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Operation Count - Gauss Elimination

In modern computers, all floating-point operations are done in hardware, so long operations may be as significant, gives an indication of the operational cost of Gaussian elimination. 

Gaussian elimination is a method of solving a linear system Ax = b (consisting of m equations in n unknowns) by bringing the augmented matrix to an upper triangular form. This elimination process is also called the forward elimination method. Then the unknowns n can be solved by back substitution.

At first a multiplier is computed and then subtracted from row Ri that multiplier times row R1. The zero that is being created in this process is not computed. So the elimination requires (n - 1) multiplications per row.

The tabular column shows the counts of each operation.

Flops refers floating point operation.

Since (n - 1) steps required, k goes from 1 to (n - 1) and thus the total number of operations in this forward elimination is

consider (n - k) = s
The above equation is obtained dropping lower power on n. The function grows about proportional to n cube.
In the back substitution, the (n -1) multiplications and subtractions are made and 1 division. Hence the number of operation in the back substitution is
The backward substitution grows more slowly than the number of operations in the forward elimination of the Gauss algorithm. Hence it is negligible for large systems.
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