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One refinement scheme is Heron's method, a special case of Newton's method. If division is much more costly than multiplication, it may be preferable to compute the inverse square root instead. Other methods are available to compute the square root digit by digit, or using Taylor series. Rational approximations of square roots may be calculated ...
Newton's method is one of many known methods of computing square roots. Given a positive number a, the problem of finding a number x such that x2 = a is equivalent to finding a root of the function f(x) = x2 − a. The Newton iteration defined by this function is given by.
Tonelli–Shanks algorithm. The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for r in a congruence of the form r2 ≡ n (mod p), where p is a prime: that is, to find a square root of n modulo p. Tonelli–Shanks cannot be used for composite moduli: finding square roots modulo ...
The traditional pen-and-paper algorithm for computing the square root is based on working from higher digit places to lower, and as each new digit pick the largest that will still yield a square . If stopping after the one's place, the result computed will be the integer square root.
The fast inverse square root is used to generalize this calculation to three-dimensional space. The inverse square root of a floating point number is used in digital signal processing to normalize a vector, scaling it to length 1 to produce a unit vector. [14] For example, computer graphics programs use inverse square roots to compute angles of ...
Square root. Notation for the (principal) square root of x. For example, √ 25 = 5, since 25 = 5 ⋅ 5, or 52 (5 squared). In mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. [1] For example, 4 and −4 are square roots of 16 ...
Steffensen's method. Newton-like root-finding algorithm that does not use derivatives. In numerical analysis, Steffensen's method is an iterative method for numerical root-finding named after Johan Frederik Steffensen that is similar to the secant method and to Newton's method. Steffensen's method achieves a quadratic order of convergence ...
th root. In mathematics, an nth root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields x: The integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root.