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The Barrett multiplication previously described requires a constant operand b to pre-compute [] ahead of time. Otherwise, the operation is not efficient. Otherwise, the operation is not efficient. It is common to use Montgomery multiplication when both operands are non-constant as it has better performance.
A base class VecExpression represents any vector-valued expression. It is templated on the actual expression type E to be implemented, per the curiously recurring template pattern. The existence of a base class like VecExpression is not strictly necessary for expression templates to work.
The C++ standard library provides a complex template class as well as complex-math functions in the <complex> header. The Go programming language has built-in types complex64 (each component is 32-bit float) and complex128 (each component is 64-bit float). Imaginary number literals can be specified by appending an "i".
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.
Splitting numbers into more than two parts results in Toom-Cook multiplication; for example, using three parts results in the Toom-3 algorithm. Using many parts can set the exponent arbitrarily close to 1, but the constant factor also grows, making it impractical.
The algorithm performs a fixed sequence of operations (up to log n): a multiplication and squaring takes place for each bit in the exponent, regardless of the bit's specific value. A similar algorithm for multiplication by doubling exists.
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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.