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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.
Constructing a square with the same area as a given oblong using the geometric mean For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side x = a b {\displaystyle x={\sqrt {ab}}} (the geometric mean of a and b ).
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...
A non-negative integer is a square number when its square root is again an integer. For example, =, so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free. For a non-negative integer n, the n th square number is n 2, with 0 2 = 0 being the zeroth one. The concept of square can be extended to some ...
In mathematics, Graeffe's method or Dandelin–Lobachesky–Graeffe method is an algorithm for finding all of the roots of a polynomial.It was developed independently by Germinal Pierre Dandelin in 1826 and Lobachevsky in 1834.
In statistical mechanics and combinatorics, if one has a number distribution of labels, then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution {n i} on a set of N total items, n i represents the number of items to be given the label i. (In statistical mechanics i is the label of the energy ...
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.
In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n, = ⌊ ⌋. For example, isqrt ( 27 ) = ⌊ 27 ⌋ = ⌊ 5.19615242270663... ⌋ = 5. {\displaystyle \operatorname {isqrt} (27)=\lfloor {\sqrt {27}}\rfloor ...
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