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Linear Algebra and Numerical Analysis

Questions on a Sequence of Polynomials. See attached file for full problem description.

Let be the sequence of polynomials defined by
, ,
1) Show that is a polynomial of degree k. Calculate the coefficient of of .
2) Show by induction that for all real .
3) Deduce that if , .
4) Show that for all whole natural numbers n, we have

where , . Give a numerical approximation of these numbers for n = 4, to the precision of your calculator.
5) Let us consider the function f defined as . Evaluate the Lagrange interpolation polynomial P of f at the points that we calculated in the previous question.
6) Estimate the error for .

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Please see attached file.

(1)

For k = 1,
For k = 2,
For k = 3,
...........
By observation, the subscript of T on the LHS is the maximum power of x on the RHS. Hence T is a polynomial of degree k.
Again by observation, the coefficient of x in T is .

(2) Let , for all real .
For k=0, P0: T0 (cos ) = 1 = cos(0* ), which is true.
For k=1, P1: T1 (cos ) = cos = cos(1* ), which is also true.

Assume that Pm-1: Tm-1(cos ) = cos((m-1) ) is true, and .................(1)
Pm: Tm(cos ) = cos(m ) is ...

Solution Summary

The expert examines linear algebra and numerical analysis polynomials.

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