2. Create a 128x128 matrix using the following commands: >> n = 128; T = 5;...

 

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2. Create a 128x128 matrix using the following commands: >> n = 128; T = 5; t linspace (0, T, n)'; 0.4; = >> gamma = >> hh = (T/n) *1/ (sqrt (2*pi*gamma^2) ) *exp(-1/(2*gamma^2) *t.^2); >> A = toeplitz (hh, hh); a. What is the condition number of this matrix? b. Let x be a square wave. Calculate b = Ax, which is the noiseless data. c. Using the 'randn' command, create a noisy vector bnoisy with zero-mean Gaussian noise with standard deviation 0.0001. Plot b and noisy on the same plot. Does this look significantly noisy? d. Use 'pinv' to solve for Ax = bnoisy. What is the norm of the error? e. Regularize the problem by adding a diagonal matrix ('diag) to A and solve (A + )x = bnoisy. Choose at least three values of and calculate the condition number of the new matrix, and the new least squares solutions. Try and get as close as possible to the true solution. Comment on what you observe. Plot all of the solutions on the same plot, compared to the true solution. Use a legend to label the plots. 2. Create a 128x128 matrix using the following commands: >> n = 128; T = 5; t linspace (0, T, n)'; 0.4; = >> gamma = >> hh = (T/n) *1/ (sqrt (2*pi*gamma^2) ) *exp(-1/(2*gamma^2) *t.^2); >> A = toeplitz (hh, hh); a. What is the condition number of this matrix? b. Let x be a square wave. Calculate b = Ax, which is the noiseless data. c. Using the 'randn' command, create a noisy vector bnoisy with zero-mean Gaussian noise with standard deviation 0.0001. Plot b and noisy on the same plot. Does this look significantly noisy? d. Use 'pinv' to solve for Ax = bnoisy. What is the norm of the error? e. Regularize the problem by adding a diagonal matrix ('diag) to A and solve (A + )x = bnoisy. Choose at least three values of and calculate the condition number of the new matrix, and the new least squares solutions. Try and get as close as possible to the true solution. Comment on what you observe. Plot all of the solutions on the same plot, compared to the true solution. Use a legend to label the plots.