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A math homework assignment for math 435, where students are required to find least-squares approximations and interpolating polynomials using legendre and chebyshev polynomials. The problems involve finding the coefficients for the approximations, locating the zeros and extreme points of chebyshev polynomials, and constructing interpolating polynomials. The document also includes problems on finding the expressions for the coefficients of chebyshev least-squares approximations and minimizing the error between polynomials.
Typology: Assignments
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Due Friday, April 10, 2009
(b) Find the quadratic least-squares approximation for f (x) = sin π(x) on [0, 1] using Chebyshev polynomials.
xj = cos
2 j − 1 2 k
π
, j = 1, 2 ,... , k
and Tk(x) has extreme points at zj = cos
jπ k
, j = 0, 1 ,... , k.
(b) What are the optimal nodes for interpolating with a polynomial of degree 2 on [2, 5]?
(c) Using the zeros of T˜ 3 (x), construct an interpolating polynomial of degree 2 for f (x) = ex^ on [2, 5].
max 1 ≤x≤ 1 |p 4 (x) − p 3 (x)|
will be minimum. What is the total error (sum of the truncation error + error by Cheysher approxima- tion)?
