QUESTION: Show= (y − y0* )(y −
y1*) . .(y − yn* ) = 5
it is Part 1 at the bottom
Homework Answers
Problem 1: Recall that the Chebyshev nodes 20, 21, ...,.are determined on the interval (-1,1) as...
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...
Part I: Show that (y − y ∗ 0 )(y − y ∗ 1 ). ....
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
Ij = COS Problem 1: Recall that the Chebyshev...
class: numerical analysis I wish if it was written in block letter Sorry I can't read...
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...
numerical methods 2+17), j = 0,1...... Problem 1: Recall that the Chebyshev nodes x0, 71,..., are...
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...
2. The polynomial p of degree n that interpolates a given function f at n+1 prescribed nodes is u...
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,...
4. The fixed point iteration X (5) converges in some interval [a.b]. Find reasonable values for...
4. The fixed point iteration X (5) converges in some interval [a.b]. Find reasonable values for a and b. 5. Exact numbers x and y are given by x = x*+el and y = y*+e2. Prove that the relative error in the quotient x/y is almost equal to the sum of relative errors in x and y 6. Given f(x) xe, find the maximum possible error in interpolating f(x) by a third degree polynomial over 113]. if Chebyshev points are...
Suppose that Y1 , Y2 ,..., Yn denote a random sample of size n from a...
Suppose
that Y1 , Y2 ,..., Yn denote a random sample of size n from a
normal population with mean μ and variance 2 .
Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
Problem 5 Let Y1 denote the minimum of a random sample of size n from a...
Problem 5 Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) e(,0x< o0, zero elsewhere X- n (Y1 0), find the cumulative distribution function (cdf) for Zn = n (Y1 - 0), and Let Zn find the limiting cdf of Zn as n >oo.
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calcula...
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calculate the ML estimator of θ. 6. Consider the pdf g(y|α) = c(1 + αy2 ), −1 < y < 1. (a) Show that g(y|α) is a pdf when c = 3 6 + 2α . (b) Calculate E(Y ) and E(Y 2 ). Referencing your calculations, explain why M1 can’t be...
Question 3 [25] , Yn denote a random sample of size n from a Let Y,...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
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...
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
Ij = COS Problem 1: Recall that the Chebyshev...
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...
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,...
4. The fixed point iteration X (5) converges in some interval [a.b]. Find reasonable values for a and b. 5. Exact numbers x and y are given by x = x*+el and y = y*+e2. Prove that the relative error in the quotient x/y is almost equal to the sum of relative errors in x and y 6. Given f(x) xe, find the maximum possible error in interpolating f(x) by a third degree polynomial over 113]. if Chebyshev points are...
Suppose
that Y1 , Y2 ,..., Yn denote a random sample of size n from a
normal population with mean μ and variance 2 .
Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
Problem 5 Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) e(,0x< o0, zero elsewhere X- n (Y1 0), find the cumulative distribution function (cdf) for Zn = n (Y1 - 0), and Let Zn find the limiting cdf of Zn as n >oo.
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
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