Question

What is the Big-O complexity for widespread algorithms of the basic arithmetic operations like multiplication, square root, logarithm, scalar and matrix product?

Are there exotic algorithms which are more efficient, in terms of Big-O complexity, but are not very widespread in practical solutions (e.g. not implemented in popular software libraries)?

Answer 1

See http://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations

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Matrix product of square matrices:

• O(N³) (naïve method)
• O(N^2.81) (Strassen’s algorithm).

There is also a O(N^2.38) Coppersmith–Winograd algorithm but I don’t think it’s wide-spread due to the huge hidden constant.

Big-int multiplication:

• Naïve: O(n²)
• Fast-Fourier transform based: O(n log n log log n) (Schönhage–Strassen algorithm).

There are also an n log n · 2^{O(log* n)} algorithm published in 2008 but that was too new to be widespread.

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Usually the naïve method is good enough for normal-sized input.