A. Show that the LU Factorization Algorithm requires 1/3 n3 -

a. Show that the LU Factorization Algorithm requires
1/3 n3 - 1/3 n multiplications/divisions and 1/3 n3 - 1/2 n2 + 1/6 n additions/subtractions
b. Show that solving Ly = b, where L is a lower-triangular matrix with lii = 1 for all i, requires
1/2 n2 - 1/2 n multiplications/divisions and 1/2 n2 - 1/2 n additions/subtractions
c. Show that solving Ax = b by first factoring A into A = LU and then solving Ly = b and Ux = y requires the same number of operations as the Gaussian Elimination Algorithm 6.1.
d. Count the number of operations required to solve m linear systems Ax(k) = b(k) for k =1, . . . ,m by first factoring A and then using the method of part (c) m times.

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