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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 ).
Octave (aka GNU Octave) is an alternative to MATLAB. Originally conceived in 1988 by John W. Eaton as a companion software for an undergraduate textbook, Eaton later opted to modify it into a more flexible tool. Development began in 1992 and the alpha version was released in 1993. Subsequently, version 1.0 was released a year after that in 1994.
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.
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However, this book is the first to focus on mathematical crankery as a whole. [1] The book consists of 57 essays, [2] loosely organized by the most common topics in mathematics for cranks to focus their attention on. [1] The "top ten" of these topics, as listed by reviewer Ian Stewart, are, in order: squaring the circle, angle trisection,