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5. Orthongonalization and least squares [2+3+3pt]. (a) Given any two nonzero vectors x and y in R^n, construct a Householder matrix H, such that Hx is a scalar multiple of y. Is the matrix H unique? (b) Use Householder matrices to compute the QR-factorization of the matrix: (c) We believe that a real number Y is approximately determined by X with the model function Y = a exp(X)+bX² + cX + d . We are given the following table of data for the values of X and Y: Using the above data points, write down 7 equations in the four unknowns a, b, c, d. The least squares solution to this system is the best fit function. Write down the normal equations for this system, solve them in MATLAB. Plot the data points (X,Y) as points and the best fit function.

(3) Use the ɛ – N definition of convergence of a sequence to verify the limit, \lim _{n \rightarrow \infty} \frac{\sqrt{3}+\sqrt{2} n^{2}}{\sqrt{2} n+\sqrt{3} n^{2}}=\sqrt{\frac{2}{3}}

| In this problem, you will prove the rate of convergence for the secant method. x_{k+1}=x_{k}-\frac{x_{k}-x_{k-1}}{f\left(x_{k}\right)-f\left(x_{k-1}\right)} f\left(x_{k}\right) can be rewritten in the form: x_{k+1}=\frac{x_{k} f\left(x_{k-1}\right)-x_{k-1} f\left(x_{k}\right)}{f\left(x_{k-1}\right)-f\left(x_{k}\right)} \psi\left(x_{k}, x_{k-1}\right)=\frac{x_{k+1}-\xi}{\left(x_{k}-\xi\right)\left(x_{k-1}-\xi\right)} where xk+1 is as in (1). Compute (for fixed value of xk-1) \varphi\left(x_{k-1}\right)=\lim _{x_{k} \rightarrow \xi} \psi\left(x_{k}, x_{k-1}\right) (c) Now compute \lim _{x_{E-1} \rightarrow \xi} \varphi\left(x_{k-1}\right) and therefore show thatlim \lim _{x_{k}, x_{k-1} \rightarrow \xi} \psi\left(x_{k}, x_{k-1}\right)=\frac{f^{\prime \prime}(\xi)}{2 f^{\prime}(\xi)} (d) Next, assume that the secant method has convergence order q, that is to say that \lim _{k \rightarrow \infty} \frac{\left|x_{k+1}-\xi\right|}{\left|x_{k}-\xi\right|^{q}}=A<\infty Using the above results, show that q – 1– 1/q = 0, and therefore that q = (1+ v5)/2. (e) Finally, show that this implies that \lim _{k \rightarrow \infty} \frac{\left|x_{k+1}-\xi\right|}{\left|x_{k}-\xi\right|^{q}}=\left(\frac{f^{\prime \prime}(\xi)}{2 f^{\prime}(\xi)}\right)^{q /(1+q)}

(11) Which of the following statements imply that {an} converges to a? (a) For every integer m > 0, there is an integer N > 0 such that |an – a| < 1/m when n> N. (b) For each 0 < ɛ < 1, there is an integer N > 0 such that |an- a| < 3ɛ when n > N. (c) For each 0 < ɛ < 1, there is an integer N> 0 such that |an- a| < 1/ɛ when n > N. (d) For each N > 0, there is ɛ > 0 such that |an – a| < 1/N when n > N + ɛ . \text { (e) For each } \varepsilon>0 \text { , there is an integer } N>0 \text { such that }\left|a_{n}-\alpha\right|<\varepsilon^{2} \text { when } n>N^{2} \text { . } (f) For each ɛ > 0, there is an integer N > 0 such that an – a < ɛ when n = N + k for all positive integer k. (g) For each ɛ > 0, there is an integer N > 0 such that |an – a| < ɛ when n = N + 2k for all positive integer k.

= 1. Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables. a. x f(x) f'(x) 0.5 0.4794 0.6 0.5646 0.7 0.6442 0.0 0.2 0.4 f(x) f'(x) 0.00000 0.74140 1.3718

(10) Find the upper and lower limits of each sequence: \text { (a) }\left\{\frac{2-(-1)^{n} n}{3 n+2}\right\} \text { (b) }\left\{\frac{2 n-1}{n} \sin \frac{n \pi}{6}\right\} \text { . }

1. Calculate the stationary states of the weighted graph represented below, by means of the eigenvector equation. (Show your work in sufficient detail. Answers not backed up by calculations will not be credited.)

(1) Use the definition of convergence of sequences to verify the following limits: \text { (a) } \lim _{n \rightarrow \infty} \frac{(-1)^{n} n}{n^{2}+1}=0 \text { (b) } \lim _{n \rightarrow \infty} n\left(\sqrt{1+\frac{1}{n}}-1\right)=\frac{1}{2} \text { . }

a. f(x) = 2 cos 2x -x b. f(x)=x² Inx+1 5. Use the most accurate three-point formula to determine each missing entry in the following tables. a. x f(x) f'(x) b. x f(x) f'(x) 1.1 9.025013 1.2 11.02318 1.3 13.46374 1.4 16.44465 8.1 8.3 8.5 8.7 16.94410 17.56492 18.19056 18.82091

4. Sharpness of condition number estimates [4pt] Let A E R^n x n be invertible. Let b E R^n\{0},and Ax = b, Ax' = b' and denote the perturbations by Ab = b' – b and Ax = x' – x . Show that the inequality obtained in Theorem 2.11 is sharp. That is, find vectors b, Ab for which \frac{\|\Delta x\|_{2}}{\|x\|_{2}}=\kappa_{2}(A) \frac{\|\Delta b\|_{2}}{\|b\|_{2}} where k2(A) is the condition number of A under the 2-norm. (Hint: consider the eigenvectors of A^T A.)