

In practice, LU decompositions are often used to solve a sequence of matrix equations, all with the same coefficient matrix:
For this situation, using the LU decomposition for A is generally more efficient than solving directly by row reduction. In this part, we explore the reason why.
The efficiency of a numerical algorithm is generally measured by counting the number of floating point operations, or flops, needed to complete the algorithm. [Any single arithmetic opertion (addition, subtraction, multiplication, or division) on two real floating point numbers is a flop.]
How does the total number of flops needed to find x compare with the number needed when you solved as in Step 1?
Of course, the increased efficiency you achieved by using the LU decomposition depended on knowing L and U, which required additional flops. But if you needed to solve other matrix equations with the same coefficient matrix, using the LU decomposition would be more efficient because you need to compute L and U only once!

