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21.4. Gaussian elimination can be used to compute the inverse A-1 of a nonsingular matrix A € Cmxm, though it is rarely really necessary to do so. (a) Describe an algorithm for computing A-1 by solving m systems of equa- tions, and show that its asymptotic operation count is 8m3/3 flops. From pages 58-59 of Trefeten et al: Operation Count floating point operations-"flops"-tha the algorithm requires. Each addi- tion, subtraction, multiplication, division, or square root counts as one flop. We make no distinction between real and complex arithmetic, although in practice on most computers there is a sizable difference. (b) Describe a variant of your algorithm, taking advantage of sparsity, that reduces the operation count to 2m³ flops. (c) Suppose one wishes to solve n systems of equations Axj = bj, or equiv- alently, a block system AX = B with B € Cmx". What is the asymptotic operation count (a function of m and n) for doing this (i) directly from the LU factorization and (ii) with a preliminary computation of A-1?

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