Short answers and true/false [25 pts]

Answer the following questions. Include a brief justification/reasoning for your answer for each question (you will not get points without a valid explanation).

How many iterations does Newton’s method require to solve a linear equation?

The 2-norm condition number of an n×n matrix with singular values σ1 ≥ σ2 ··· ≥ σn is …

Using the initial guess x0 = [0,1,0]T , the power method applied to the matrix

will converge to what eigenvalue if exact arithmetic (no roundoff error) is used?

True or False: For the nodes x0 = −1, x1 = 0, x2 = 1, the Lagrange interpolation polynomial L0(x) corresponding to node

True or false: The n + 1 point Lagrange polynomial

interpolates the function exp(x).

The Legendre polynomial of degree n is orthogonal to every polynomial with degree (n − 1) or less.

True or False: The three point (i.e., non-composite) Simpsons rule and the three point Gauss(-Legendre) quadrature formulas are equally accurate approximations of

Let, and let In be the result of the composite Simpson’s rule approximation to I with n + 1 quadrature points, and denote en = |I − In|.

For n large, what does e2n/en converge to?

[12 pts] Newton’s method computes the new iterate xk+1 as the x-intercept of the “line of best fit” through the point (xk, f(xk)), i.e., the line that passes through (xk, f(xk)) and whose first derivative is f ‘(xk). We will define a new method which finds the “quadratic of best fit” and uses it to compute the new iterate.

Find the quadratic of best fit through the point (xk, f(xk)), i.e., find the quadratic that goes through (xk, f(xk)) and whose first and second derivatives at xk agree with f ‘(xk) and f ‘’(xk), respectively.

Write down a “quadratic Newton” method for finding roots by computing the x-intercept for the quadratic of best fit. To make the answer unique, use the root that is closes to xk.

[12 pts] In this problem you will derive the most accurate quadrature rule possible that uses some values of the derivative of the function in addition to the values of the function.

For an arbitrary/generic smooth function f(x), find the best values for the weights w−1, w0, and w1 in the quadrature rule:

Verify in some way that the weights you obtained are correct.

Use this rule to estimate and compare to the answer from

(the non-composite) Simpson’s rule. Which rule is more accurate for this specific problem?

[12 pts] Consider the inner product

= (1)

Calculate orthogonal polynomials φ0, φ1, and φ2 in this inner product.

Calculate the best quadratic approximation to f(x) = x3 in the norm induced by this inner-product.

Consider the function). Prove or disprove that

(2)

is the best quadratic approximation to f in the inner-product defined above.

[12 pts] In this problem, we look at the midpoint rule for approximation to a definite integral

We take h = (b − a)/n and xi = a + ih.

Sketch a graph to show what this approximation is finding.

Determine if the formula is exact if f is constant on each sub interval.

Determine if the formula is exact if f is linear on each sub interval. (d) Expand f ∈ C2[a,b] in a Taylor series about the midpoint

xi−1/2 = (xi + xi−1)/2

of the sub interval. What is the order of the error in each sub interval? What can you say about the total error of the rule?

[9 pts] Consider the Matlab code:

A=[7 −sqrt (3); −sqrt (3) 5];

x=[0 ,1] ’;

n=100; % Large integer

for k=1:n

x=A∗x ; x=x/norm(x ,2);

end x

What does the output of the Matlab code converge to as n becomes larger and larger? Call this limit x0.

Approximately how big does n need to be for ||x − x0||2 to be smaller than 1/210 ≈ 10−3? You do not need to give an exact number, in numerical analysis we only need an error estimate.

If we print x’∗A∗x

at the end of the program, what would we get as output for very large n and why?

(a) [9 pts] Find a Householder transformation to transform the matrix A, given below, into a tridiagonal matrix T. Write down the Householder matrix H as well as T.

Why is this tridiagonalization useful for finding the eigenvalues of A?

If we apply the power method to T, give bounds for the eigenvalue that λ(k) will converge to. (Do not actually compute this eigenvalue).

[9 pts] If

and (0) = (1, −1, 2)t. Answer the following questions. If you use MATLAB/Python, include your code and output.

Find the first three iterations obtained by the Power method.

Find the first three iterations using the Inverse Power method.

Find the first three iterations obtained by the symmetric power method.

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