Consider you have two computational methods to solve a specific problem (also the corresponding software) and want to see which method is more efficient for the specific problem. Is it possible to use two different computers (one for each computation) or we necessarily should carry out the two computations on one computer (or two identical computers)?
Profiling tools such help in this regards. gnu prof, or gprof in linux shall fit the bill. Thereafter many factors shall come into play, which would include but not limited to r/w time, or i/o time, compute time, memory access time and so on.
To conclude, any good profiling tool shall help.
Comparison of different numerical techniques is indeed an extremely tricky business and all experts in the field who deserve this definition are always extremely cautious in drawing strong conclusions on this issue. This is also because, on modern computing architectures, the time needed to carry out arithmetic computations is of the same order of the time needed to move data between different memory layers, so that even algorithms that require more operations can be faster than those requiring less, if appropriate effort is done to optimize memory handling. All this being said, it is customary in numerical analysis to leave memory handling aside as a software engineering issue (this does not mean it is not important, simply, it depends on too many non mathematical factors to be debated in a context independent way) and to focus on quantities that are the same independently of the computer platform or the operating system or the programming language used for implementation. For example, one may look at number of floating point operations, function evaluations, number of degrees of freedom (e.g. size of the vectors) involved in the computation, number of iterations of a linear or nonlinear solver. The other thing to be kept in mind, which is often forgotten, is that efficiency is related to the ratio error/(computational cost). Clearly, there are many different ways of estimating both the numerator and denominator in this ratio, but unless this estimate is attempted one cannot talk about efficiency. Just measuring the CPU time of two methods (even implemented in the same framework and run on the same computer) does not say which is more efficient if there is no comparison of the accuracy of the results on some agreed and relevant benchmarks.
I would recommend doing quantification of computational performance on single machine instead of using two different computers.
Using two different computers requires all the hardware and software elements of CPU's to be same. If you are sure your two computers are identical in all such aspects then you can do your analysis on multiple computers.
Strongly recommend you to use a single mchine that you can run all the algoriths, and have a comparison.
Its better you use one computer for both algorithms (your best one), unless you are running your algorithms on an FPGA or DSP board, than you can compare the performance with a computer.
If the algorithms are not very complicated, you can try to estimate the theoretical complexity based on big O notation. That might give an intuition of the computational effort being invested in each algorithm.
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