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_325.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_326.gif]


3 (b).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_327.gif],  of degree n = 2.

[Graphics:../Images/ChebyshevPolyMod_gr_328.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_329.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_330.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_331.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_332.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_333.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_334.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_335.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_336.gif]


3 (c).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_337.gif], of degree n = 3.

[Graphics:../Images/ChebyshevPolyMod_gr_338.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_339.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_340.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_341.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_342.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_343.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_344.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_345.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_346.gif]


3 (d).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_347.gif], of degree n = 4.

[Graphics:../Images/ChebyshevPolyMod_gr_348.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_349.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_350.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_351.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_352.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_353.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_354.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_355.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_356.gif]


3 (e).  Construct the Chebyshev interpolation polynomial [Graphics:../Images/ChebyshevPolyMod_gr_357.gif], of degree n = 5.

[Graphics:../Images/ChebyshevPolyMod_gr_358.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_359.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_360.gif]

Now graph the function and polynomial, and interpolation nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_361.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_362.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_363.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_364.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_365.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_366.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004