Please Answer ALL Parts, and thoroughly as this is quite difficult and all the o
ID: 3283208 • Letter: P
Question
Please Answer ALL Parts, and thoroughly as this is quite difficult and all the other answers i've seen don't answer the questions fully or correctly.
(c) Use the zeros of the monic Chebyshev polynomial T?4(x) to construct an interpolating polynomial of degre 3 for f(x) = xlnx on [1, 3]. Find the bound of the maximum error of the interpolating polynomial on the entire interval [1,3].
(d) Use 4 evenly spaced nodes {xi} on [1,3] with x0 = 1 and x3 = 3 to find a polynomial interpolation of degree 3 for f(x) = xlnx on [1,3]. Find also the bound of the maximum error for the interpolating polynomial on the entire interval [1,3].
(e) Compare the two error bounds on (c) and (d). Is the conclusion consistent with the conclusion on Section 8.4 about the minimizing property of monic Chebyshev polynomials.
Explanation / Answer
given f(x)=x?lnx
obtain the first derivative f'(x) then equate to zero.
f'(x)=x?(1/x)+lnx?1=0
1+lnx=0
lnx=?1
e?1=x
x=1/e
Solving for f(x) at x=1/e
f(x)=(1/e)?ln(1/e)
f(x)=(1/e)?(?1)
f(x)=?1/e
A polynomial P for which P(xi) = yi when 0 ? i ? n is said to interpolate the given set of data points. The points xi are called nodes or interpolating points.
Consider the function f (x) = -1/e on [0, 3] and the nodes. We have x0 = 0, x1 = 1, x2 = 2 and x3 = 3.
The Lagrange polynomial of order 3, connecting the four points, is given by
P3(x) = L0(x)f(x0) + L1(x)f(x1) + L2(x)f(x2) + L3(x)f(x3)
L0(x) = (x ? x1)(x ? x2)(x ? x3) /(x0 ? x1)(x0 ? x2)(x0 ? x3) ,
L1(x) = (x ? x0)(x ? x2)(x ? x3) / (x1 ? x0)(x1 ? x2)(x1 ? x3) ,
L2(x) = (x ? x0)(x ? x1)(x ? x3) / (x2 ? x0)(x2 ? x1)(x2 ? x3) ,
L3(x) = (x ? x0)(x ? x1)(x ? x2) / (x3 ? x0)(x3 ? x1)(x3 ? x2) .