Matlab Exercises The condition number of a matrix A, defined by cond A = ||A||2||A accuracy of a computed solution of a system Ax = b. If the entries of A and b are accurate to about ||2, allows one to estimate the r significant digits, and cond Ack, then the computed solution of Ax = b should usually be accurate to at least rk significant digits. Solve the following two problems: Compute the Hilbert matrix H of order k=5 and k = 12 using the hilb command. - For k = 5, solve the system Hx = b for a suitable b to find the last column of the inverse of H. Use the backslash command. To find a suitable b think of the solution as x = H-lb. - For k = 5 and k = 12, solve the system Hx = y using the backslash command, where y is generated by applying H to a random vector x*, i.e., y = Hx*. Compute the 2-norm of the difference between x* and x, i.e., d = ||x* - *||2 and print d to the screen. Print also the condition number (command: cond (H)) of H to the screen. Describe what you observe. . Find the determinant (command: det (A)) and the condition number (command: cond (A)) of the Hilbert matrix H of order k (command: hilb(k), for k=1,2,..., 10. Plot the determinant and the condition number as a function of k using a logarithmic scale for the vertical axis. Matlab Exercises The condition number of a matrix A, defined by cond A = ||A||2||A accuracy of a computed solution of a system Ax = b. If the entries of A and b are accurate to about ||2, allows one to estimate the r significant digits, and cond Ack, then the computed solution of Ax = b should usually be accurate to at least rk significant digits. Solve the following two problems: Compute the Hilbert matrix H of order k=5 and k = 12 using the hilb command. - For k = 5, solve the system Hx = b for a suitable b to find the last column of the inverse of H. Use the backslash command. To find a suitable b think of the solution as x = H-lb. - For k = 5 and k = 12, solve the system Hx = y using the backslash command, where y is generated by applying H to a random vector x*, i.e., y = Hx*. Compute the 2-norm of the difference between x* and x, i.e., d = ||x* - *||2 and print d to the screen. Print also the condition number (command: cond (H)) of H to the screen. Describe what you observe. . Find the determinant (command: det (A)) and the condition number (command: cond (A)) of the Hilbert matrix H of order k (command: hilb(k), for k=1,2,..., 10. Plot the determinant and the condition number as a function of k using a logarithmic scale for the vertical axis
