Part I: Show that (y − y ∗ 0 )(y − y ∗ 1 ). . .(y − y ∗ n ) = 5 n+1 2 n Tn+1(x), where x = y/5
Part II: It can be shown that there exists R > 0 such that |f (n) (y)| ≤ Rn for all y ∈ [−5, 5]. Assuming this, show that limn→∞ max{|f(y) − Pn(y)|, y ∈ [−5, 5]} = 0
Solution:-
Given that
The interval [-1, 1] as the zeros of
on the interval t = 5, 5
degree m that interpolates f at these polats. Then the error estimate for polynomial interpolation gives for
for some
........(A)
from (1):-
For m = 0
m = 0, j = 0
m = 1:-
From (1)
j = 1, m = 1
j = 0, m = 1
For m = 2:-
From (1)
j = 0, m = 2
j = 1, m = 2
j = 2, m = 2
So, proceeding thus we have
.......(2)
Part I:-
So, from (A) we have
where
Part II:-
It can be shown that there exist R > 0
such that
and we know,
error estimate
[by part (1) and ]
[]
= 0
as is increases very fast as n grows large
Hence
Thanks for supporting...
Please give positive rating...
class: numerical analysis I wish if it was written in block letter Sorry I can't read cursive = COS Problem 1: Recall that the Chebyshev nodes x4, x1,...,xy are determined on the interval (-1,1] as the zeros of Tn+1(x) = cos((n + 1) arccos(x)) and are given by 2j +10 Xj j = 0,1, ... 1 n+1 2 Consider now interpolating the function f(x) = 1/(1 + x2) on the interval (-5,5). We have seen in lecture that if equispaced...
Problem 1: Recall that the Chebyshev nodes 20, 21, ...,.are determined on the interval (-1,1) as the zeros of Tn+1(x) cos((n + 1) arccos(x)) and are given by 2; +17 Tj = COS , j = 0,1,...n. n+1 2 Consider now interpolating the function f(x) = 1/(1 + x2) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...
QUESTION: Show= (y − y0* )(y − y1*) . .(y − yn* ) = 5 it is Part 1 at the bottom We were unable to transcribe this image(7+17) Problem 1: Recall that the Chebyshev nodes x7, x1,...,x* are determined on the interval (-1,1] [-1, 1) as the zeros of Tn+1(x) = cos((n + 1) arccos(x)) and are given by 2j +12 X; - cos j = 0,1, ... n. n+1 2 Consider now interpolating the function f(x) = 1/(1+x2)...
numerical methods 2+17), j = 0,1...... Problem 1: Recall that the Chebyshev nodes x0, 71,..., are determined on the interval (-1,1) as the zeros of Tn+1(x) = cos((n +1) arccos(x)) and are given by 2j +17 X; = cos in +12 Consider now interpolating the function f(x) = 1/(1+22) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...
Please answer problem 4, thank you. 2. The polynomial p of degree n that interpolates a given function f at n+1 prescribed nodes is uniquely defined. Hence, there is a mapping f -> p. Denote this mapping by L and show that rl Show that L is linear; that is, 3. Prove that the algorithm for computing the coefficients ci in the Newton form of the interpolating polynomial involves n long operations (multiplications and divisions 4. Refer to Problem 2,...
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...
Please help me solve this. #4 a&b. thanks We were unable to transcribe this imageeilymai retine, or you m yur ova rodinT ur pl nda listing of your code. (b) Repeat part (a) with a polynomial of degree 12 that interpolates f at 13 scaled Chebyshev points, x 5 cos 12 Again you may use MATLAB's polyfit and polyval routines to fit the polyno- mial to the function at these points and evaluate the result at points throughout the interval,...
1. (Taylor Polynomial for cos(ax)) For f(x)cos(ar) do the following. (a) Find the Taylor polynomials T(x) about 0 for f(x) for n 1,2,3,4,5 (b) Based on the pattern in part (a), if n is an even number what is the relation between Tn (x) and TR+1()? (c) You might want to approximate cos(az) for all in 0 xS /2 by a Taylor polynomial about 0. Use the Taylor polynomial of order 3 to approximate f(0.25) when a -2, i.e. f(x)...
Week 7: Nonlinear equations 1. Let f(x) --9. The equation (x)0 has a solution in [0, 1] i) Find the interpolation polynomial that interpolates f at x,-0, x2 1 0.5 and x3-1. ii) Use this polynomial to find an approximation to the solution of the equation f(x)0 Week 7: Nonlinear equations 1. Let f(x) --9. The equation (x)0 has a solution in [0, 1] i) Find the interpolation polynomial that interpolates f at x,-0, x2 1 0.5 and x3-1. ii)...
(6) Show that F(x, y) = (x+y)i + (**)is conservative. (a) Then find such that S = F (potential function). (5) Use the results in part(a) to cakulae ( F. ds along C which the curve y = a* from (0,0) to (2,16). (2) Use Green's Theorem to evaluate 1. F. ds. F(1,y) =(yº+sin(26))i + (2xy2 + cos y)and C is the unit circle oriented counter clockwise (6) Evaluate the surface integral || 9. ds. F(x,y,z) = xi +++where S...