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Consider the following Gaussian-elimination-Gauss-Jordan hybrid method for solving the system (6.4). First, apply the Gaussian-elimination technique to reduce the system to triangular form. Then use the nth equation to eliminate the coefficients of xn in each of the first n − 1 rows. After this is completed use the (n − 1)st equation to eliminate the coefficients of xn−1 in the first n − 2 rows, etc. The system will eventually appear as the reduced system in Exercise 12.
a. Show that this method requires
n3/3 + 3/2 n2 - 5/6 n multiplications/divisions
and
n3/3 + n2/2 - 5/6 n additions/subtractions.
b. Make a table comparing the required operations for the Gaussian elimination, Gauss-Jordan, and hybrid methods, for n = 3, 10, 50, 100. |
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