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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
Note that C99 and C++ do not implement complex numbers in a code-compatible way – the latter instead provides the class std:: complex. All operations on complex numbers are defined in the <complex.h> header. As with the real-valued functions, an f or l suffix denotes the float complex or long double complex variant of the function.
In computing, a roundoff error, [1] also called rounding error, [2] is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. [3]
replacing integer multiplication by a constant with a combination of shifts, adds or subtracts; replacing integer division by a constant with a multiplication, taking advantage of the limited range of machine integers. [3] This method also works if divisor is a non-integer sufficiently greater than 1, e.g. √2 or π. [4]
Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and cyan denotes left shift. (A), (B) and (C) show recursion with z=10 to obtain intermediate values. The Karatsuba algorithm is a fast multiplication algorithm.
For multiplication, the most straightforward algorithms used for multiplying numbers by hand (as taught in primary school) require (N 2) operations, but multiplication algorithms that achieve O(N log(N) log(log(N))) complexity have been devised, such as the Schönhage–Strassen algorithm, based on fast Fourier transforms, and there are also ...
In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size n is in NC if there exist constants c and k such that it can be solved in time O ((log n ) c ) using O ...
This class of problem is associated with Rank revealing QR factorizations and D optimal experimental design. [39] Minimal addition chains for sequences. [40] The complexity of minimal addition chains for individual numbers is unknown. [41] Modal logic S5-Satisfiability; Pancake sorting distance problem for strings [42]