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In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation.
A third method drastically reduces the number of operations to perform modular exponentiation, while keeping the same memory footprint as in the previous method. It is a combination of the previous method and a more general principle called exponentiation by squaring (also known as binary exponentiation).
In expressions such as , the notation for exponentiation is usually to write the exponent as a superscript to the base number .But many environments — such as programming languages and plain-text e-mail — do not support superscript typesetting.
Horner's method can be used to convert between different positional numeral systems – in which case x is the base of the number system, and the a i coefficients are the digits of the base-x representation of a given number – and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater.
The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and ...
Exponentiation with Montgomery reduction O ( M ( n ) k ) {\displaystyle O(M(n)\,k)} On stronger computational models, specifically a pointer machine and consequently also a unit-cost random-access machine it is possible to multiply two n -bit numbers in time O ( n ).
Python uses the ** operator for exponentiation. Python uses the + operator for string concatenation. Python uses the * operator for duplicating a string a specified number of times. The @ infix operator is intended to be used by libraries such as NumPy for matrix multiplication. [104] [105]
To convert, the program reads each symbol in order and does something based on that symbol. The result for the above examples would be (in reverse Polish notation) "3 4 +" and "3 4 2 1 − × +", respectively. The shunting yard algorithm will correctly parse all valid infix expressions, but does not reject all invalid expressions.