In each of the problems below compute the average case time complexity (or expected time) of the given code. State your answer using asymptotic notation. Show your work for this problem by stating what the different cases are, the probability of each case, and how long each case takes. Also show the calculation of the expected time. a) def foo (n): #randomly choose a number between 0 and n-1 in constant time k np.random.randint (n) if k np. sqrt (n): for i in range(n): print (i) print('Never mind...') else: b) def bogosearch (numbers, target): """search by randomly guessing. numbers is an array of n numbers""" n len (numbers) while True: #randomly choose a number between 0 and n-1 in constant time guess = np.random.randint (n) if numbers [guess] == target: return guess In this part, you may assume that the numbers are distinct and that the target is in the array. Hint: if 0 < b < 1, then Σ1p.bp-1 = (1-b)².

7. Prove or disprove: f(n) + g(n) = 0 (min(f(n), g(n)))

You solve a non-singular system of 10,000 linear equations with 10,000 unknowns using the Gauss-Jordan algorithm without pivoting with single precision numbers and arithmetics on a computer that natively can do single precision operations very fast, but can operate in double and half precision as well. Your solution has a residual infinity-norm that is unacceptably large. In which order should you apply the following strategies to lower the residual norm? If a strategy is not / no longer helpful, do not list it as an option. a) use half precision numbers and arithmetics instead of single precision; b) use double precision numbers and arithmetics instead of single precision; c) use partial pivoting; d) use pivoting when encountering a zero in the pivot position.

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