You are required to design a sequence detector circuit which detects all non-overlapped instances of the input pattern "10110" in a string of bits coming through an input line X

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 { . }

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.)

1. Matrix condition numbers, [2+1+2pt+2pt (extra credit)] Let us explore matrix norms and condition numbers. (a) For the following matrix given by calculate ||A||1, ||A||2, ||A||0, as well as the condition numbers for each norm by hand. Is A well or poorly conditioned? (b) Recall the formulas from Theorems 2.7 and 2.8 in the text book. If you assume that taking the absolute value and determining the maximum does not contribute to the overall computational cost, how many flops (floating point operations) are needed to calculate ||A||1 and \|A\|_{\infty} \text { for } A \in \mathbf{R}^{n \times n} ? By what factor will the calculation time increase when you double the matrix size? (c) Now implement a simple code that calculates || A||1 and ||A||. for a matrix of any size n > 1. Try to do this without using loops?! Using system sizes of n1 =2nk, k = 1,... , 7, determine how long your code takes to calculate || A||1 and || A||. for a matrix A E R"i×ni with random entries and report the results. Can you confirm the estimate from (b)?100, nk+1 = 4(d) (extra credit) MATLAB has the build-in function norm to calculate matrix norms. Calculate for the system sizes in (c) ||A||1 and ||A||oo. using both your implementation and MATLAB's norm function, determine for each n; how long each code takes and plot the results in one graph. On average, by what factor is MATLAB's implementation faster than yours? Please also hand in your code.