About how long does a 1 Gflop computer take to solve a system of 100 linear equations (with 100 variables)?
About how long does a 1 Gflop computer take to solve 10 systems of 100 linear equations, with the same coefficient matrix but 10 different righthand sides?
Algorithm flop counts allow for very accurate prediction of running time on a given computer.
Since matrix multiplication is associative, the flop count for multiplying three or more matrices doesn't depend on the order in which you multiply them.
Suppose $A\in \mathbf{R}^{n \times n}$ is lower triangular.
The flop count for computing $Ab$ is the same (order) as the flop count for computing $A^{-1}b$.
Suppose $A\in \mathbf{R}^{m \times n}$, and we need to compute $x$ that minimizes $\|Ax-b\|^2_2 + (\rho/2)\|x\|_2^2$, where $\rho >0$.
For $m \geq n$, the flop count (order) is
For $m \leq n$, the flop count (order) is
Suppose $m\geq n$, we have already solved the problem for one value of $\rho$, and now we want to compute the solution for $k$ other values of $\rho$ (this gives the so-called regularization path). The flop count (order) for doing this is