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Many iterative square root algorithms require an initial seed value. The seed must be a non-zero positive number; it should be between 1 and , the number whose square root is desired, because the square root must be in that range. If the seed is far away from the root, the algorithm will require more iterations.
For example, if one computes the integer square root of 2000000 using the algorithm above, one obtains the sequence In total 13 iteration steps are needed. Although Heron's method converges quadratically close to the solution, less than one bit precision per iteration is gained at the beginning.
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.
A square root of a number x is a number r which, when squared, becomes x: =. Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:
Fast inverse square root, sometimes referred to as Fast InvSqrt () or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates , the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number in IEEE 754 floating-point format. The algorithm is best known for its implementation in 1999 in Quake III ...
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 ...
These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y. This equation was first studied extensively in India starting with Brahmagupta, [1] who found an integer solution to + = in his Brāhmasphuṭasiddhānta circa 628. [2]
Nested radical. In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include. which arises in discussing the regular pentagon, and more complicated ones such as.