Oberon, Tania and Tina are carrying out some computations involving matrices. Tina reminds everyone that the computation

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Oberon Tania And Tina Are Carrying Out Some Computations Involving Matrices Tina Reminds Everyone That The Computation 1 (135.64 KiB) Viewed 39 times

Oberon Tania And Tina Are Carrying Out Some Computations Involving Matrices Tina Reminds Everyone That The Computation 2 (142.9 KiB) Viewed 39 times

Oberon, Tania and Tina are carrying out some computations involving matrices. Tina reminds everyone that the computational complexities of some common operations with n-by-n matrices are given in the table below. Operations Matrix-matrix multiplication Matrix-vector multiplication LU factorization Cholesky factorization Back/forward substitution Tridiagonal solve Flops 2n3 2n2 2n3/3 + O(na) 2n3/3 mg /3+2(m2) = mg /3 n2 3 n en Tania reminds everyone that the storage requirement of n-by-n matrices with special structures are given in the table below. n-by-n matrix No special structure Symmetric п Upper/lower triangular Tridiagonal Toeplitz Sparse with k nonzero elements Storage requirement n2 n(n + 1)/2 = n2/2 n(n + 1)/2 = n2/2 3n – 2 ~ 3n 2n – 1 ~ 2n 3k
(a) [1 mark] Oberon has a 2.4 GHz workstation with 4 cores where each core can do 4 floating point operations per clock cycle. Consider the n-by-n linear system Az = b where the invertible matrix A is tridiagonal. Estimate the largest value of n such that the linear system can be solved in 90 minutes. no Number Give your answer to at least 2 significant figures. (b) [1 mark] Tania wants to estimate the size n of the largest n-by-n symmetric matrix that can be stored in 4 Gb RAM using double precision floating point arithmetic. Oberon says to use the conversion 1 Gb = 230 bytes. n Number Give your answer to at least 2 significant figures. (c) [1 mark] Tina's laptop can compute 10 flops per second. Suppose that > 10,000 and it took Tina 119 hours to solve an n-by-n linear system where the invertible matrix is symmetric and positive definite. Assuming that Tina takes into account the structure of the input matrix, estimate how long it would take Tina to multiply two n-by-n matrices with no special structure. 714 hours 238 hours 119 hours 23.7 seconds 11.85 seconds 0.0008709 seconds
(d) [2 marks] Suppose again that n > 10,000. It took Tania's laptop 23.7 seconds to solve an n-by-n linear system where the invertible matrix is triangular. Oberon claims it would take 500 times longer to do this with the same matrix for 500 different right-hand side vectors, while Tania claims it would take roughly the same amount of time. Who do you agree with? Explain your answer. 들 Equation 12 Editor A A. TX BI U S x x Styles Font Size Words: 0 END OF QUESTION