Number of operations in the Gaussian elimination method

To estimate the computational time required in a Gaussian elimination procedure we need to evaluate the number of arithmetic operations during the forward reduction and back substitution processes. Obviously multiplication and division take much longer time than addition and subtraction and hence the total time required for the latter operations, especially in large systems of equations, is relatively small and can be ignored. Let us consider a system of simultaneous algebraic equations, the representative calculation for forward reduction at stage is expressed as 

The augmented coefficient matrix at this stage can be shown as 

As Equation (6.7) shows in each of ( - k) rows in the rectangular sub-matrix we need to evaluate (n k) multipliers and carry out (n - k + I) multiplications, therefore the total number of operations required is calculated as 

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