This project computes the two largest eigenvalues of a 50x50 matrix. You will use the usual Power Method to compute the largest eigenvalue. For the next largest eigenvalue, you can use an “Annihilation or Deflation or Shifting technique” discussed in class and also in our book. The matrix A is tridiagonal. Its main diagonal has ones on it. The super diagonal (the diagonal above the main diagonal) has negative-ones on it. The sub-diagonal (below the main diagonal) has negative-ones also. Our starting vector xo has all ones. Our tolerance is 0.01.Turn in the following on one page1) Draw Gershgorin Circles that contain the eigenvalues of A. Use “insert” from the menu to locate and draw circles. 2) Based on part (1), what is the spectral radius of A. Print your answer here: _______________3) With a starting vector xo= [1 1 1 1….1]T, apply the usual power method to estimate lmax , the dominant- eigenvalue of matrix A. Use a tolerance of 0.01. Print your answer with 4 decimals: ____________4) Print the number of iterations required to converge. ____________5) Use the deflation technique discussed in class to compute the second largest eigenvalue. Tolerance = 0.01 Print the second largest eigenvalue with 4 decimals: _________________6) Print the number of iterations required to converge. ___________7) Print your computer program here. As discussed in class, the main body of your program is 3 lines, so your program should not be long.

Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox, which at that time was a major threat to public health.Question 1. Who was Daniel Bernoulli?Why was he working on this project?Bernoulli's model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity.Consider the cohort of individuals born in a given year𝑡=0t=0, and let𝑛(𝑡)n(t) be the number of these individuals surviving 𝑡t years later.Let 𝑥(𝑡)x(t)be the number of members of this cohort who have not had smallpox by year 𝑡t and who are therefore still susceptible. Let 𝛽β be the rate at which susceptibles contract smallpox, and let 𝜈ν be the rate at which people who contract smallpox die from the disease. Finally, let 𝜇(𝑡)μ(t) be the death rate from all causes other than smallpox. Then 𝑑𝑥𝑑𝑡dxdt, the rate at which the number of susceptibles declines, is given by𝑑𝑥𝑑𝑡=−[𝛽+𝜇(𝑡)]𝑥dxdt=−[β+μ(t)]x (1)The first term on the right of Eq.1 is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. Also,𝑑𝑛𝑑𝑡=−𝜈𝛽𝑥−𝜇(𝑡)𝑛dndt=−νβx−μ(t)n, (2)where 𝑑𝑛𝑑𝑡dndt is the death rate of the entire cohort, and the two terms on the right side are the death rates due to smallpox and to all other causes, respectively.Introduce a new variable 𝑧(𝑡)=𝑥(𝑡)𝑛(𝑡).z(t)=x(t)n(t).Question 2.Formulatetheinitial value problem for 𝑧(𝑡)z(t).Question 3. Find the complete general solution to differential equation in Q.2 equation.Question 4. Find the solution to the initial value problem.Question 5. Draw the slope field( by hand, or with DField oruse MATLAB, for example, ode45) for this equation and draw some representative solutions. What does this picture tell youabout the solution to this initial value problem.Question 6.Does the solutionin Q.5 agree with your analysis in Q. 4?Question 5. Bernoulli estimated that 𝜈=𝛽=1/8ν=β=1/8.With that in mind, determine the proportion of 20-year-olds who have not had smallpox.On the basis of the model just described and the best mortality data available at the time, Bernoulli calculated that if deaths due to smallpox could be eliminated, 𝜈=0ν=0, then the life expectancy among inoculated will increase by almost 10%.Question 6. Check the internet for the information about what was the life expectancy in 1760 in the country where Bernoulli lived. How would the inoculation program change it?As a result of this study,Bernoullisupported the inoculation program.