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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.
In the above case, the reduce or slash operator moderates the multiply function. The expression ×/2 3 4 evaluates to a scalar (1 element only) result through reducing an array by multiplication. The above case is simplified, imagine multiplying (adding, subtracting or dividing) more than just a few numbers together.
The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either exp {\displaystyle \exp } or log {\displaystyle \log } in the complex domain can be computed with some complexity, then that complexity is ...
A function definition starts with the name of the type of value that it returns or void to indicate that it does not return a value. This is followed by the function name, formal arguments in parentheses, and body lines in braces. In C++, a function declared in a class (as non-static) is called a member function or method.
When multiplication is repeated, the resulting operation is known as exponentiation. For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 2 3, a two with a superscript three. In this example, the number two is the base, and three is the exponent. [26]
In class-based programming, methods are defined within a class, and objects are instances of a given class. One of the most important capabilities that a method provides is method overriding - the same name (e.g., area) can be used for multiple different kinds of classes. This allows the sending objects to invoke behaviors and to delegate the ...
Java is a high-level, class-based, object-oriented programming language that is designed to have as few implementation dependencies as possible. It is a general-purpose programming language intended to let programmers write once, run anywhere (), [16] meaning that compiled Java code can run on all platforms that support Java without the need to recompile. [17]
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.